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SSC/Structure/BiPolytropes/Analytic15

2025-01-24T00:17:31Z

Xxsrm: /* Step 6: Envelope Solution */

__FORCETOC__ 


=BiPolytrope with n<sub>c</sub> = 1 and n<sub>e</sub> = 5=
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<font size="-1">[[H_BookTiledMenu#MoreModels|<b>Murphy (1983)<br /><br />Analytic</b>]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (1, 5)</font>
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[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 12 April 2015: I became aware of the published discussions of this system by Murphy (1983) and Murphy & Fiedler (1985b) in March of 2015 after searching the internet for previous analyses of radial oscillations in polytropes and, then, reading through Horedt's (2004) §2.8.1 discussion of composite polytropes.]]Here we construct a [[SSC/Structure/BiPolytropes#BiPolytropes|system of bipolytropic configurations]] in which the core has an <math>~n_c=1</math> polytropic index and the envelope has an <math>~n_e=5</math> polytropic index. As in the case of our [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|separately discussed, "mirror image" bipolytropic configurations having <math>~(n_c, n_e) = (5, 1)</math>]], this system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. Bipolytropes of this type were first constructed by {{ Murphy83afull }}, and attributes of their physical structure were further discussed by {{ MF85afull }}; [[SSC/Structure/BiPolytropes/Analytic15#Key_References|additional, closely related references are given below]]. In the discussion that follows, we will be heavily referencing {{ Murphy83a }} — hereafter, {{ Murphy83ahereafter }}.
 <br />
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic15|Part I:   Steps 2 thru 7]]<br /> </td>
<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt2|Part II:  Analytic Solution of Interface Relation]]<br /> </td>
<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt3|III:  Modeling]]<br /> </td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/MurphyUVplane|IV:  Murphy's UV Plane]]<br /> </td>
</tr>
</table>

==Steps 2 & 3==
Based on the discussion [[SSC/Structure/Polytropes/Analytic#n_=_1_Polytrope|presented elsewhere of the structure of an isolated n = 1 polytrope]], the core of this bipolytrope will have the following properties:
<div align="center">
<math>
\theta(\xi) = \frac{\sin\xi}{\xi} ~~~~\Rightarrow ~~~~ \theta_i = \frac{\sin\xi_i}{\xi_i} ;
</math>

<math>
\frac{d\theta}{d\xi} = \biggl[ \frac{\cos\xi}{\xi}- \frac{\sin\xi}{\xi^2}\biggr] ~~~~\Rightarrow ~~~~
\biggl(\frac{d\theta}{d\xi}\biggr)_i = \biggl[\frac{\cos\xi_i}{\xi_i}- \frac{\sin\xi_i}{\xi_i^2} \biggr] \, .
</math>
</div>
The first zero of the function <math>~\theta(\xi)</math> and, hence, the surface of the corresponding isolated <math>~n=1</math> polytrope is located at <math>~\xi_s = \pi</math>. Hence, the interface between the core and the envelope can be positioned anywhere within the range, <math>~0 < \xi_i < \pi</math>.

==Step 4: Throughout the core (0 ≤ ξ ≤ ξ<sub>i</sub>)==

<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="center" colspan="3">
Specify: <math>~K_c</math> and <math>~\rho_0 ~\Rightarrow</math>
</td>
<td colspan="2">
 
</td>
</tr>
<tr>
<td align="right">
<math>~\rho</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\rho_0 \theta^{n_c}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\rho_0 \biggl( \frac{\sin\xi}{\xi} \biggr)</math>
</td>
</tr>

<tr>
<td align="right">
<math>~P</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~K_c \rho_0^{2} \biggl( \frac{\sin\xi}{\xi}\biggr)^{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{K_c}{2\pi G} \biggr]^{1/2} \xi</math>
</td>
</tr>

<tr>
<td align="right">
<math>~M_r</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi \biggl[ \frac{K_c}{2\pi G} \biggr]^{3/2} \rho_0 \biggl[\sin\xi - \xi \cos\xi \biggr]</math>
</td>
</tr>
</table>

</div>

==Step 5: Interface Conditions==

<div align="center">
<table border="0" cellpadding="3">
<tr>
<td colspan="3">
 
</td>
<td align="left" colspan="2">
Setting <math>~n_c=1</math> and <math>~n_e=5~~~~~\Rightarrow</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\rho_e}{\rho_0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \phi_i^{-n_e}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i \phi_i^{-5}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\rho_0^{1/n_c - 1/n_e}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} \theta^{1 - n_c/n_e}_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\rho_0^{4}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-6} \theta^{4}_i\biggr]^{1/5}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\eta_i}{\xi_i}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n_c + 1}{n_e+1} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \theta_i^{(n_c-1)/2} \phi_i^{(1-n_e)/2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{1}{3} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \phi_i^{-2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n_c + 1}{n_e + 1} \biggr]^{1/2} \theta_i^{- (n_c + 1)/2} \phi_i^{(n_e+1)/2} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{1}{3} \biggr)^{1/2} \theta_i^{- 1} \biggl( \frac{d\theta}{d\xi} \biggr)_i \phi_i^3</math>
</td>
</tr>
</table>
</div>

<span id="Alternative"><font color="red">Alternative:</font> In our</span> [[SSC/Structure/BiPolytropes#UVplane|introductory description of how to build a bipolytropic structure]], we pointed out that, instead of employing these last two fitting conditions, Chandrasekhar [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]] found it useful to employ, instead, the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions, which in the present case produces,
<div align="center">
<math>
\frac{\eta_i \phi_i^{5}}{(d\phi/d\eta)_i} = \frac{\xi_i \theta_i}{(d\theta/d\xi)_i} \biggl( \frac{\mu_e}{\mu_c}\biggr) \, ,
</math>
</div>
and the product of the <math>3^\mathrm{rd}</math> and <math>4^\mathrm{th}</math> expressions, which in the present case generates,
<div align="center">
<math>
\frac{3\eta_i (d\phi/d\eta)_i}{ \phi_i } = \frac{\xi_i (d\theta/d\xi)_i}{ \theta_i } \biggl( \frac{\mu_e}{\mu_c}\biggr) \, .
</math>
</div>
In what follows we will sometimes refer to the first of these two expressions as Chandrasekhar's "U-constraint" and we will sometimes refer to the second as Chandrasekhar's "V-constraint." As is explained in [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar's_U_and_V_Functions|an accompanying discussion]], {{ Murphy83a }} followed Chandrasekhar's lead and extracted fitting conditions from this last pair of expressions. In seeking the most compact analytic solution, we have found it advantageous to invoke our standard <math>3^\mathrm{rd}</math> fitting expression in tandem with the Chandrasekhar's V-constraint.

==Step 6: Envelope Solution==

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 20 April 2015: There is a type-setting error in this function expression as published in the upper left-hand column of the second page of the article by Murphy (1983); the sine function in the denominator should be sine-squared, as presented here.]]Following the work of {{ Murphy83a }} and of {{ MF85a }}, we will adopt for the envelope's structure the F-Type solution of the n = 5 Lane-Emden function discovered by {{ Srivastava62full }} and described in an [[SSC/Structure/Polytropes/Analytic#Srivastava's_F-Type_Solution|accompanying discussion]], namely,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\phi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{B_0^{-1}\sin[\ln(A_0\eta)^{1/2})]}{\eta^{1/2}\{3-2\sin^2[\ln(A_0\eta)^{1/2}]\}^{1/2}} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{B_0^{-1}\sin\Delta}{\eta^{1/2}(3-2\sin^2\Delta)^{1/2}} \, ,</math>
</td>
</tr>

</table>
</div>
[[File:CommentButton02.png|right|100px|Note that our homology factor and scaling coefficient serve virtually the same roles as the homology factor, A, and scaling coefficient, B, used by Murphy (1983), but they are not mathematically identical so we have added a subscript "0" to highlight the distinctions.]]
where <math>~A_0</math> is a "homology factor" and <math>~B_0</math> is an overall scaling coefficient — the values of both will be determined presently from the interface conditions — and we have introduced the notation,
<div align="center">
<math>~\Delta \equiv \ln(A_0\eta)^{1/2} = \frac{1}{2} (\ln A_0 + \ln\eta) \, .</math>
</div>

The first derivative of Srivastava's function is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{d\phi}{d\eta}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{B_0^{-1}[3\cos\Delta-3\sin\Delta + 2\sin^3\Delta] }{2\eta^{3/2}(3-2\sin^2\Delta)^{3/2}} \, .
</math>
</td>
</tr>
</table>
</div>

As has been explained in the context of our [[SSC/Structure/Polytropes/Analytic#Srivastava's_F-Type_Solution|more general discussion of Srivastava's function]], if we ignore, for the moment, the additional "<math>m\pi</math>" phase shift that can be attached to a determination of the angle, <math>~\Delta</math>, the physically viable interval for the dimensionless radial coordinate is, <math>~e^{2\pi} \ge A_0\eta \ge \eta_\mathrm{crit} \equiv e^{2\tan^{-1}(1+2^{1/3})} \, .</math>

For this bipolytropic configuration, it is worth emphasizing how the dimensionless radial coordinate of the <math>~n_e = 5</math> envelope, <math>~\eta</math>, is related to the dimensionless radial coordinate of the <math>~n_c = 1</math> core, <math>~\xi</math>. Referring to the general setup procedure for constructing any bipolytropic configuration that has been [[SSC/Structure/BiPolytropes#TableSetup|presented in tabular form in a separate discussion]], it is clear that in order for the radial coordinate, <math>~r</math>, to carry a consistent meaning throughout the model, we must have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r = \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~\biggl( \frac{K_c}{3K_e} \biggr)\rho_e^{4/5} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{\eta}{\xi}\biggr)^2 \, .</math>
</td>
</tr>
</table>
</div>

Referring back to the [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|already established interface conditions, above]], to relate <math>~\rho_e</math> to <math>~\rho_0</math>, and to re-express the ratio, <math>~K_e/K_c</math>, we therefore have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl(\frac{\eta}{\xi}\biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{3} \biggl[ \rho_0^{4}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-6} \theta^{4}_i\biggr]^{-1/5}
\biggl[ \rho_0\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i \phi_i^{-5} \biggr]^{4/5} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \frac{\eta}{\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{1}{3}\biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \phi_i^{-2} \, .</math>
</td>
</tr>
</table>
</div>
While this result is not a surprise because the right-hand-side is the same expression that was presented, above, as the interface condition for the ratio, <math>~\eta_i/\xi_i</math>, it is nevertheless useful because it shows that the same relation works throughout the system — not just at the interface — and it clearly defines how we can swap back and forth between the two dimensionless radial coordinates when examining the structure and characteristics of this composite bipolytropic structure.

===First Constraint===
Calling upon Chandrasekhar's V-constraint, as just defined above — see also [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar.27s_U_and_V_Functions|our accompanying discussion]] for elaboration on [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy's (1983)] "V<sub>5F</sub>" and "V<sub>1E</sub>" function notations — one fitting condition at the interface is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~- \frac{2\xi_i }{ 3\theta_i } \biggl( \frac{d\theta}{d\xi} \biggr)_i \biggl( \frac{\mu_e}{\mu_c}\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2\biggl[ \frac{\eta (- d\phi/d\eta)}{\phi} \biggr]_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{[3\sin\Delta_i - 2\sin^3\Delta_i -3\cos\Delta_i ] }{\sin\Delta_i (3-2\sin^2\Delta_i)}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3 - 2\sin^2\Delta_i -3\cot\Delta_i}{(3-2\sin^2\Delta_i)} \, .</math>
</td>
</tr>
</table>
</div>
The left-hand side of this expression is inherently positive over the physically viable radial coordinate range, <math>~0 \ge \xi_i \ge \pi</math> and its value is known once the radial coordinate of the edge of the core has been specified. So, defining the interface parameter,
<div align="center">
<math>~
\kappa_i \equiv - \frac{2\theta_i^' \xi_i}{3\theta_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)
\, ,</math>
</div>
we will recast the first constraint into, what will henceforth be referred to as, the
<div align="center" id="KeyInterfaceRelation">
<table border="1" cellpadding="8" align="center">
<tr><td align="center">

<font color="#770000">'''Key Nonlinear Interface Relation'''</font>
<br />
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\kappa_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3 - 2\sin^2\Delta_i -3\cot\Delta_i}{(3-2\sin^2\Delta_i)} \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

In a [[SSC/Structure/BiPolytropes/Analytic15/Pt2#Analytic_Solution_of_Key_Interface_Relation|separate subsection (Part II) of this chapter]], we present a closed-form analytic solution, <math>\Delta_i(\kappa_i)</math>, to this nonlinear equation.



===Second Constraint===

====Obtained from Third Interface Condition====
Our [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|3<sup>rd</sup> interface condition, as detailed above]], states that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\eta_i}{\xi_i}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3^{-1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \phi_i^{-2} \, .</math>
</td>
</tr>

</table>
</div>

If we now choose to normalize the interface amplitude such that, <math>~\phi_i = 1</math>, then this condition establishes two relations: First, from the 3<sup>rd</sup> interface condition alone,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\eta_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3^{-1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \xi_i \, ;</math>
</td>
</tr>

</table>
</div>
and, second, from the definition of Srivastava's function, <math>~\phi</math>, we deduce that the overall scaling parameter is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~B_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sin^2\Delta_i}{\eta_i(3-2\sin^2\Delta_i)} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sqrt{3}}{\xi_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \biggl( \frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1} \, . </math>
</td>
</tr>

</table>
</div>
Notice that, after the solution, <math>~\Delta_i(\kappa_i)</math>, of the [[SSC/Structure/BiPolytropes/Analytic15#KeyInterfaceRelation|key nonlinear interface relation]] has been determined, the first of these two relations also permits us to write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~A_0\eta_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~e^{2\Delta_i}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ A_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \xi_i^{-1} e^{2\Delta_i} \, .</math>
</td>
</tr>
</table>
</div>
Throughout the envelope, therefore, the angle,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Delta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\ln(A_0\eta)^{1/2} = \frac{1}{2} \ln\biggl[ \xi \cdot \xi_i^{-1} e^{2\Delta_i} \biggr]
= \Delta_i + \ln\biggl( \frac{\xi}{\xi_i} \biggr)^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

====Obtained from Chandrasekhar's U-constraint====
We shall now demonstrate that the same expression for the scaling coefficient, <math>~B_0</math>, can alternatively be obtained from Chandrasekhar's U-constraint, without assuming that <math>~\phi_i = 1</math>, after taking into account the result that already has been obtained from the V-constraint. As [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|described above]], the U-constraint is an alternative interface condition that may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\xi_i \theta_i}{(-d\theta/d\xi)_i} \biggl( \frac{\mu_e}{\mu_c}\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\eta_i \phi_i^{5}}{(-d\phi/d\eta)_i} \, ,</math>
</td>
</tr>
</table>
</div>
which, in the particular case being examined here, becomes — again, see [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar.27s_U_and_V_Functions|our accompanying discussion]] for elaboration on the "U<sub>5F</sub>" and "U<sub>1E</sub>" function notations used by {{ Murphy83a }} —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{2\xi_i^2}{3\kappa_i} \biggl( \frac{\mu_e}{\mu_c}\biggr)^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( U_\mathrm{5F} \biggr)_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2B_0^{-4} \sin^4\Delta_i}{(3-2\sin^2\Delta_i)(3 - 2\sin^2\Delta_i - 3\cot\Delta_i)} \, .
</math>
</td>
</tr>
</table>
</div>

Now, from our [[SSC/Structure/BiPolytropes/Analytic15#First_Constraint|discussion, above, of the first constraint]], we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(3 - 2\sin^2\Delta_i - 3\cot\Delta_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3-2\sin^2\Delta_i)\kappa_i \, .</math>
</td>
</tr>
</table>
</div>

Hence, Chandrasekhar's U-constraint becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{2\xi_i^2}{3\kappa_i} \biggl( \frac{\mu_e}{\mu_c}\biggr)^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2B_0^{-4} \sin^4\Delta_i}{(3-2\sin^2\Delta_i)^2 \kappa_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ B_0^4 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\sin^4\Delta_i}{\xi_i^2 (3-2\sin^2\Delta_i)^2} \biggl( \frac{\mu_e}{\mu_c}\biggr)^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ B_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sqrt{3}}{\xi_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \biggl( \frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1} \, ,</math>
</td>
</tr>
</table>
</div>
which, as predicted, is identical to what we learned from the third interface condition, alone.

===Comment on Murphy's Scalings===

The {{ Murphy83a }} derivations also include an homology factor, <math>~A</math>, and an overall scaling factor, <math>~B</math>, but they are calculated differently from our <math>~A_0</math> and <math>~B_0</math>. In the righthand column of the third page of his paper, Murphy states that,
<div align="center">
<math>~A = \frac{\xi_J}{\zeta_J} \, ,</math>
</div>
which, when translated into our notation <math>~(\zeta_J \rightarrow \xi_i </math> and <math>~\xi_J \rightarrow A_0\eta_\mathrm{root})</math> gives,
<div align="center">
<math>~A = \frac{A_0 \eta_\mathrm{root}}{\xi_i} \, .</math>
</div>
Now, in our derivation, <math>~\eta_\mathrm{root}</math> is synonymous with the location of the envelope interface, <math>~\eta_i</math>, as expressed in terms of the dimensionless radial coordinate associated with Srivastava's Lane-Emden function, so we can equally well state that,
<div align="center">
<math>~A = \frac{A_0 \eta_i}{\xi_i} \, .</math>
</div>
Recalling that <math>~\phi_i = 1</math>, we know from the [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|interface conditions detailed above]] that,
<div align="center">
<math>~\frac{\eta_i}{\xi_i} = \frac{1}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>
</div>
Hence Murphy's homology factor, <math>~A</math>, is related to our homology factor, <math>~A_0</math> via the expression,
<div align="center">
<math>~A = \frac{A_0}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>
</div>
It is usually the value of this quantity, rather than simply our derived value of <math>~A_0</math>, that is tabulated below — both [[SSC/Structure/BiPolytropes/Analytic15#Murphy.27s_Example_Model_Characteristics|here]] and [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29|here]] — as we make quantitative comparisons between the characteristics of our derived models and those published by {{ Murphy83a }} and by {{ MF85a }}.

In the lefthand column of the fourth page of his paper, {{ Murphy83a }} defines the coefficient <math>B</math> in such a way that the ''value'' of the envelope function, <math>\phi_{5F}</math>, equals the ''value'' of the core function, <math>\theta_{1E}</math>, at the interface. Specifically, he sets,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\zeta_J}{\sin\zeta_J} \biggr] \biggl[ \frac{A^{1/2} \sin(\ln\sqrt{A\zeta_J}) }{(A\zeta_J)^{1/2} \{2 + \cos[\ln(A\zeta_J)]\}^{1/2}}\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\zeta_J}{\sin\zeta_J} \biggr] \biggl[ \frac{\sin(\ln\sqrt{A\zeta_J}) }{\zeta_J^{1/2} \{3 - 2\sin^2(\ln\sqrt{A\zeta_J}) \}^{1/2}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
Switching to our terminology, that is, setting,
<div align="center">
<math>\ln\sqrt{A\zeta_J} \rightarrow \Delta_i</math>     and, as before,     
<math>~\zeta_J \rightarrow \xi_i \, ,</math>
</div>
gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\xi_i}{\sin\xi_i} \biggr] \biggl[ \frac{\sin\Delta_i }{\xi_i^{1/2} (3 - 2\sin^2\Delta_i )^{1/2}}\biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\theta_i^{-1} \biggl[ \xi_i^{-1} \biggl(\frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1} \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>
Hence, in terms of the definition of ''our'' scaling coefficient, <math>~B_0</math>, derived above, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{B_0}{3^{1/4}} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{1/2} \theta_i^{-1} \, .</math>
</td>
</tr>
</table>
</div>

As we make quantitative comparisons between the characteristics of our derived models and those published by [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)] and by [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy & Fiedler (1985a)], below, we usually will tabulate the value of this quantity,
rather than simply our derived value of <math>~B_0</math>.

==Step 7: Identifying the Surface==

Because Shrivastava's function — and, along with it, the envelope's density — drops to zero when,
<div align="center">
<math>\Delta = \Delta_s \equiv \pi \, ,</math>
</div>
we know that the radius, <math>~\xi_s</math>, of the bipolytropic configuration is given by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Delta_i + \ln\biggl( \frac{\xi_s}{\xi_i} \biggr)^{1/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \ln\biggl( \frac{\xi_s}{\xi_i} \biggr)^{1/2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi - \Delta_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \xi_s</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi_i e^{2(\pi - \Delta_i )} \, .
</math>
</td>
</tr>
</table>
</div>

In terms of the natural radial coordinate of the envelope, this is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\eta_s \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3^{-1/2} \xi_i e^{2(\pi - \Delta_i)} \, .</math>
</td>
</tr>
</table>
</div>

=Key References=
* [http://adsabs.harvard.edu/abs/1962ApJ...136..680S S. Srivastava (1968, ApJ, 136, 680)] ''A New Solution of the Lane-Emden Equation of Index n = 5''
* [http://adsabs.harvard.edu/abs/1978AuJPh..31..115B H. A. Buchdahl (1978, Australian Journal of Physics, 31, 115)]: ''Remark on the Polytrope of Index 5'' — the result of this work by Buchdahl has been [[SSC/Structure/BiPolytropes/Analytic51#Buchdahl1978|highlighted inside our discussion of bipolytropes with <math>~(n_c, n_e) = (5, 1)</math>]].
* [http://adsabs.harvard.edu/abs/1980PASAu...4...37M J. O. Murphy (1980a, Proc. Astr. Soc. of Australia, 4, 37)]: ''A Finite Radius Solution for the Polytrope Index 5''
* [http://adsabs.harvard.edu/abs/1980PASAu...4...41M J. O. Murphy (1980b, Proc. Astr. Soc. of Australia, 4, 41)]: ''On the F-Type and M-Type Solutions of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1981PASAu...4..205M J. O. Murphy (1981, Proc. Astr. Soc. of Australia, 4, 205)]: ''Physical Characteristics of a Polytrope Index 5 with Finite Radius''
* [http://adsabs.harvard.edu/abs/1982PASAu...4..376M J. O. Murphy (1982, Proc. Astr. Soc. of Australia, 4, 376)]: ''A Sequence of E-Type Composite Analytical Solutions of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1983AuJPh..36..453M J. O. Murphy (1983, Australian Journal of Physics, 36, 453)]: ''Structure of a Sequence of Two-Zone Polytropic Stellar Models with Indices 0 and 1''
* [http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175)]: ''Composite and Analytical Solutions of the Lane-Emden Equation with Polytropic Indices n = 1 and n = 5''
* [http://adsabs.harvard.edu/abs/1985PASAu...6..219M J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219)]: ''Physical Structure of a Sequence of Two-Zone Polytropic Stellar Models''
* [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222)]: ''Radial Pulsations and Vibrational Stability of a Sequence of Two-Zone Polytropic Stellar Models''

=Related Discussions=
* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Polytropes emdeded in an external medium]]
* [[SSC/Structure/BiPolytropes#BiPolytropes|Constructing BiPolytropes]]

{{ SGFfooter }}

SSC/Structure/BiPolytropes/Analytic15

2025-01-24T00:14:25Z

Xxsrm: /* Step 5: Interface Conditions */

__FORCETOC__ 


=BiPolytrope with n<sub>c</sub> = 1 and n<sub>e</sub> = 5=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#MoreModels|<b>Murphy (1983)<br /><br />Analytic</b>]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (1, 5)</font>
|}
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 12 April 2015: I became aware of the published discussions of this system by Murphy (1983) and Murphy & Fiedler (1985b) in March of 2015 after searching the internet for previous analyses of radial oscillations in polytropes and, then, reading through Horedt's (2004) §2.8.1 discussion of composite polytropes.]]Here we construct a [[SSC/Structure/BiPolytropes#BiPolytropes|system of bipolytropic configurations]] in which the core has an <math>~n_c=1</math> polytropic index and the envelope has an <math>~n_e=5</math> polytropic index. As in the case of our [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|separately discussed, "mirror image" bipolytropic configurations having <math>~(n_c, n_e) = (5, 1)</math>]], this system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. Bipolytropes of this type were first constructed by {{ Murphy83afull }}, and attributes of their physical structure were further discussed by {{ MF85afull }}; [[SSC/Structure/BiPolytropes/Analytic15#Key_References|additional, closely related references are given below]]. In the discussion that follows, we will be heavily referencing {{ Murphy83a }} — hereafter, {{ Murphy83ahereafter }}.
 <br />
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic15|Part I:   Steps 2 thru 7]]<br /> </td>
<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt2|Part II:  Analytic Solution of Interface Relation]]<br /> </td>
<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt3|III:  Modeling]]<br /> </td>
<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/MurphyUVplane|IV:  Murphy's UV Plane]]<br /> </td>
</tr>
</table>

==Steps 2 & 3==
Based on the discussion [[SSC/Structure/Polytropes/Analytic#n_=_1_Polytrope|presented elsewhere of the structure of an isolated n = 1 polytrope]], the core of this bipolytrope will have the following properties:
<div align="center">
<math>
\theta(\xi) = \frac{\sin\xi}{\xi} ~~~~\Rightarrow ~~~~ \theta_i = \frac{\sin\xi_i}{\xi_i} ;
</math>

<math>
\frac{d\theta}{d\xi} = \biggl[ \frac{\cos\xi}{\xi}- \frac{\sin\xi}{\xi^2}\biggr] ~~~~\Rightarrow ~~~~
\biggl(\frac{d\theta}{d\xi}\biggr)_i = \biggl[\frac{\cos\xi_i}{\xi_i}- \frac{\sin\xi_i}{\xi_i^2} \biggr] \, .
</math>
</div>
The first zero of the function <math>~\theta(\xi)</math> and, hence, the surface of the corresponding isolated <math>~n=1</math> polytrope is located at <math>~\xi_s = \pi</math>. Hence, the interface between the core and the envelope can be positioned anywhere within the range, <math>~0 < \xi_i < \pi</math>.

==Step 4: Throughout the core (0 ≤ ξ ≤ ξ<sub>i</sub>)==

<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="center" colspan="3">
Specify: <math>~K_c</math> and <math>~\rho_0 ~\Rightarrow</math>
</td>
<td colspan="2">
 
</td>
</tr>
<tr>
<td align="right">
<math>~\rho</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>\rho_0 \theta^{n_c}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\rho_0 \biggl( \frac{\sin\xi}{\xi} \biggr)</math>
</td>
</tr>

<tr>
<td align="right">
<math>~P</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~K_c \rho_0^{2} \biggl( \frac{\sin\xi}{\xi}\biggr)^{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{K_c}{2\pi G} \biggr]^{1/2} \xi</math>
</td>
</tr>

<tr>
<td align="right">
<math>~M_r</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4\pi \biggl[ \frac{K_c}{2\pi G} \biggr]^{3/2} \rho_0 \biggl[\sin\xi - \xi \cos\xi \biggr]</math>
</td>
</tr>
</table>

</div>

==Step 5: Interface Conditions==

<div align="center">
<table border="0" cellpadding="3">
<tr>
<td colspan="3">
 
</td>
<td align="left" colspan="2">
Setting <math>~n_c=1</math> and <math>~n_e=5~~~~~\Rightarrow</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\rho_e}{\rho_0}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \phi_i^{-n_e}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i \phi_i^{-5}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{K_e}{K_c} \biggr) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\rho_0^{1/n_c - 1/n_e}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} \theta^{1 - n_c/n_e}_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\rho_0^{4}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-6} \theta^{4}_i\biggr]^{1/5}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\eta_i}{\xi_i}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n_c + 1}{n_e+1} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \theta_i^{(n_c-1)/2} \phi_i^{(1-n_e)/2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{1}{3} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \phi_i^{-2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{n_c + 1}{n_e + 1} \biggr]^{1/2} \theta_i^{- (n_c + 1)/2} \phi_i^{(n_e+1)/2} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{1}{3} \biggr)^{1/2} \theta_i^{- 1} \biggl( \frac{d\theta}{d\xi} \biggr)_i \phi_i^3</math>
</td>
</tr>
</table>
</div>

<span id="Alternative"><font color="red">Alternative:</font> In our</span> [[SSC/Structure/BiPolytropes#UVplane|introductory description of how to build a bipolytropic structure]], we pointed out that, instead of employing these last two fitting conditions, Chandrasekhar [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]] found it useful to employ, instead, the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions, which in the present case produces,
<div align="center">
<math>
\frac{\eta_i \phi_i^{5}}{(d\phi/d\eta)_i} = \frac{\xi_i \theta_i}{(d\theta/d\xi)_i} \biggl( \frac{\mu_e}{\mu_c}\biggr) \, ,
</math>
</div>
and the product of the <math>3^\mathrm{rd}</math> and <math>4^\mathrm{th}</math> expressions, which in the present case generates,
<div align="center">
<math>
\frac{3\eta_i (d\phi/d\eta)_i}{ \phi_i } = \frac{\xi_i (d\theta/d\xi)_i}{ \theta_i } \biggl( \frac{\mu_e}{\mu_c}\biggr) \, .
</math>
</div>
In what follows we will sometimes refer to the first of these two expressions as Chandrasekhar's "U-constraint" and we will sometimes refer to the second as Chandrasekhar's "V-constraint." As is explained in [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar's_U_and_V_Functions|an accompanying discussion]], {{ Murphy83a }} followed Chandrasekhar's lead and extracted fitting conditions from this last pair of expressions. In seeking the most compact analytic solution, we have found it advantageous to invoke our standard <math>3^\mathrm{rd}</math> fitting expression in tandem with the Chandrasekhar's V-constraint.

==Step 6: Envelope Solution==

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 20 April 2015: There is a type-setting error in this function expression as published in the upper left-hand column of the second page of the article by Murphy (1983); the sine function in the denominator should be sine-squared, as presented here.]]Following the work of {{ Murphy83a }} and of {{ MF85a }}, we will adopt for the envelope's structure the F-Type solution of the n = 5 Lane-Emden function discovered by {{ Srivastava62full }} and described in an [[SSC/Structure/Polytropes/Analytic#Srivastava's_F-Type_Solution|accompanying discussion]], namely,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\phi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{B_0^{-1}\sin[\ln(A_0\eta)^{1/2})]}{\eta^{1/2}\{3-2\sin^2[\ln(A_0\eta)^{1/2}]\}^{1/2}} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{B_0^{-1}\sin\Delta}{\eta^{1/2}(3-2\sin^2\Delta)^{1/2}} \, ,</math>
</td>
</tr>

</table>
</div>
[[File:CommentButton02.png|right|100px|Note that our homology factor and scaling coefficient serve virtually the same roles as the homology factor, A, and scaling coefficient, B, used by Murphy (1983), but they are not mathematically identical so we have added a subscript "0" to highlight the distinctions.]]where <math>~A_0</math> is a "homology factor" and <math>~B_0</math> is an overall scaling coefficient — the values of both will be determined presently from the interface conditions — and we have introduced the notation,
<div align="center">
<math>~\Delta \equiv \ln(A_0\eta)^{1/2} = \frac{1}{2} (\ln A_0 + \ln\eta) \, .</math>
</div>

The first derivative of Srivastava's function is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\frac{d\phi}{d\eta}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{B_0^{-1}[3\cos\Delta-3\sin\Delta + 2\sin^3\Delta] }{2\eta^{3/2}(3-2\sin^2\Delta)^{3/2}} \, .
</math>
</td>
</tr>
</table>
</div>

As has been explained in the context of our [[SSC/Structure/Polytropes/Analytic#Srivastava's_F-Type_Solution|more general discussion of Srivastava's function]], if we ignore, for the moment, the additional "<math>m\pi</math>" phase shift that can be attached to a determination of the angle, <math>~\Delta</math>, the physically viable interval for the dimensionless radial coordinate is, <math>~e^{2\pi} \ge A_0\eta \ge \eta_\mathrm{crit} \equiv e^{2\tan^{-1}(1+2^{1/3})} \, .</math>

For this bipolytropic configuration, it is worth emphasizing how the dimensionless radial coordinate of the <math>~n_e = 5</math> envelope, <math>~\eta</math>, is related to the dimensionless radial coordinate of the <math>~n_c = 1</math> core, <math>~\xi</math>. Referring to the general setup procedure for constructing any bipolytropic configuration that has been [[SSC/Structure/BiPolytropes#TableSetup|presented in tabular form in a separate discussion]], it is clear that in order for the radial coordinate, <math>~r</math>, to carry a consistent meaning throughout the model, we must have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r = \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~\biggl( \frac{K_c}{3K_e} \biggr)\rho_e^{4/5} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{\eta}{\xi}\biggr)^2 \, .</math>
</td>
</tr>
</table>
</div>

Referring back to the [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|already established interface conditions, above]], to relate <math>~\rho_e</math> to <math>~\rho_0</math>, and to re-express the ratio, <math>~K_e/K_c</math>, we therefore have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl(\frac{\eta}{\xi}\biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{3} \biggl[ \rho_0^{4}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-6} \theta^{4}_i\biggr]^{-1/5}
\biggl[ \rho_0\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i \phi_i^{-5} \biggr]^{4/5} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \frac{\eta}{\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{1}{3}\biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \phi_i^{-2} \, .</math>
</td>
</tr>
</table>
</div>
While this result is not a surprise because the right-hand-side is the same expression that was presented, above, as the interface condition for the ratio, <math>~\eta_i/\xi_i</math>, it is nevertheless useful because it shows that the same relation works throughout the system — not just at the interface — and it clearly defines how we can swap back and forth between the two dimensionless radial coordinates when examining the structure and characteristics of this composite bipolytropic structure.

===First Constraint===
Calling upon Chandrasekhar's V-constraint, as just defined above — see also [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar.27s_U_and_V_Functions|our accompanying discussion]] for elaboration on [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy's (1983)] "V<sub>5F</sub>" and "V<sub>1E</sub>" function notations — one fitting condition at the interface is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~- \frac{2\xi_i }{ 3\theta_i } \biggl( \frac{d\theta}{d\xi} \biggr)_i \biggl( \frac{\mu_e}{\mu_c}\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2\biggl[ \frac{\eta (- d\phi/d\eta)}{\phi} \biggr]_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{[3\sin\Delta_i - 2\sin^3\Delta_i -3\cos\Delta_i ] }{\sin\Delta_i (3-2\sin^2\Delta_i)}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3 - 2\sin^2\Delta_i -3\cot\Delta_i}{(3-2\sin^2\Delta_i)} \, .</math>
</td>
</tr>
</table>
</div>
The left-hand side of this expression is inherently positive over the physically viable radial coordinate range, <math>~0 \ge \xi_i \ge \pi</math> and its value is known once the radial coordinate of the edge of the core has been specified. So, defining the interface parameter,
<div align="center">
<math>~
\kappa_i \equiv - \frac{2\theta_i^' \xi_i}{3\theta_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)
\, ,</math>
</div>
we will recast the first constraint into, what will henceforth be referred to as, the
<div align="center" id="KeyInterfaceRelation">
<table border="1" cellpadding="8" align="center">
<tr><td align="center">

<font color="#770000">'''Key Nonlinear Interface Relation'''</font>
<br />
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\kappa_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{3 - 2\sin^2\Delta_i -3\cot\Delta_i}{(3-2\sin^2\Delta_i)} \, .</math>
</td>
</tr>
</table>

</td></tr>
</table>
</div>

In a [[SSC/Structure/BiPolytropes/Analytic15/Pt2#Analytic_Solution_of_Key_Interface_Relation|separate subsection (Part II) of this chapter]], we present a closed-form analytic solution, <math>\Delta_i(\kappa_i)</math>, to this nonlinear equation.



===Second Constraint===

====Obtained from Third Interface Condition====
Our [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|3<sup>rd</sup> interface condition, as detailed above]], states that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\eta_i}{\xi_i}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3^{-1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \phi_i^{-2} \, .</math>
</td>
</tr>

</table>
</div>

If we now choose to normalize the interface amplitude such that, <math>~\phi_i = 1</math>, then this condition establishes two relations: First, from the 3<sup>rd</sup> interface condition alone,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\eta_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3^{-1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \xi_i \, ;</math>
</td>
</tr>

</table>
</div>
and, second, from the definition of Srivastava's function, <math>~\phi</math>, we deduce that the overall scaling parameter is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~B_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sin^2\Delta_i}{\eta_i(3-2\sin^2\Delta_i)} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sqrt{3}}{\xi_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \biggl( \frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1} \, . </math>
</td>
</tr>

</table>
</div>
Notice that, after the solution, <math>~\Delta_i(\kappa_i)</math>, of the [[SSC/Structure/BiPolytropes/Analytic15#KeyInterfaceRelation|key nonlinear interface relation]] has been determined, the first of these two relations also permits us to write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~A_0\eta_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~e^{2\Delta_i}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ A_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \xi_i^{-1} e^{2\Delta_i} \, .</math>
</td>
</tr>
</table>
</div>
Throughout the envelope, therefore, the angle,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Delta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\ln(A_0\eta)^{1/2} = \frac{1}{2} \ln\biggl[ \xi \cdot \xi_i^{-1} e^{2\Delta_i} \biggr]
= \Delta_i + \ln\biggl( \frac{\xi}{\xi_i} \biggr)^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

====Obtained from Chandrasekhar's U-constraint====
We shall now demonstrate that the same expression for the scaling coefficient, <math>~B_0</math>, can alternatively be obtained from Chandrasekhar's U-constraint, without assuming that <math>~\phi_i = 1</math>, after taking into account the result that already has been obtained from the V-constraint. As [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|described above]], the U-constraint is an alternative interface condition that may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\xi_i \theta_i}{(-d\theta/d\xi)_i} \biggl( \frac{\mu_e}{\mu_c}\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\eta_i \phi_i^{5}}{(-d\phi/d\eta)_i} \, ,</math>
</td>
</tr>
</table>
</div>
which, in the particular case being examined here, becomes — again, see [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar.27s_U_and_V_Functions|our accompanying discussion]] for elaboration on the "U<sub>5F</sub>" and "U<sub>1E</sub>" function notations used by {{ Murphy83a }} —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{2\xi_i^2}{3\kappa_i} \biggl( \frac{\mu_e}{\mu_c}\biggr)^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( U_\mathrm{5F} \biggr)_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2B_0^{-4} \sin^4\Delta_i}{(3-2\sin^2\Delta_i)(3 - 2\sin^2\Delta_i - 3\cot\Delta_i)} \, .
</math>
</td>
</tr>
</table>
</div>

Now, from our [[SSC/Structure/BiPolytropes/Analytic15#First_Constraint|discussion, above, of the first constraint]], we know that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(3 - 2\sin^2\Delta_i - 3\cot\Delta_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(3-2\sin^2\Delta_i)\kappa_i \, .</math>
</td>
</tr>
</table>
</div>

Hence, Chandrasekhar's U-constraint becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{2\xi_i^2}{3\kappa_i} \biggl( \frac{\mu_e}{\mu_c}\biggr)^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2B_0^{-4} \sin^4\Delta_i}{(3-2\sin^2\Delta_i)^2 \kappa_i}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ B_0^4 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{3\sin^4\Delta_i}{\xi_i^2 (3-2\sin^2\Delta_i)^2} \biggl( \frac{\mu_e}{\mu_c}\biggr)^{-2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ B_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sqrt{3}}{\xi_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \biggl( \frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1} \, ,</math>
</td>
</tr>
</table>
</div>
which, as predicted, is identical to what we learned from the third interface condition, alone.

===Comment on Murphy's Scalings===

The {{ Murphy83a }} derivations also include an homology factor, <math>~A</math>, and an overall scaling factor, <math>~B</math>, but they are calculated differently from our <math>~A_0</math> and <math>~B_0</math>. In the righthand column of the third page of his paper, Murphy states that,
<div align="center">
<math>~A = \frac{\xi_J}{\zeta_J} \, ,</math>
</div>
which, when translated into our notation <math>~(\zeta_J \rightarrow \xi_i </math> and <math>~\xi_J \rightarrow A_0\eta_\mathrm{root})</math> gives,
<div align="center">
<math>~A = \frac{A_0 \eta_\mathrm{root}}{\xi_i} \, .</math>
</div>
Now, in our derivation, <math>~\eta_\mathrm{root}</math> is synonymous with the location of the envelope interface, <math>~\eta_i</math>, as expressed in terms of the dimensionless radial coordinate associated with Srivastava's Lane-Emden function, so we can equally well state that,
<div align="center">
<math>~A = \frac{A_0 \eta_i}{\xi_i} \, .</math>
</div>
Recalling that <math>~\phi_i = 1</math>, we know from the [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|interface conditions detailed above]] that,
<div align="center">
<math>~\frac{\eta_i}{\xi_i} = \frac{1}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>
</div>
Hence Murphy's homology factor, <math>~A</math>, is related to our homology factor, <math>~A_0</math> via the expression,
<div align="center">
<math>~A = \frac{A_0}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .</math>
</div>
It is usually the value of this quantity, rather than simply our derived value of <math>~A_0</math>, that is tabulated below — both [[SSC/Structure/BiPolytropes/Analytic15#Murphy.27s_Example_Model_Characteristics|here]] and [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29|here]] — as we make quantitative comparisons between the characteristics of our derived models and those published by {{ Murphy83a }} and by {{ MF85a }}.

In the lefthand column of the fourth page of his paper, {{ Murphy83a }} defines the coefficient <math>B</math> in such a way that the ''value'' of the envelope function, <math>\phi_{5F}</math>, equals the ''value'' of the core function, <math>\theta_{1E}</math>, at the interface. Specifically, he sets,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\zeta_J}{\sin\zeta_J} \biggr] \biggl[ \frac{A^{1/2} \sin(\ln\sqrt{A\zeta_J}) }{(A\zeta_J)^{1/2} \{2 + \cos[\ln(A\zeta_J)]\}^{1/2}}\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\zeta_J}{\sin\zeta_J} \biggr] \biggl[ \frac{\sin(\ln\sqrt{A\zeta_J}) }{\zeta_J^{1/2} \{3 - 2\sin^2(\ln\sqrt{A\zeta_J}) \}^{1/2}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
Switching to our terminology, that is, setting,
<div align="center">
<math>\ln\sqrt{A\zeta_J} \rightarrow \Delta_i</math>     and, as before,     
<math>~\zeta_J \rightarrow \xi_i \, ,</math>
</div>
gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\xi_i}{\sin\xi_i} \biggr] \biggl[ \frac{\sin\Delta_i }{\xi_i^{1/2} (3 - 2\sin^2\Delta_i )^{1/2}}\biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\theta_i^{-1} \biggl[ \xi_i^{-1} \biggl(\frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1} \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>
Hence, in terms of the definition of ''our'' scaling coefficient, <math>~B_0</math>, derived above, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{B_0}{3^{1/4}} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{1/2} \theta_i^{-1} \, .</math>
</td>
</tr>
</table>
</div>

As we make quantitative comparisons between the characteristics of our derived models and those published by [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)] and by [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy & Fiedler (1985a)], below, we usually will tabulate the value of this quantity,
rather than simply our derived value of <math>~B_0</math>.

==Step 7: Identifying the Surface==

Because Shrivastava's function — and, along with it, the envelope's density — drops to zero when,
<div align="center">
<math>\Delta = \Delta_s \equiv \pi \, ,</math>
</div>
we know that the radius, <math>~\xi_s</math>, of the bipolytropic configuration is given by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\pi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\Delta_i + \ln\biggl( \frac{\xi_s}{\xi_i} \biggr)^{1/2}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \ln\biggl( \frac{\xi_s}{\xi_i} \biggr)^{1/2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\pi - \Delta_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \xi_s</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\xi_i e^{2(\pi - \Delta_i )} \, .
</math>
</td>
</tr>
</table>
</div>

In terms of the natural radial coordinate of the envelope, this is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\eta_s \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~3^{-1/2} \xi_i e^{2(\pi - \Delta_i)} \, .</math>
</td>
</tr>
</table>
</div>

=Key References=
* [http://adsabs.harvard.edu/abs/1962ApJ...136..680S S. Srivastava (1968, ApJ, 136, 680)] ''A New Solution of the Lane-Emden Equation of Index n = 5''
* [http://adsabs.harvard.edu/abs/1978AuJPh..31..115B H. A. Buchdahl (1978, Australian Journal of Physics, 31, 115)]: ''Remark on the Polytrope of Index 5'' — the result of this work by Buchdahl has been [[SSC/Structure/BiPolytropes/Analytic51#Buchdahl1978|highlighted inside our discussion of bipolytropes with <math>~(n_c, n_e) = (5, 1)</math>]].
* [http://adsabs.harvard.edu/abs/1980PASAu...4...37M J. O. Murphy (1980a, Proc. Astr. Soc. of Australia, 4, 37)]: ''A Finite Radius Solution for the Polytrope Index 5''
* [http://adsabs.harvard.edu/abs/1980PASAu...4...41M J. O. Murphy (1980b, Proc. Astr. Soc. of Australia, 4, 41)]: ''On the F-Type and M-Type Solutions of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1981PASAu...4..205M J. O. Murphy (1981, Proc. Astr. Soc. of Australia, 4, 205)]: ''Physical Characteristics of a Polytrope Index 5 with Finite Radius''
* [http://adsabs.harvard.edu/abs/1982PASAu...4..376M J. O. Murphy (1982, Proc. Astr. Soc. of Australia, 4, 376)]: ''A Sequence of E-Type Composite Analytical Solutions of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1983AuJPh..36..453M J. O. Murphy (1983, Australian Journal of Physics, 36, 453)]: ''Structure of a Sequence of Two-Zone Polytropic Stellar Models with Indices 0 and 1''
* [http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175)]: ''Composite and Analytical Solutions of the Lane-Emden Equation with Polytropic Indices n = 1 and n = 5''
* [http://adsabs.harvard.edu/abs/1985PASAu...6..219M J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219)]: ''Physical Structure of a Sequence of Two-Zone Polytropic Stellar Models''
* [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222)]: ''Radial Pulsations and Vibrational Stability of a Sequence of Two-Zone Polytropic Stellar Models''

=Related Discussions=
* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Polytropes emdeded in an external medium]]
* [[SSC/Structure/BiPolytropes#BiPolytropes|Constructing BiPolytropes]]

{{ SGFfooter }}

SSC/Structure/BiPolytropes

2025-01-24T00:14:00Z

Xxsrm: /* Solution Steps */

__FORCETOC__
=BiPolytropes=
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:#ffff99;" |
<font size="-1">[[H_BookTiledMenu#MoreModels|<b>Composite<br />Polytropes</b>]]<br />(Bipolytropes)</font>
|}

==Background==
While [[SSC/Structure/Polytropes#Polytropic_Spheres|polytropic spheres]] can give us useful ''general'' insight into the internal structure of stars, they do not faithfully represent the detailed structure of most stars, in part, because the equation of state that is most relevant to the densest regions of a star usually is different from the equation of state that is relevant to the star's envelope. As is highlighted in the following excerpt from [http://adsabs.harvard.edu/abs/1931MNRAS..91..472C Cowling (1931)], {{ Milne30full }} was among the first to suggest that more realism could be achieved by constructing what is now commonly referred to as a bipolytrope: a composite model in which the envelope, obeying one equation of state — described by polytropic index <math>n_e</math> — is fitted to a core obeying a different equation of state — described by polytropic index <math>n_c</math>.

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Text extracted<sup>&dagger;</sup> from p. 472 of [http://adsabs.harvard.edu/abs/1931MNRAS..91..472C T. G. Cowling (1931)]<p></p>
"''Note on the Fitting of Polytropic Models in the Theory of Stellar Structure''"<p></p>
MNRAS, 91, 472-478 © Royal Astronomical Society
</td></tr>
<tr>
<td align="center">
[[Image:CowlingRE_Milne1931.jpg|500px|center|Cowling (1931)]]

</td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Text displayed here, as a single digital image, with presentation order & layout modified from the original publication.</td></tr>
</table>
</div>

{{ Milne30 }} envisioned that the temperature, {{Math/VAR_Temperature01}}, and pressure, {{Math/VAR_Pressure01}}, should be "continuous across the surface of fit." Given that {{Math/VAR_Temperature01}} and {{Math/VAR_Pressure01}} are continuous, he envisioned that the gas density, {{Math/VAR_Density01}}, should be continuous across the surface of fit as well. {{ SC42full }} later pointed out — see the following journal article excerpt — that, if the core and envelope have different molecular weights, it is the ratio {{Math/VAR_Density01}}/{{Math/MP_MeanMolecularWeight}} that should be continuous across the interface. A discontinuous jump in {{Math/MP_MeanMolecularWeight}} at the interface — for example, switching from a pure helium core to an envelope whose dominant element is hydrogen — will therefore lead to a discontinuous jump in {{Math/VAR_Density01}} at the interface.

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Text excerpts<sup>&dagger;</sup> from §2 (pp. 161-162) of <br />{{ SC42figure }}<br />© American Astronomical Society
</td></tr>
<tr>
<td align="center">
[[Image:SchonbergChandra1942.jpg|600px|center|Schönberg & Chandrasekhar (1942)]]


</td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Text displayed here, as a single digital image, with presentation order & layout modified from the original publication.</td></tr>
</table>
</div>

{{ SC42 }} furthermore pointed out — again, see the relevant article excerpt — that, when constructing a bipolytropic model, the mathematical function that specifies the mass enclosed inside a spherical radius <math>r</math> for the ''core'', <math>M(r)|_\mathrm{c}</math>, must give the same value as the mathematical function that specifies the mass enclosed inside a spherical radius <math>r</math> for the ''envelope'', <math>M(r)|_\mathrm{e}</math> ''at the radial location of the interface'', <math>r_i</math>.

==Setup==

Here we lay the mathematical foundation for building a spherically symmetric structure in which the core and the envelope are described by different barotropic equations of state. The <math>2^\mathrm{nd}</math> and <math>3^\mathrm{rd}</math> columns of Table 1 provide the relevant set of relations for a structure in which the core and envelope obey polytropic equations of state that have, respectively, polytropic indexes <math>n_c</math> and <math>n_e</math>. Drawing on our [[SSC/Structure/Polytropes#Lane-Emden_Equation|related discussion of isolated polytropes]], it is clear that the structural profile in both regions should be given by a solution of the
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{Math/EQ_SSLaneEmden01}}
</div>
If either region is governed by an isothermal, rather than polytropic, equation of state then, as we have discussed in the context of both [[SSC/Structure/IsothermalSphere#Isothermal_Sphere|isolated]] and [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|pressure-bounded isothermal spheres]] and as is shown by equation (374) of [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]], the structural profile should be given instead by a solution of the related equation,
<div align="center">
<math>
\frac{1}{\chi^2} \frac{d}{d\chi} \biggl( \chi^2 \frac{d\psi}{d\chi} \biggr) = e^{-\psi} \, .
</math>
</div>
The <math>1^\mathrm{st}</math> column of Table 1 provides the relevant set of associated structural relations for an isothermal core.

<div align="center" id="TableSetup">
<table border="1" cellpadding="5" width="80%">
<tr>
<td align="center" colspan="3">
<font size="+1"><b>Table 1:</b> Setup</font>
</td>
</tr>
<tr>
<td align="center" colspan="2">
<font size="+1" color="darkblue">
'''Core''':
</font>
Choose isothermal equation of state or polytropic index <math>n_c</math>
</td>
<td align="center">
<font size="+1" color="darkblue">
'''Envelope'''
</font>
</td>
</tr>

<tr>
<td align="center">
Isothermal <math>(n_c = \infty)</math>
</td>
<td align="center">
<math>n = n_c</math>
</td>
<td align="center">
<math>n = n_e</math>
</td>
</tr>

<tr>
<td align="center">
<math>
\frac{1}{\chi^2} \frac{d}{d\chi} \biggl( \chi^2 \frac{d\psi}{d\chi} \biggr) = e^{-\psi}
</math>

sol'n:
<math>
\psi(\chi)
</math>
</td>
<td align="center">
<math>
\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\theta}{d\xi} \biggr) = - \theta^{n_c}
</math>

sol'n:
<math>
\theta(\xi)
</math>
</td>
<td align="center">
<math>
\frac{1}{\eta^2} \frac{d}{d\eta} \biggl( \eta^2 \frac{d\phi}{d\eta} \biggr) = - \phi^{n_e}
</math>

sol'n:
<math>
\phi(\eta)
</math>
</td>
</tr>

<tr>
<td align="center">


<table border="0" cellpadding="3">
<tr>
<td align="center" colspan="3">
Specify: <math>c_s^2</math> and <math>\rho_0 ~\Rightarrow</math>
</td>
</tr>
<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0 e^{-\psi}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>c_s^2 \rho_0 e^{-\psi}</math>
</td>
</tr>

<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{c_s^2}{4\pi G\rho_0} \biggr]^{1/2} \chi</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{c_s^6}{4\pi G^3\rho_0} \biggr]^{1/2} \biggl( \chi^2 \frac{d\psi}{d\chi} \biggr)</math>
</td>
</tr>
</table>


</td>
<td align="center">


<table border="0" cellpadding="3">
<tr>
<td align="center" colspan="3">
Specify: <math>K_c</math> and <math>\rho_0 ~\Rightarrow</math>
</td>
</tr>
<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_0 \theta^{n_c}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
</td>
</tr>
</table>


</td>
<td align="center">


<table border="0" cellpadding="3">
<tr>
<td align="center" colspan="3">
Knowing: <math>K_e</math> and <math>\rho_e ~\Rightarrow</math>
</td>
</tr>
<tr>
<td align="right">
<math>\rho</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\rho_e \phi^{n_e}</math>
</td>
</tr>

<tr>
<td align="right">
<math>P</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}</math>
</td>
</tr>

<tr>
<td align="right">
<math>r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta</math>
</td>
</tr>

<tr>
<td align="right">
<math>M_r</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{(3-n_e)/(2n_e)} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
</td>
</tr>
</table>


</td>
</tr>
<tr>
<td align="left" colspan="3">
<font color="red">'''NOTE typed on 13 May 2019:'''</font>  Prior to this date, in both the middle and right-hand colums, the RHS of the ''polytropic'' version of the Lane-Emden equation incorrectly had a ''positive'' sign. This was a type-setting error that has now been corrected.
</td>
</tr>
</table>
</div>

===Polytropic Core===
Using the notation established in Table 1, if the core obeys a polytropic equation of state, the variable <math>\xi</math> will denote the dimensionless radial coordinate through the core and the relevant solution is a function, <math>\theta(\xi)</math>, whose value goes to <math>1</math> and whose first derivative, <math>d\theta/d\xi</math>, goes to <math>0</math> at <math>\xi = 0</math>. Then, given a value of the central density, <math>\rho_0</math>, the density throughout the core is,
<div align="center">
<math>\rho(\xi) = \rho_0 \theta(\xi)^{n_c}</math> ;
</div>
and, given a value of the polytropic constant in the core, <math>K_c</math>, the pressure throughout the core is,
<div align="center">
<math>P(\xi) = K_c \rho_0^{1+1/n_c} \theta(\xi)^{n_c + 1}</math>.
</div>
Likewise, given <math>\rho_0</math> and <math>K_c</math>, the radial coordinate <math>r</math> (in dimensional rather than dimensionless units) and the mass enclosed within this radius, <math>M_r</math>, are given by the bottom two expressions shown in the <math>2^\mathrm{nd}</math> column of Table 1.

The structure of an [[SSC/Structure/Polytropes#Lane-Emden_Equation|isolated polytrope]] would be described by following the function <math>\theta(\xi)</math> all the way out to the surface, that is, to the radial location <math>\xi_\mathrm{surf}</math> where <math>\theta(\xi)</math> first drops to zero. (Analytic solutions of this type are presented elsewhere for [[SSC/Structure/Polytropes#n_.3D_0_Polytrope|<math>n = 0</math>]], [[SSC/Structure/Polytropes#n_.3D_1_Polytrope|<math>n = 1</math>]], and [[SSC/Structure/Polytropes#n_.3D_5_Polytrope|<math>n = 5</math>]].) In constructing a bipolytrope, we will instead follow <math>\theta(\xi)</math> out to a radius <math>\xi_i < \xi_\mathrm{surf}</math>, then build an envelope whose inner radius — or ''base'' — is at the ''interface'' radius, <math>r_i</math>, that corresponds to <math>\xi_i</math>. For any choice of the pair of polytropic indexes, <math>n_c</math> and <math>n_e</math>, a series of bipolytropes can then be constructed by choosing a variety of different interface radii.

===Polytropic Envelope===
Again following the notation of Table 1, we will use <math>\eta</math> to identify the dimensionless radial coordinate through the envelope, and the relevant solution of the Lane-Emden equation will be the function, <math>\phi(\eta)</math>. The ''particular'' solution that we seek for the envelope will differ from the solution obtained in the core not only because the governing polytropic index is different but also because the envelope solution will be constrained by different boundary conditions: The value of the function <math>\phi(\eta)</math> as well as its first derivative, <math>d\phi/d\eta</math>, must be specified at some radial location within the envelope. Because the envelope does not extend all the way to the center of the structure, it makes more sense to choose a solution in which <math>\phi</math> is set to unity at the inner edge of the envelope — that is, at the radial location <math>\eta_i</math> that corresponds to <math>r_i</math> — rather than at <math>\eta = 0</math>. By setting <math>\phi = 1</math> at the base of the envelope, it should be clear from the expression,
<div align="center">
<math>
\rho = \rho_e \phi^{n_e} \, ,
</math>
</div>
(taken from the <math>3^\mathrm{rd}</math> column of Table 1) that <math>\rho_e</math> represents the value of the gas density at the base of the envelope. The value of <math>\rho_e</math> can be obtained from knowledge of the gas density at the outer edge of the core (''i.e.'', at <math>r_i</math>) combined with the specified molecular-weight jump condition at the interface. Looking ahead at the first interface condition for a polytropic core catalogued in Table 2, we see more specifically that, after setting <math>\phi_i = 1</math>,
<div align="center">
<math>
\rho_e = \rho_0 \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \, .
</math>
</div>

As we will show in connection with Table 3, the value of <math>d\phi/d\eta</math> at the base of the envelope (''i.e.'', at <math>r_i</math>) also will be set by the properties of the core at the interface — specifically, by the values of <math>(d\theta/d\xi)_i</math> and <math>\theta_i</math>.

==Interface Conditions==
===The Schönberg-Chandrasekhar Condition===
The <math>2^\mathrm{nd}</math> and <math>3^\mathrm{rd}</math> columns of Table 1 provide the mathematical relations that are needed to implement the interface matching conditions specified by equation (1) of {{ SC42 }} (see the above article excerpt) when the core obeys a polytropic equation of state. For example, in order to ensure that,
<div align="center">
<math>
P(r_i)|_c = P(r_i)|_e \, ,
</math>
</div>
the expression for <math>P</math> given in the <math>2^\mathrm{nd}</math> column and ''evaluated at the interface'' is set equal to the expression for <math>P</math> given in the <math>3^\mathrm{rd}</math> column and ''evaluated at the interface''. This results in the <math>2^\mathrm{nd}</math> relation shown in the left-hand column of Table 2, namely,
<div align="center">
<math>
K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}_i = K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}_i \, .
</math>
</div>
Likewise, in order to ensure that,
<div align="center">
<math>
M_r(r_i)|_c = M_r(r_i)|_e \, ,
</math>
</div>
the expression for <math>M_r</math> given in the <math>2^\mathrm{nd}</math> column of Table 1 and ''evaluated at the interface'' is set equal to the expression for <math>M_r</math> given in the <math>3^\mathrm{rd}</math> column of Table 1 and ''evaluated at the interface''. This results in the <math>4^\mathrm{th}</math> relation shown in the left-hand column of Table 2. The <math>3^\mathrm{rd}</math> relation shown in the left-hand column of Table 2 ensures that the dimensionless coordinate used to identify the surface of the core, <math>\xi_i</math>, and the dimensionless coordinate used to identify the base of the envelope, <math>\eta_i</math>, refer to exactly the same physical (dimensional) radial location, <math>r_i</math>. It is obtained by setting the expression for <math>r</math> given in the <math>2^\mathrm{nd}</math> column of Table 1 and ''evaluated at the interface'' equal to the expression for <math>r</math> given in the <math>3^\mathrm{rd}</math> column of Table 1 and ''evaluated at the interface''.

Finally, drawing on the expressions for <math>\rho</math> that are given in the <math>2^\mathrm{nd}</math> and <math>3^\mathrm{rd}</math> columns of Table 1, we note that the <math>1^\mathrm{st}</math> relation shown in the left-hand column of Table 2 ensures that,
<div align="center">
<math>
\frac{\rho(r_i)}{\mu}\biggr|_c = \frac{\rho(r_i)}{\mu}\biggr|_e \, .
</math>
</div>
This relation also ensures that, as desired,
<div align="center">
<math>
T(r_i)|_c = T(r_i)|_e \, ,
</math>
</div>
if, as assumed by {{ SC42 }} (again, see the above article excerpt), <math>T</math> is related to <math>P</math> and <math>\rho</math> through the ideal-gas equation of state. More specifically, according to the ideal-gas equation of state,
<div align="center">
<math>
\frac{1}{T} = \biggl[ \frac{H}{kP} \biggr] \frac{\rho}{\mu} \, .
</math>
</div>
Hence, if <math>P</math> is continuous across the interface, as has already been assured, then the continuity of <math>T</math> across the interface will be ensured by guaranteeing the continuity of <math>\rho/\mu</math> across the interface.

<div align="center" id="Table2">
<table border="1" cellpadding="5">
<tr>
<td align="center" colspan="2">
<font size="+1"><b>Table 2:</b> Interface Conditions</font>
</td>
</tr>
<tr>
<td align="center" colspan="1">
<font size="+1" color="darkblue">
'''Polytropic Core'''
</font>
</td>
<td align="center">
<font size="+1" color="darkblue">
'''Isothermal Core'''
</font>
</td>
</tr>

<tr>
<td align="center">


<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\biggl( \frac{\rho_0}{\mu_c} \biggr) \theta^{n_c}_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_e}{\mu_e} \biggr) \phi^{n_e}_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>[ (n_c + 1)K_c ]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(\xi^2 \frac{d\theta}{d\xi} \biggr)_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>[ (n_e + 1)K_e ]^{3/2} \rho_e^{(3-n_e)/(2n_e)} \biggl(\eta^2 \frac{d\phi}{d\eta} \biggr)_i</math>
</td>
</tr>
</table>


</td>
<td align="center">


<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\biggl( \frac{\rho_0}{\mu_c} \biggr) e^{-\psi_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\rho_e}{\mu_e} \biggr) \phi^{n_e}_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>c_s^2 \rho_0 e^{-\psi_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \frac{c_s^2}{4\pi G\rho_0} \biggr]^{1/2} \chi_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl[ \frac{c_s^6}{4\pi G^3\rho_0} \biggr]^{1/2} \biggl( \chi^2 \frac{d\psi}{d\chi} \biggr)_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>4\pi \biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{(3-n_e)/(2n_e)} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_i</math>
</td>
</tr>
</table>


</td>
</tr>

<tr>
<td align="left">
After setting <math>\mu_c=\mu_e=1</math>, these four relations become identical to equations 482-485 (p. 172) of [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]].
</td>
<td align="left">
After setting <math>n_e = 1</math>, these four relations are essentially identical to, respectively, equations (13), (10), (12), & (11) of {{ Beech88full }}; adopting <math>n_e=3</math> instead, they serve as the foundation of earlier work by {{ HC41full }}.
</td>
</tr>
</table>
</div>

<span id="TohlineGeneralization">

===The Tohline Generalization===
</span>
[Introduced '''<font color="red">10 June 2013</font>'''] It should be pointed out that, while the interface conditions shown in Table 2 and the solution steps that follow do ensure that the gas pressure is continuous across the interface and allow for a discontinuity in the mass-density across the interface, they do not actually force the temperature to be continuous across the interface. More generally, pressure continuity is ensured if,
<div align="center">
<math>
\frac{\rho(r_i)}{\mu/T(r_i)}\biggr|_c = \frac{\rho(r_i)}{\mu/T(r_i)}\biggr|_e \, .
</math>
</div>
So a discontinuity across the interface will arise if the ratio of the molecular weights, <math>\mu_c/\mu_e</math>, is not unity, or if there is a discontinuity in the temperature across the interface — that is, if <math>T(r_i)|_e \ne T(r_i)|_c</math>, or both. Because they were using bipolytropes to model optically thick stellar interiors, {{ SC42 }} argued that the temperature should also be continuous across the interface and, hence, that a discontinuity in the density would be introduced at the interface from a discontinuity in the molecular weight. [[SSC/Structure/LimitingMasses#Relationship_Between_the_Bonnor-Ebert_and_Sch.C3.B6nberg-Chandrasekhar_Critical_Masses|In a separate chapter where we discuss the relationship between the Schönberg-Chandrasekhar critical mass and the Bonnor-Ebert critical mass]], we will argue that a discontinuous drop in the density is introduced by a substantial jump in the gas temperature at the interface. In making this alternate assumption, the structural equations describing the bipolytropic model will remain unchanged; we will only need to replace the ratio <math>\mu_c/\mu_e</math> by the ratio <math>T(r_i)|_e/T(r_i)|_c</math>.

==Solution Steps==

* Step 1: Choose <math>n_c</math> and <math>n_e</math>.
* Step 2: Adopt boundary conditions at the center of the core (<math>\theta = 1</math> and <math>d\theta/d\xi = 0</math> at <math>\xi=0</math>), then solve the Lane-Emden equation to obtain the solution, <math>\theta(\xi)</math>, and its first derivative, <math>d\theta/d\xi</math> throughout the core; the radial location, <math>\xi = \xi_s</math>, at which <math>\theta(\xi)</math> first goes to zero identifies the natural surface of an [[SSC/Structure/Polytropes#Lane-Emden_Equation|isolated polytrope]] that has a polytropic index <math>n_c</math>.
* Step 3 Choose the desired location, <math>0 < \xi_i < \xi_s</math>, of the outer edge of the core.
* Step 4: Specify <math>K_c</math> and <math>\rho_0</math>; the structural profile of, for example, <math>\rho(r)</math>, <math>P(r)</math>, and <math>M_r(r)</math> is then obtained throughout the core — over the radial range, <math>0 \le \xi \le \xi_i</math> and <math>0 \le r \le r_i</math> — via the relations shown in the <math>2^\mathrm{nd}</math> column of Table 1.
* Step 5: Specify the ratio <math>\mu_e/\mu_c</math> and adopt the boundary condition, <math>\phi_i = 1</math>; then use the interface conditions as rearranged and presented in Table 3 to determine, respectively:
** The gas density at the base of the envelope, <math>\rho_e</math>;
** The polytropic constant of the envelope, <math>K_e</math>, relative to the polytropic constant of the core, <math>K_c</math>;
** The ratio of the two dimensionless radial parameters at the interface, <math>\eta_i/\xi_i</math>;
** The radial derivative of the envelope solution at the interface, <math>(d\phi/d\eta)_i</math>.
* Step 6: The last sub-step of solution step 5 provides the boundary condition that is needed — in addition to our earlier specification that <math>\phi_i = 1</math> — to derive the desired ''particular'' solution, <math>\phi(\eta)</math>, of the Lane-Emden equation that is relevant throughout the envelope; knowing <math>\phi(\eta)</math> also provides the relevant structural first derivative, <math>d\phi/d\eta</math>, throughout the envelope.
* Step 7: The surface of the bipolytrope will be located at the radial location, <math>\eta = \eta_s</math> and <math>r=R</math>, at which <math>\phi(\eta)</math> first drops to zero.
* Step 8: The structural profile of, for example, <math>\rho(r)</math>, <math>P(r)</math>, and <math>M_r(r)</math> is then obtained throughout the envelope — over the radial range, <math>\eta_i \le \eta \le \eta_s</math> and <math>r_i \le r \le R</math> — via the relations provided in the <math>3^\mathrm{rd}</math> column of Table 1.

<div align="center" id="Table3">
<table border="1" cellpadding="5">
<tr>
<td align="center" colspan="2">
<font size="+1"><b>Table 3:</b> Sub-steps of Solution Step 5</font> <br>
(derived from the relations in Table 2)
</td>
</tr>
<tr>
<td align="center" colspan="1">
<font size="+1" color="darkblue">
'''Polytropic Core'''
</font>
</td>
<td align="center">
<font size="+1" color="darkblue">
'''Isothermal Core'''
</font>
</td>
</tr>

<tr>
<td align="center">


<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\frac{\rho_e}{\rho_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \phi_i^{-n_e}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) \rho_0^{1/n_e - 1/n_c}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} \theta^{1 - n_c/n_e}_i</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\eta_i}{\xi_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{(n_c + 1)}{(n_e+1)} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \theta_i^{(n_c-1)/2} \phi_i^{(1-n_e)/2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \frac{n_c + 1}{n_e + 1} \biggr]^{1/2} \theta_i^{- (n_c + 1)/2} \phi_i^{(n_e+1)/2} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>
</td>
</tr>
</table>


</td>
<td align="center">


<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>\biggl( \frac{\rho_e}{\rho_0} \biggr)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) e^{-\psi_i} \phi_i^{-n_e}</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{K_e \rho_0^{1/n_e} }{c_s^2} </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} e^{+\psi_i/n_e} </math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{\eta_i}{\chi_i}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>(n_e + 1)^{-1/2}\biggl( \frac{\mu_e}{\mu_c} \biggr) e^{-\psi_i/2} \phi_i^{(1-n_e)/2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>- \biggl(\frac{d\phi}{d\eta} \biggr)_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>(n_e + 1)^{-1/2} e^{+\psi_i/2} \phi_i^{(n_e+1)/2} \biggl(\frac{d\psi}{d\chi} \biggr)_i</math>
</td>
</tr>
</table>


</td>
</tr>
</table>
</div>

<span id="UVplane">
Taking the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions on the left-hand side of Table 3 produces,</span>
<div align="center">
<math>
\frac{\eta_i \phi_i^{n_e}}{(d\phi/d\eta)_i} = \frac{\xi_i \theta_i^{n_c}}{(d\theta/d\xi)_i} \biggl( \frac{\mu_e}{\mu_c}\biggr) \, .
</math>
</div>
''Multiplying'' the <math>3^\mathrm{rd}</math> expression by the <math>4^\mathrm{th}</math> expression generates,
<div align="center">
<math>
(n_e+1)\frac{\eta_i (d\phi/d\eta)_i}{ \phi_i } = (n_c+1)\frac{\xi_i (d\theta/d\xi)_i}{ \theta_i } \biggl( \frac{\mu_e}{\mu_c}\biggr) \, .
</math>
</div>
These are two relations that Chandrasekhar found to be useful in his analysis of "composite [polytropic] configurations" — after setting <math>\mu_e=\mu_c</math> they match, respectively, equations 486 & 489 in § 28 of [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]].

==Isothermal Core==

The relations shown in the right-hand columns of Table 2 and Table 3 provide analogous conditions that must be satisfied in order to ensure the proper continuity of physical parameters across the interface of a composite structure that has an isothermal, rather than a polytropic, core. The ''interface conditions'' shown in Table 2 were derived by properly associating the relations in the <math>1^\mathrm{st}</math> column of Table 1 with corresponding relations in the <math>3^\mathrm{rd}</math> column of Table 1. Strictly speaking, a structure whose core obeys an isothermal equation of state can also be considered a bipolytrope, but its mathematical description must be handled separately because its effective polytropic index is <math>n_c=\infty</math>.

=Example Solutions=

* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Analytic solution]] for <math>n_c = 5, ~n_e = 1</math>.
* [[SSC/Structure/BiPolytropes/Analytic00|Analytic solution]] for <math>n_c = 0, ~n_e = 0</math>.

=Related Discussions=
* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Polytropes emdeded in an external medium]]
* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]
* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert spheres]]
** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia
** [http://www.astro.umd.edu/~cychen/MATLAB/ASTR320/matlabFrom320spring2011/Bonnor-EbertSphere/html/BonnorEbert.html A MATLAB script to determine the Bonnor-Ebert Mass coefficient] developed by [http://www.astro.umd.edu/people/cychen.html Che-Yu Chen] as a graduate student in the University of Maryland Department of Astronomy
* [[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|Schönberg-Chandrasekhar limiting mass]]
* [[SSC/Structure/LimitingMasses#Relationship_Between_the_Bonnor-Ebert_and_Sch.C3.B6nberg-Chandrasekhar_Critical_Masses|Relationship between Bonnor-Ebert and Schönberg-Chandrasekhar limiting masses]]

* [http://www.researchgate.net/publication/2228492_Low_order_p-modes_in_a_bipolytropic_model_of_the_Sun Oscillations in a BiPolytropic Model of the Sun]
* [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Schoenberg-Chandrasekhar Limit: A BiPolytropic Approximation (Beech 1988b)]
* [http://adsabs.harvard.edu/abs/1988Ap%26SS.146..299B BiPolytropic Model for Low-Mass Stars (Beech 1988a)]
* An analysis (published in French) of the secular stability of configurations containing an isothermal core, with relevance to the Schönberg-Chandrasekhar mass limit; [http://adsabs.harvard.edu/abs/1967AnAp...30..975G M. Gabriel and P. Ledoux (1967, Annals d'Astrophysique, 30, 975)]

=See Also=

* [http://adsabs.harvard.edu/abs/1983ApJ...275..713R Rappaport, Verbunt, & Joss (1983, ApJ, 275, 713)] — ''A New Technique for Calculations of Binary Stellar Evolution, with Application to Magnetic Braking''.

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Xxsrm: Created page with "__FORCETOC__  =Approximate Power-Series Expressions= ==Broadly Used Mathematical Expressions (shown here without proof)== ===Binomial=== <div align="center"> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~(1 \pm x)^n</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~\pm ~nx + \biggl[\frac{n(n-1)}{2!}\biggr]x^2 ~\pm~ \biggl[\frac{n(n-1)(n-2)}{3!}\biggr]x^3..."

__FORCETOC__

=Approximate Power-Series Expressions=

==Broadly Used Mathematical Expressions (shown here without proof)==
===Binomial===
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(1 \pm x)^n</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 ~\pm ~nx + \biggl[\frac{n(n-1)}{2!}\biggr]x^2
~\pm~ \biggl[\frac{n(n-1)(n-2)}{3!}\biggr]x^3
+ \biggl[\frac{n(n-1)(n-2)(n-3)}{4!}\biggr]x^4
~~\pm ~~ \cdots
</math>
     for <math>~(x^2 < 1)</math>
</td>
</tr>
</table>
</div>

<table border="1" align="center" cellpadding="8" width="70%">
<tr>
<th align="center" bgcolor="yellow">
LaTeX mathematical expressions cut-and-pasted directly from
<br />
NIST's ''Digital Library of Mathematical Functions''
</th>
</tr>
<tr>
<td align="left">
As a primary point of reference, note that according to [http://dlmf.nist.gov/1.2 §1.2 of NIST's ''Digital Library of Mathematical Functions''], the binomial theorem states that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(a+b)^{n}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a^{n}+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^{2}+\dots+\binom{n}{n-1}ab^{n-1}+b^{n},
</math>
</td>
</tr>
</table>
where, for nonnegative integer values of <math>~k</math> and <math>~n</math> and <math>~k \le n</math>, the notation,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\binom{n}{k}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{n!}{(n-k)!k!}=\binom{n}{n-k}.
</math>
</td>
</tr>
</table>

----

'''Our Example:'''  Setting <math>~a = 1</math> gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(1+b)^{n}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1+\binom{n}{1}b+\binom{n}{2}b^{2}+\binom{n}{3}b^{3}+\binom{n}{4}b^{4}+\dots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1+\frac{n!}{(n-1)!}~b + \frac{n!}{(n-2)! 2!}~b^{2} + \frac{n!}{(n-3)! 3!}~b^{3} + \frac{n!}{(n-4)! 4!}~b^{4} + \dots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1+ nb + \biggl[ \frac{n(n-1)}{2!}\biggr] b^{2} + \biggl[ \frac{n(n-1)(n-2)}{3!} \biggr] b^{3} + \biggl[ \frac{n(n-1)(n-2)(n-3)}{4!} \biggr] b^{4} + \dots
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>

Note, for example, that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(1+x)^{-1}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 - x +x^2 - x^3 + x^4 - x^5 + \cdots \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~(1+x)^{-2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 - 2x + 3x^2 - 4x^3 + 5x^4 - 6x^5 + \cdots \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~(1+x)^{-3}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 - 3x + \biggl[ \frac{3\cdot 4 }{ 2} \biggr]x^2 - \biggl[ \frac{ 3\cdot 4 \cdot 5}{ 2\cdot 3} \biggr]x^3
+ \biggl[ \frac{3\cdot 4 \cdot 5 \cdot 6 }{2\cdot 3 \cdot 4 } \biggr]x^4 - \biggl[ \frac{3\cdot 4 \cdot 5 \cdot 6 \cdot 7}{2\cdot 3 \cdot 4 \cdot 5 } \biggr]x^5 + \cdots </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 - 3x + 6x^2 - 10x^3 + 15x^4 - 21x^5 + \cdots \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~(1+x)^{-4}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 - 4x + \biggl[ \frac{4\cdot 5 }{ 2} \biggr]x^2 - \biggl[ \frac{ 4\cdot 5 \cdot 6}{ 2\cdot 3} \biggr]x^3
+ \biggl[ \frac{4\cdot 5 \cdot 6 \cdot 7 }{2\cdot 3 \cdot 4 } \biggr]x^4 - \biggl[ \frac{4\cdot 5 \cdot 6 \cdot 7 \cdot 8}{2\cdot 3 \cdot 4 \cdot 5 } \biggr]x^5 + \cdots </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 - 4x + 10 x^2 - 20x^3
+ 35x^4 - 56x^5 + \cdots \, .</math>
</td>
</tr>
</table>

See also:
* [http://mathworld.wolfram.com/BinomialTheorem.html Wolfram's presentation]

===Exponential===
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e^x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots
</math>
</td>
</tr>
</table>
</div>

==Expressions with Astrophysical Relevance==
===Polytropic Lane-Emden Function===
We seek a power-series expression for the polytropic, Lane-Emden function, <math>~\Theta_\mathrm{H}(\xi)</math> — expanded about the coordinate center, <math>~\xi = 0</math> — that approximately satisfies the Lane-Emden equation,
<div align="center">
{{ Math/EQ_SSLaneEmden01 }}
</div>

A general power-series should be of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Theta_H</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\theta_0 + a\xi + b\xi^2 + c\xi^3 + d\xi^4 + e\xi^5 + f\xi^6 + g\xi^7 + h\xi^8 + \cdots
</math>
</td>
</tr>
</table>
</div>

First derivative:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\Theta_H}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a + 2b\xi + 3c\xi^2 + 4d\xi^3 + 5e\xi^4 + 6f\xi^5 + 7g\xi^6 + 8h\xi^7 + \cdots
</math>
</td>
</tr>
</table>
</div>

Left-hand-side of Lane-Emden equation:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2a}{\xi} + 2\cdot 3b + 2^2\cdot 3c\xi + 2^2\cdot 5d\xi^2 + 2\cdot 3\cdot 5e\xi^3 + 2\cdot 3\cdot 7f\xi^4 + 2^3\cdot 7g\xi^5 + 2^3\cdot 3^2h\xi^6 + \cdots
</math>
</td>
</tr>
</table>
</div>

Right-hand-side of Lane-Emden equation (adopt the normalization, <math>~\theta_0=1</math>, then use the [[#Binomial|binomial theorem]] recursively):
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Theta_H^n</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 ~+ ~nF + \biggl[\frac{n(n-1)}{2!}\biggr]F^2
~+~ \biggl[\frac{n(n-1)(n-2)}{3!}\biggr]F^3
+ \biggl[\frac{n(n-1)(n-2)(n-3)}{4!}\biggr]F^4
~~+ ~~ \cdots
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~F</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
a\xi + b\xi^2 + c\xi^3 + d\xi^4 + e\xi^5 + f\xi^6 + g\xi^7 + h\xi^8 + \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a\xi\biggl[1 + \frac{b}{a}\xi + \frac{c}{a}\xi^2 + \frac{d}{a}\xi^3 + \frac{e}{a}\xi^4 + \frac{f}{a}\xi^5 + \frac{g}{a}\xi^6 + \frac{h}{a}\xi^7 + \cdots\biggr] \, .
</math>
</td>
</tr>
</table>
</div>
<font color="red">First approximation</font>:  Assume that <math>~e=f=g=h=0</math>, in which case the LHS contains terms only up through <math>~\xi^2</math>. This means that we must ignore all terms on the RHS that are of higher order than <math>~\xi^2</math>; that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Theta_H^n</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
1 ~+ ~nF + \biggl[\frac{n(n-1)}{2!}\biggr]F^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
1 ~+ ~n(a\xi+b\xi^2) + \biggl[\frac{n(n-1)}{2!}\biggr]a^2\xi^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
1 ~+~na\xi + ~\biggl[n b + \frac{n(n-1)a^2}{2}\biggr]\xi^2\, .
</math>
</td>
</tr>
</table>
</div>
Expressions for the various coefficients can now be determined by equating terms on the LHS and RHS that have like powers of <math>~\xi</math>. Remembering to include a negative sign on the RHS, we find:
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
<td align="center">Term</td>
<td align="center">LHS</td>
<td align="center">RHS</td>
<td align="center">Implication</td>
</tr>

<tr>
<td align="right">
<math>~\xi^{-1}:</math>
</td>
<td align="center">
<math>~2a</math>
</td>
<td align="center">
<math>~0</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~a=0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^{0}:</math>
</td>
<td align="center">
<math>~2\cdot 3 b</math>
</td>
<td align="center">
<math>~-1</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~b=- \frac{1}{6}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^{1}:</math>
</td>
<td align="center">
<math>~2^2\cdot 3 c</math>
</td>
<td align="center">
<math>~-na</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~c=0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^{2}:</math>
</td>
<td align="center">
<math>~2^2\cdot 5 d</math>
</td>
<td align="center">
<math>~-\biggl[n b + \frac{n(n-1)a^2}{2}\biggr]</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~d=+\frac{n}{120}</math>
</td>
</tr>
</table>
</div>

By including higher and higher order terms in the series expansion for <math>~\Theta_H</math>, and proceeding along the same line of deductive reasoning, one finds:

* Expressions for the four coefficients, <math>~a, b, c, d</math>, remain unchanged.
* The coefficient is zero for all other terms that contain ''odd'' powers of <math>~\xi</math>; specifically, for example, <math>~e = g = 0</math>.
* The coefficients of <math>~\xi^6</math> and <math>~\xi^8</math> are, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{n}{378}\biggl(\frac{n}{5}-\frac{1}{8} \biggr) \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~h</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{n(122n^2 -183n + 70)}{3265920} \, .</math>
</td>
</tr>
</table>
</div>

In summary, the desired, approximate power-series expression for the polytropic Lane-Emden function is:
<div align="center" id="PolytropicLaneEmden">
<table border="1" width="80%" cellpadding="8" align="center">
<tr><th align="center">For Spherically Symmetric Configurations</th></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 - \frac{\xi^2}{6} + \frac{n}{120} \xi^4 - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^6 + \biggl[ \frac{n(122n^2 -183n + 70)}{3265920} \biggr] \xi^8 + \cdots
</math>
</td>
</tr>
</table>
</td></tr></table>
</div>

NOTE:  For cylindrically symmetric, rather than spherically symmetric, configurations, the analogous power-series expression appears as equation (15) in the article by [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. P. Ostriker (1964, ApJ, 140, 1056)] titled, ''The Equilibrium of Polytropic and Isothermal Cylinders''.

===Isothermal Lane-Emden Function===



Here we seek a power-series expression for the isothermal, Lane-Emden function — expanded about the coordinate center — that approximately satisfies the [[SSC/Structure/IsothermalSphere#Chandrasekhar|isothermal Lane-Emden equation]]; making the variable substitution (sorry for the unnecessary complication!), <math>~\psi(\xi) \leftrightarrow w(r)</math>, the governing ODE is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~e^{-w} \, . </math>
</td>
</tr>
</table>
</div>

A general power-series should be of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
w_0 + ar + br^2 + cr^3 + dr^4 + er^5 + fr^6 + gr^7 + hr^8 +\cdots
</math>
</td>
</tr>
</table>
</div>

Derivatives:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dw}{dr}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a + 2br + 3cr^2 + 4dr^3 + 5er^4 + 6fr^5 + 7gr^6 + 8hr^7 +\cdots \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{d^2w}{dr^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2b + 2\cdot 3cr + 2^2\cdot 3dr^2 + 2^2\cdot 5er^3 + 2\cdot 3 \cdot 5fr^4 + 2\cdot 3 \cdot 7gr^5 + 2^3\cdot 7hr^6 +\cdots \, .
</math>
</td>
</tr>
</table>
</div>

Put together, then, the left-hand-side of the isothermal Lane-Emden equation becomes:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2b + 2\cdot 3cr + 2^2\cdot 3dr^2 + 2^2\cdot 5er^3 + 2\cdot 3 \cdot 5fr^4 + 2\cdot 3 \cdot 7gr^5 + 2^3\cdot 7hr^6
+ \frac{2}{r}\biggl[ a + 2br + 3cr^2 + 4dr^3 + 5er^4 + 6fr^5 + 7gr^6 + 8hr^7 \biggr] + \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2a}{r} + r^0(6b) + r^1(2^2\cdot 3c) + r^2(2^2\cdot 3d + 2^3d) + r^3(2^2\cdot 5e + 2\cdot 5e)
+ r^4(2\cdot 3\cdot 5 f + 2^2\cdot 3f) + r^5(2\cdot 3\cdot 7 g+ 2\cdot 7g) + r^6(2^3 \cdot 7 h + 2^4 h) + \cdots
</math>
</td>
</tr>
</table>
</div>

Drawing on the [[#Exponential|above power-series expression for an exponential function]], and adopting the convention that <math>~w_0 = 0</math>, the right-hand-side becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e^{-w}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
e^{0}\cdot e^{-ar} \cdot e^{-br^2} \cdot e^{-cr^3} \cdot e^{-dr^4} \cdot e^{-er^5} \cdot e^{-fr^6} \cdot e^{-gr^7} \cdot e^{-hr^8} \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ 1 -ar + \frac{a^2r^2}{2!} - \frac{a^3r^3}{3!} + \frac{a^4r^4}{4!} - \frac{a^5r^5}{5!} + \frac{a^6r^6}{6!} + \cdots \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl[ 1 -br^2 + \frac{b^2r^4}{2!} - \frac{b^3r^6}{3!} + \cdots \biggr] \times \biggl[ 1 -cr^3 + \frac{c^2r^6}{2!} + \cdots \biggr]
\times \biggl[1 - dr^4\biggr] \times \biggl[1 - er^5\biggr]\times \biggl[1 - fr^6\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
\biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + \frac{a^4r^4}{24} - \frac{a^5r^5}{5\cdot 24} + \frac{a^6r^6}{30\cdot 24} \biggr]
\times \biggl[ 1 -cr^3 + \frac{c^2r^6}{2} -br^2 + bcr^5 + \frac{b^2r^4}{2} - \frac{b^3r^6}{6} \biggr]
\times \biggl[1 - dr^4 - er^5 - fr^6\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\biggl\{
\biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + \frac{a^4r^4}{24} - \frac{a^5r^5}{5\cdot 24} + \frac{a^6r^6}{30\cdot 24} \biggr]
- dr^4 \biggl[ 1 -ar + \frac{a^2r^2}{2} \biggr] - er^5 \biggl[ 1 -ar \biggr] - fr^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl[ 1 -br^2 -cr^3 + \frac{b^2r^4}{2} + bcr^5 + r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\biggl[
1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + \frac{a^4r^4}{24} - \frac{a^5r^5}{5\cdot 24} + \frac{a^6r^6}{30\cdot 24}
- dr^4 + adr^5 - \frac{a^2d r^6}{2} - er^5 + aer^6 - fr^6
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl[ 1 -br^2 -cr^3 + \frac{b^2r^4}{2} + bcr^5 + r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\biggl[
1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + r^4\biggl(\frac{a^4}{24} - d\biggr) + r^5\biggl(ad - e-\frac{a^5}{5\cdot 24}\biggr)
+ r^6 \biggl(\frac{a^6}{30\cdot 24} - \frac{a^2d}{2} + ae - f \biggr)
\biggr]
\times \biggl[ 1 -br^2 -cr^3 + \frac{b^2r^4}{2} + bcr^5 + r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + r^4\biggl(\frac{a^4}{24} - d\biggr) + r^5\biggl(ad - e-\frac{a^5}{5\cdot 24}\biggr)
+ r^6 \biggl(\frac{a^6}{30\cdot 24} - \frac{a^2d}{2} + ae - f \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~-br^2\biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + r^4\biggl(\frac{a^4}{24} - d\biggr) \biggr]
-cr^3 \biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} \biggr]
+ \frac{b^2r^4}{2}\biggl[ 1 -ar + \frac{a^2r^2}{2} \biggr]
+ bcr^5\biggl[1 -ar \biggr]
+ r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr)
</math>
</td>
</tr>
</table>
</div>

Expressions for the various coefficients can now be determined by equating terms on the LHS and RHS that have like powers of <math>~r</math>. Beginning with the highest order terms, we initially find,
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
<td align="center">Term</td>
<td align="center">LHS</td>
<td align="center">RHS</td>
<td align="center">Implication</td>
</tr>

<tr>
<td align="right">
<math>~r^{-1}:</math>
</td>
<td align="center">
<math>~2a</math>
</td>
<td align="center">
<math>~0</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~a=0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r^{0}:</math>
</td>
<td align="center">
<math>~6b</math>
</td>
<td align="center">
<math>~1</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~b = + \frac{1}{6}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r^{1}:</math>
</td>
<td align="center">
<math>~2^2\cdot 3c</math>
</td>
<td align="center">
<math>~-a</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~c = -\frac{a}{2^2\cdot 3} =0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r^{2}:</math>
</td>
<td align="center">
<math>~(2^2\cdot 3d + 2^3d)</math>
</td>
<td align="center">
<math>~\frac{a^2}{2} - b</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~d = \frac{1}{20}\biggl( \frac{a^2}{2} - b \biggr) = - \frac{1}{120}</math>
</td>
</tr>
</table>
</div>

With this initial set of coefficient values in hand, we can rewrite (and significantly simplify) our approximate expression for the RHS, namely,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e^{-w}</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
1 -d r^4 -e r^5 -f r^6
-br^2 ( 1 -d r^4 ) + \frac{b^2r^4}{2} - \frac{b^3r^6}{6}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 -br^2+ r^4 \biggl(\frac{b^2}{2} -d \biggr) -e r^5
+r^6\biggl( bd - \frac{b^3}{6} -f \biggr) \, .
</math>
</td>
</tr>
</table>
</div>

Continuing, then, with equating terms with like powers on both sides of the equation, we find,
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
<td align="center">Term</td>
<td align="center">LHS</td>
<td align="center">RHS</td>
<td align="center">Implication</td>
</tr>

<tr>
<td align="right">
<math>~r^{3}:</math>
</td>
<td align="center">
<math>~30e</math>
</td>
<td align="center">
<math>~0</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~e=0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r^{4}:</math>
</td>
<td align="center">
<math>~(2\cdot 3\cdot 5 f + 2^2\cdot 3f)</math>
</td>
<td align="center">
<math>~\biggl(\frac{b^2}{2} -d \biggr) </math>
</td>
<td align="left">
<math>~\Rightarrow ~~~f = \frac{1}{2\cdot 3\cdot 7}\biggl(\frac{1}{2^3\cdot 3^2}+\frac{1}{2^3\cdot 3 \cdot 5}\biggr) = \frac{1}{2\cdot 3^3\cdot 5 \cdot 7}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r^{5}:</math>
</td>
<td align="center">
<math>~(2\cdot 3\cdot 7 g+ 2\cdot 7g)</math>
</td>
<td align="center">
<math>~-e</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~g = 0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r^{6}:</math>
</td>
<td align="center">
<math>~(2^3 \cdot 7 h + 2^4 h)</math>
</td>
<td align="center">
<math>~\biggl( bd - \frac{b^3}{6} -f \biggr)</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~
h = -\frac{1}{2^3\cdot 3^2}\biggl( \frac{1}{2^4\cdot 3^2 \cdot 5} + \frac{1}{2^4\cdot 3^4} + \frac{1}{2\cdot 3^3\cdot 5\cdot 7}\biggr)
= -\frac{61}{2^{6} \cdot 3^6\cdot 5\cdot 7}
</math>
</td>
</tr>
</table>
</div>

Result:
<div align="center" id="IsothermalLaneEmden">
<table border="1" width="80%" cellpadding="8" align="center">
<tr><th align="center">For Spherically Symmetric Configurations</th></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w(r)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{r^2}{6} - \frac{r^4}{120} + \frac{r^6}{1890} - \frac{61 r^8}{1,632,960} + \cdots \, .</math>
</td>
</tr>
</table>
</td></tr></table>
</div>

See also:
* Equation (377) from §22 in Chapter IV of [[Appendix/References#C67|C67]].

NOTE:  For cylindrically symmetric, rather than spherically symmetric, configurations, an analytic expression for the function, <math>~w(r)</math>, is presented as equation (56) in a paper by [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. P. Ostriker (1964, ApJ, 140, 1056)] titled, ''The Equilibrium of Polytropic and Isothermal Cylinders''.

===Displacement Function for Polytropic LAWE===
The [[SSC/Stability/Polytropes#Adiabatic_.28Polytropic.29_Wave_Equation|LAWE for polytropic spheres]] may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{(n+1)}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{dx}{d\xi} +
\frac{(n+1)}{\theta}\biggl[\frac{\sigma_c^2}{6\gamma } -
\frac{\alpha}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] x </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\theta \frac{d^2x}{d\xi^2} + \biggl[4\theta - (n+1)\xi \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{1}{\xi}\frac{dx}{d\xi} +
\frac{(n+1)}{6} \biggl[\frac{\sigma_c^2}{\gamma } -
\frac{6\alpha}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] x \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>~\theta(\xi)</math> is the polytropic Lane-Emden function describing the configuration's unperturbed radial density distribution, and <math>~\gamma</math>, <math>~\sigma_c^2</math>, and <math>~\alpha \equiv (3-4/\gamma)</math> are constants. Here we seek a power-series expression for the displacement function, <math>~x(r)</math>, expanded about the center of the configuration, that approximately satisfies this LAWE.

First we note that, near the center, an accurate [[#PolytropicLaneEmden|power-series expression for the polytropic Lane-Emden function]] is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 - \frac{\xi^2}{6} + \frac{n}{120} \xi^4 - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^6 + \cdots
</math>
</td>
</tr>
</table>

Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~-\frac{d\theta}{d\xi}</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
\frac{1}{3} \biggl[ \xi - \frac{n}{10} \xi^3 + \frac{n}{21} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^5 \biggr]
\, .</math>
</td>
</tr>
</table>
</div>
Therefore, near the center of the configuration, the LAWE may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~6~\theta \frac{d^2x}{d\xi^2} + \biggl\{ 12~\theta
- (n+1)\xi \biggl[ \xi - \frac{n}{10} \xi^3 + \frac{n}{21} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^5 \biggr] \biggr\} \frac{2}{\xi}\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~ -
(n+1) \biggl\{ \frac{\sigma_c^2}{\gamma } -
\frac{2\alpha}{\xi} \biggl[ \xi - \frac{n}{10} \xi^3 + \frac{n}{21} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^5 \biggr] \biggr\} x </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~ ~6\biggl[ 1 - \frac{\xi^2}{6} + \frac{n}{120} \xi^4 \biggr] \frac{d^2x}{d\xi^2}
+ \biggl\{ 12 \biggl[ 1 - \frac{\xi^2}{6} + \frac{n}{120} \xi^4 \biggr]
- (n+1)\biggl[ \xi^2 - \frac{n}{10} \xi^4 \biggr] \biggr\} \frac{2}{\xi}\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~ -
(n+1) \biggl\{ \mathfrak{F}
+ 2\alpha \biggl[ \frac{n}{10} \xi^2 - \frac{n}{21} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^4 \biggr] \biggr\} x </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~ ~\biggl( 6 - \xi^2 + \frac{n}{20} \xi^4 \biggr) \frac{d^2x}{d\xi^2}
+ \biggl[ 12 - (n+3)\xi^2 + \frac{n(n+2)}{10} \xi^4 \biggr] \frac{2}{\xi}\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~ -
(n+1) \biggl[ \mathfrak{F}
+ \frac{n\alpha}{5} \xi^2 - \frac{2n\alpha}{21} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^4 \biggr] x \, ,</math>
</td>
</tr>
</table>
</div>
where, <math>\mathfrak{F} \equiv (\sigma_c^2/\gamma - 2\alpha)</math> and, for present purposes, we have kept terms in the series no higher than <math>~\xi^4</math>.

====Displacement Finite at Center====
Let's adopt a power-series expression for the displacement function of a form that is finite at the center of the configuration, namely,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 + a\xi + b\xi^2 + c\xi^3 + d\xi^4 + e\xi^5 + f\xi^6\cdots
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{1}{\xi}\frac{dx}{d\xi}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{a}{\xi} + 2b + 3 c\xi + 4d\xi^2 + 5e\xi^3 + 6f\xi^4 +\cdots
</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2x}{d\xi^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2b + 6c\xi + 12d\xi^2 + 20e\xi^3 + 30f\xi^4 + \cdots
</math>
</td>
</tr>
</table>
</div>
Substituting these expressions into the LAWE gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( 6 - \xi^2 + \frac{n}{20} \xi^4 \biggr) \biggl( 2b + 6c\xi + 12d\xi^2 + 20e\xi^3 + 30f\xi^4 \biggr)
+ \biggl[ 12 - (n+3)\xi^2 + \frac{n(n+2)}{10} \xi^4 \biggr] \biggl( \frac{2a}{\xi} + 4b + 6 c\xi + 8d\xi^2 + 10e\xi^3 + 12f\xi^4 \biggr)</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~ -
(n+1) \biggl[ \mathfrak{F}
+ \frac{n\alpha}{5} \xi^2 - \frac{2n\alpha}{21} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^4 \biggr] \biggl( 1 + a\xi + b\xi^2 + c\xi^3 + d\xi^4 \biggr)</math>
</td>
</tr>
</table>
</div>

Expressions for the various coefficients can now be determined by equating terms on the LHS and RHS that have like powers of <math>~\xi</math>.
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
<td align="center">Term</td>
<td align="center">LHS</td>
<td align="center">RHS</td>
<td align="center">Implication</td>
</tr>

<tr>
<td align="right">
<math>~\xi^{-1}:</math>
</td>
<td align="center">
<math>~24a</math>
</td>
<td align="center">
<math>~0</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~a=0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^{0}:</math>
</td>
<td align="center">
<math>~(12b + 48b)</math>
</td>
<td align="center">
<math>~-(n+1)\mathfrak{F}</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~b = - \frac{(n+1)\mathfrak{F}}{60}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^{1}:</math>
</td>
<td align="center">
<math>~[36c+72c-2a(n+3)]</math>
</td>
<td align="center">
<math>~-a(n+1)\mathfrak{F}</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~108c = 2a(n+3)-a(n+1)\mathfrak{F} \Rightarrow~~c=0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\xi^{2}:</math>
</td>
<td align="center">
<math>~[72d-2b+96d-4b(n+3)]</math>
</td>
<td align="center">
<math>~\biggl[-b(n+1)\mathfrak{F}-\frac{n(n+1)\alpha}{5}\biggr]</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~d = - (n+1)\biggl\{ \frac{n\alpha +\mathfrak{F}[(4n+14)-(n+1)\mathfrak{F} ]}{10080} \biggr\}</math>
</td>
</tr>
</table>
</div>

In summary, the desired, approximate power-series expression for the polytropic displacement function is:
<div align="center" id="PolytropicDisplacement">
<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x(\xi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 - \frac{(n+1)\mathfrak{F}}{60} \xi^2- (n+1)\biggl\{ \frac{n\alpha +\mathfrak{F}[(4n+14)-(n+1)\mathfrak{F} ]}{10080} \biggr\} \xi^4 + \cdots
</math>
</td>
</tr>
</table>
</td></tr></table>
</div>

===Displacement Function for Isothermal LAWE===
The [[SSC/Stability/Isothermal#Taff_and_Van_Horn_.281974.29|LAWE for isothermal spheres]] may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2 x}{dr^2} + \biggl[4 - r \biggl(\frac{dw }{dr}\biggr) \biggr] \frac{1}{r}\frac{dx}{dr}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \frac{\sigma_c^2}{6\gamma} - \frac{\alpha}{r} \biggl(\frac{dw }{dr}\biggr)\biggr] x \, ,
</math>
</td>
</tr>
</table>
</div>
where, <math>~w(r)</math> is the isothermal Lane-Emden function describing the configuration's unperturbed radial density distribution, and <math>~\gamma</math>, <math>~\sigma_c^2</math>, and <math>~\alpha \equiv (3-4/\gamma)</math> are constants. Here we seek a power-series expression for the displacement function, <math>~x(r)</math>, expanded about the center of the configuration, that approximately satisfies this LAWE.

First we note that, near the center, an accurate [[#Isothermal_Lane-Emden_Function|power-series expression for the isothermal Lane-Emden function]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~w(r)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{r^2}{6} - \frac{r^4}{120} + \frac{r^6}{1890} - \frac{61 r^8}{1,632,960} + \cdots \, .</math>
</td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{dw}{dr}</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\frac{r}{3} - \frac{r^3}{30} + \frac{r^5}{315} \, .</math>
</td>
</tr>
</table>
</div>
Therefore, near the center of the configuration, the LAWE may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2 x}{dr^2} + \biggl[4 - \biggl(\frac{r^2}{3} - \frac{r^4}{30} + \frac{r^6}{315}\biggr) \biggr] \frac{1}{r}\frac{dx}{dr}</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
- \frac{1}{6} \biggl[ \frac{\sigma_c^2}{\gamma} - 2\alpha \biggl(1 - \frac{r^2}{10} + \frac{r^4}{105}\biggr) \biggr] x \, .
</math>
</td>
</tr>
</table>
</div>

Let's now adopt a power-series expression for the displacement function of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 + ar + br^2 + cr^3 + dr^4 + \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{1}{r}\frac{dx}{dr}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{a}{r} + 2b + 3 cr + 4dr^2 + \cdots
</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2x}{dr^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2b + 6cr + 12dr^2 + \cdots
</math>
</td>
</tr>
</table>
</div>
Substituting these expressions into the LAWE gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~2b + 6cr + 12dr^2 + \biggl[4 - \biggl(\frac{r^2}{3} - \frac{r^4}{30} + \frac{r^6}{315}\biggr) \biggr] \biggl[ \frac{a}{r} + 2b + 3 cr + 4dr^2 \biggr] </math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
- \frac{1}{6} \biggl[ \frac{\sigma_c^2}{\gamma} - 2\alpha \biggl(1 - \frac{r^2}{10} + \frac{r^4}{105}\biggr) \biggr] \biggl( 1 + ar + br^2 + cr^3 + dr^4 \biggr) \, .
</math>
</td>
</tr>
</table>
</div>
Keeping terms only up through <math>~r^2</math> leads to the following simplification:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
2b + 6cr + 12dr^2
+ 4 \biggl[ \frac{a}{r} + 2b + 3 cr + 4dr^2 \biggr]
- \frac{r^2}{3} \biggl[ \frac{a}{r} + 2b \biggr]
</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~
- \frac{\mathfrak{F} }{6} \biggl( 1 + ar + br^2 \biggr)
- \frac{\alpha}{3} \biggl(\frac{r^2}{10} \biggr)
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\mathfrak{F} \equiv \frac{\sigma_c^2}{\gamma} - 2\alpha \, .</math>
</div>

Finally, balancing terms of like powers on both sides of the equation leads us to conclude the following:

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
<td align="center">Term</td>
<td align="center">LHS</td>
<td align="center">RHS</td>
<td align="center">Implication</td>
</tr>

<tr>
<td align="right">
<math>~r^{-1}:</math>
</td>
<td align="center">
<math>~4a</math>
</td>
<td align="center">
<math>~0</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~a = 0 </math>
</td>
</tr>

<tr>
<td align="right">
<math>~r^{0}:</math>
</td>
<td align="center">
<math>~2b + 8b</math>
</td>
<td align="center">
<math>~- \frac{\mathfrak{F}}{6}</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~b = - \frac{\mathfrak{F}}{60}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r^{1}:</math>
</td>
<td align="center">
<math>~6c + 12c - \frac{a}{3}</math>
</td>
<td align="center">
<math>~-\frac{a\mathfrak{F}}{6}</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~c=0</math>
</td>
</tr>

<tr>
<td align="right">
<math>~r^{2}:</math>
</td>
<td align="center">
<math>~12d + 16d - \frac{2b}{3}</math>
</td>
<td align="center">
<math>~-\frac{\mathfrak{F}b}{6} - \frac{\alpha}{30}</math>
</td>
<td align="left">
<math>~\Rightarrow ~~~
28d = \frac{1}{30}\biggl[ 5b (4- \mathfrak{F} ) - \alpha \biggr] ~
\Rightarrow~
d = \frac{1}{10080}\biggl[ \mathfrak{F}(\mathfrak{F} -4) - 12\alpha \biggr]
</math>
</td>
</tr>
</table>
</div>

In summary, the desired, approximate power-series expression for the isothermal displacement function is:
<div align="center" id="IsothermalDisplacement">
<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
1 - \frac{\mathfrak{F}}{60} r^2 + \frac{1}{10080}\biggl[ \mathfrak{F}(\mathfrak{F} -4) - 12\alpha \biggr] r^4 + \cdots
</math>
</td>
</tr>
</table>
</td></tr></table>
</div>

===Maclaurin Spheroid Index Symbols===
In our accompanying discussion of the equilibrium properties of models along the [[Apps/MaclaurinSpheroidSequence#Equilibrium_Angular_Velocity|Maclaurin spheroid sequence]], we find the "Index Symbols" expressions,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>A_1</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr](1-e^2)^{1/2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>A_3</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{2}{e^2} \biggl[(1-e^2)^{-1/2} -\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2} \, ,
</math>
</td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>e</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>\biggl[1 - \biggl(\frac{c}{a}\biggr)^2\biggr]^{1 / 2}</math>
      (always positive).
</td>
</tr>
</table>

Our aim, here, is to derive a power-series expression for these two index symbols (a) in the case of nearly spherical configurations <math>(e \ll 1)</math>, and (b) in the case of an infinitesimally thin disk <math>(c/a \ll 1)</math>.

====Nearly Spherical Configurations====

On p. 457 of [<b>[[Appendix/References#CRC|<font color="red">CRC</font>]]</b>], we find that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\sin^{-1}e}{e}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + \biggl[\frac{1}{2\cdot 3}\biggr]e^2
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]e^4
+ \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]e^6
+ \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]e^8
+ \cdots
</math>
      for,      
<math>
\biggl(e^2 < 1, -\tfrac{\pi}{2} < \sin^{-1}e < \tfrac{\pi}{2}\biggr) \, .
</math>
</td>
</tr>
</table>
Also, from the [[#Binomial|above binomial-theorem expression]], we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>(1 - e^2)^{1 / 2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 - \tfrac{1}{2}e^2 + \biggl[\frac{\tfrac{1}{2}(\tfrac{1}{2}-1)}{2!}\biggr]e^4
- \biggl[\frac{\tfrac{1}{2}(\tfrac{1}{2}-1)(\tfrac{1}{2}-2)}{3!}\biggr]e^6
+ \biggl[\frac{\tfrac{1}{2}(\tfrac{1}{2}-1)(\tfrac{1}{2}-2)(\tfrac{1}{2}-3)}{4!}\biggr]e^8
- \cdots
</math>
     for <math>~(e^4 < 1)</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 - \biggl[ \frac{1}{2} \biggr]e^2 - \biggl[\frac{1}{2^3}\biggr]e^4
- \biggl[\frac{1}{2^4}\biggr]e^6
- \biggl[\frac{5}{2^7 }\biggr]e^8
- \cdots
</math>
</td>
</tr>
</table>
So we can write,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathcal{F}_1 \equiv (1 - e^2)^{1 / 2}\biggl[ \frac{\sin^{-1}e}{e}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
1 - \biggl[ \frac{1}{2} \biggr]e^2 - \biggl[\frac{1}{2^3}\biggr]e^4
- \biggl[\frac{1}{2^4}\biggr]e^6
- \biggl[\frac{5}{2^7 }\biggr]e^8
- \cdots
\biggr\}
\times ~ \biggl\{
1 + \biggl[\frac{1}{2\cdot 3}\biggr]e^2
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]e^4
+ \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]e^6
+ \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]e^8
+ \cdots
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
1 + \biggl[\frac{1}{2\cdot 3}\biggr]e^2
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]e^4
+ \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]e^6
+ \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]e^8
\biggr\}
-\frac{e^2}{2}
\biggl\{
1 + \biggl[\frac{1}{2\cdot 3}\biggr]e^2
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]e^4
+ \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]e^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \frac{e^4}{2^3}\biggl\{
1 + \biggl[\frac{1}{2\cdot 3}\biggr]e^2
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]e^4
\biggr\}
- \frac{e^6}{2^4}\biggl\{
1 + \biggl[\frac{1}{2\cdot 3}\biggr]e^2
\biggr\}
- \biggl[\frac{5}{2^7}\biggr]e^8
+ \mathcal{O}(e^{10})
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + e^2 \biggl[ \frac{1}{2\cdot 3} - \frac{1}{2} \biggr]
+ e^4 \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}-\frac{1}{2^2\cdot 3} - \frac{1}{2^3} \biggr]
+ e^6 \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7} - \frac{1}{2}\cdot \frac{1\cdot 3 }{2\cdot 4\cdot 5}
- \frac{1}{2^3} \cdot \frac{1}{2\cdot 3} - \frac{1}{2^4}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ e^8 \biggl[\frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9} - \frac{1}{2} \cdot \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}
-\frac{1}{2^3} \cdot \frac{1\cdot 3 }{2\cdot 4\cdot 5} - \frac{1}{2^4} \cdot \frac{1}{2\cdot 3} - \frac{5}{2^7}\biggr]
+ \mathcal{O}(e^{10})
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 - e^2 \biggl[ \frac{1}{3} \biggr]
+ e^4 \biggl[\frac{3^2 - 2\cdot 5 - 3\cdot 5}{2^3 \cdot 3 \cdot 5}\biggr]
+ e^6 \biggl[ \frac{ 3\cdot 5^2 - 3^2\cdot 7 - 5\cdot 7 - 3\cdot 5\cdot 7}{2^4 \cdot 3 \cdot 5 \cdot 7}\biggr]
+ e^8 \biggl[\frac{ 5^2 \cdot 7^2 - 2^2 \cdot 3^2\cdot 5^2 - 2\cdot 3^3 \cdot 7 - 2^2\cdot 3\cdot 5\cdot 7
- 3^2 \cdot 5^2 \cdot 7 }{2^7 \cdot 3^2 \cdot 5 \cdot 7} \biggr]
+ \mathcal{O}(e^{10})
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 - e^2 \biggl[ \frac{1}{3} \biggr]
- e^4 \biggl[\frac{2}{3 \cdot 5}\biggr]
- e^6 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]
- e^8 \biggl[\frac{ 2^4 }{3^2 \cdot 5 \cdot 7} \biggr]
+ \mathcal{O}(e^{10})
</math>
</td>
</tr>
</table>

Hence,

<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>e^2 A_1</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\mathcal{F}_1 - (1-e^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
e^2 \biggl[ \frac{2}{3} \biggr]
- e^4 \biggl[\frac{2}{3 \cdot 5}\biggr]
- e^6 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]
- e^8 \biggl[\frac{ 2^4 }{3^2 \cdot 5 \cdot 7} \biggr]
+ \mathcal{O}(e^{10})
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~ A_1</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{2}{3}
- e^2 \biggl[\frac{2}{3 \cdot 5}\biggr]
- e^4 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]
- e^6 \biggl[\frac{ 2^4 }{3^2 \cdot 5 \cdot 7} \biggr]
+ \mathcal{O}(e^{8}) \, .
</math>
</td>
</tr>
</table>

And,
<table align="center" border=0 cellpadding="3">

<tr>
<td align="right">
<math>\biggl(\frac{e^2}{2}\biggr) A_3</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 - \mathcal{F}_1
</math>
</td>
</tr>

<tr>
<td align="right">
 

</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
e^2 \biggl[ \frac{1}{3} \biggr]
+ e^4 \biggl[\frac{2}{3 \cdot 5}\biggr]
+ e^6 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]
+ e^8 \biggl[\frac{ 2^4 }{3^2 \cdot 5 \cdot 7} \biggr]
+ \mathcal{O}(e^{10})
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow~~~ A_3</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{2}{3}
+ e^2 \biggl[\frac{2^2}{3 \cdot 5}\biggr]
+ e^4 \biggl[ \frac{ 2^4}{3 \cdot 5 \cdot 7}\biggr]
+ e^6 \biggl[\frac{ 2^5 }{3^2 \cdot 5 \cdot 7} \biggr]
+ \mathcal{O}(e^{8}) \, .
</math>
</td>
</tr>
</table>

This looks okay, in the sense that <math>(2A_1 + A_3) = 2</math>.

====Infinitesimally Thin Axisymmetric Disk====
As <math>e \rightarrow 1</math> — that is, in the case of an infinitesimally thin, axisymmetric disk — the preferred small parameter is,

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\frac{c}{a}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 - e^2)^{1 / 2} \ll 1 \, .
</math>
</td>
</tr>
</table>

Recognizing as well that,

<table align="center" border="0" cellpadding="5">

<tr>
<td align="right">
<math>
\sin^{-1} e
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\cos^{-1}(1 - e^2)^{1 / 2} = \cos^{-1}\biggl(\frac{c}{a}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
\Rightarrow ~~~ \mathcal{F}_2 \equiv \frac{\sin^{-1} e}{e^3} \cdot (1 - e^2)^{1 / 2}
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{c}{a}\biggr)\biggl[\cos^{-1}\biggl(\frac{c}{a}\biggr)\biggr]\biggl[ 1 - \frac{c^2}{a^2}\biggr]^{-3 / 2} \, ,
</math>
</td>
</tr>
</table>

the expressions for the pair of relevant index symbols may be rewritten as,
<table align="center" border=0 cellpadding="3">
<tr>
<td align="right">
<math>A_1</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\mathcal{F}_2 - \biggl(\frac{c}{a}\biggr)^2 \biggl(1 - \frac{c^2}{a^2}\biggr)^{-1} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{A_3}{2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(1 - \frac{c^2}{a^2}\biggr)^{-1} - \mathcal{F}_2 \, .
</math>
</td>
</tr>
</table>

Pulling again from p. 457 of [<b>[[Appendix/References#CRC|<font color="red">CRC</font>]]</b>], we find that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\cos^{-1}\biggl(\frac{c}{a}\biggr)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} - \frac{c}{a}\biggl\{1 + \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^2
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
\biggr\}
</math>
      for,      
<math>
\biggl(\frac{c^2}{a^2} < 1, 0 < \cos^{-1}\biggl(\frac{c}{a}\biggr) < \pi\biggr)\, .
</math>
</td>
</tr>
</table>

<table border="1" align="center" width="90%" cellpadding="8"><tr><td align="left">
<div align="center"><b>LAGNIAPPE:</b></div>

According to the [[#Binomial|above binomial-theorem expression]], we find for <math>(c^2/a^2 < 1)</math>,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{1}{e} = \biggl[ 1 - \frac{c^2}{a^2} \biggr]^{-1 / 2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + \frac{1}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{-\tfrac{1}{2}(-\tfrac{1}{2}-1)}{2!}\biggr]\biggl(\frac{c}{a}\biggr)^4
- \biggl[\frac{-\tfrac{1}{2}(-\tfrac{1}{2}-1)(-\tfrac{1}{2}-2)}{3!}\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{-\tfrac{1}{2}(-\tfrac{1}{2}-1)(-\tfrac{1}{2}-2)(-\tfrac{1}{2}-3)}{4!}\biggr]\biggl(\frac{c}{a}\biggr)^8
- \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + \frac{1}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{3}{2^2 \cdot 2!}\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[\frac{3\cdot 5}{2^3 \cdot 3!}\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{3\cdot 5\cdot 7}{2^4 \cdot 4!}\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + \frac{1}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{3}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[\frac{5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
</math>
</td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\sin^{-1}e}{e} = \cos^{-1}\biggl(\frac{c}{a}\biggr) \biggl[ 1 - \frac{c^2}{a^2} \biggr]^{-1 / 2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2}\biggl\{
1 + \frac{1}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{3}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[\frac{5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- ~\frac{c}{a}\biggl\{1 + \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^2
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
\times ~\biggl\{
1 + \frac{1}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{3}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[\frac{5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2}\biggl\{
1 + \biggl[\frac{1}{2}\biggr] \biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{3}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[\frac{5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^8
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- ~\biggl\{\biggl(\frac{c}{a}\biggr) + \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^3
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^5
+ \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^7
+ \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]\biggl(\frac{c}{a}\biggr)^9
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- ~\frac{1}{2} \biggl\{\biggl(\frac{c}{a}\biggr)^3 + \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^5
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^7
+ \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^9
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- ~\biggl[\frac{3}{2^3 }\biggr] \biggl\{\biggl(\frac{c}{a}\biggr)^5 + \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^7
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^9
\biggr\}
- ~\biggl[\frac{5}{2^4 }\biggr] \biggl\{\biggl(\frac{c}{a}\biggr)^7 + \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^9
\biggr\}
- ~\biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^9
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \frac{\pi}{2}\biggl[\frac{1}{2}\biggr] \biggl(\frac{c}{a}\biggr)^2 + \frac{\pi}{2}\biggl[\frac{3}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \frac{\pi}{2}\biggl[\frac{5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \frac{\pi}{2}\biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^8
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \biggl(\frac{c}{a}\biggr) - \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^3
- \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^7
- \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]\biggl(\frac{c}{a}\biggr)^9
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \frac{1}{2} \biggl(\frac{c}{a}\biggr)^3 - \biggl[\frac{1}{2^2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[\frac{1\cdot 3 }{2^2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^7
- \biggl[ \frac{1 \cdot 3\cdot 5}{2^2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^9
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \biggl[\frac{3}{2^3 }\biggr] \biggl(\frac{c}{a}\biggr)^5 - \biggl[\frac{3}{2^3 }\biggr] \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^7
- \biggl[\frac{3}{2^3 }\biggr] \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^9

- \biggl[\frac{5}{2^4 }\biggr] \biggl(\frac{c}{a}\biggr)^7 - \biggl[\frac{5}{2^4 }\biggr] \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^9

- \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^9
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr) \, .
</math>
</td>
</tr>
</table>

(continue expression simplification)

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{\sin^{-1}e}{e} = \cos^{-1}\biggl(\frac{c}{a}\biggr) \biggl[ 1 - \frac{c^2}{a^2} \biggr]^{-1 / 2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \frac{\pi}{2}\biggl[\frac{1}{2}\biggr] \biggl(\frac{c}{a}\biggr)^2 + \frac{\pi}{2}\biggl[\frac{3}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \frac{\pi}{2}\biggl[\frac{5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \frac{\pi}{2}\biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^8
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \biggl(\frac{c}{a}\biggr) - \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^3
- \biggl[\frac{3 }{2^3\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[ \frac{5}{2^4\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^7
- \biggl[ \frac{5 \cdot 7}{2^7\cdot 3^2}\biggr]\biggl(\frac{c}{a}\biggr)^9
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \frac{1}{2} \biggl(\frac{c}{a}\biggr)^3 - \biggl[\frac{1}{2^2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[\frac{ 3 }{2^4 \cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^7
- \biggl[ \frac{5}{2^5 \cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^9
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \biggl[\frac{3}{2^3 }\biggr] \biggl(\frac{c}{a}\biggr)^5 - \biggl[\frac{1}{2^4}\biggr]\biggl(\frac{c}{a}\biggr)^7
- \biggl[\frac{3^2 }{2^6\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^9

- \biggl[\frac{5}{2^4 }\biggr] \biggl(\frac{c}{a}\biggr)^7 - \biggl[\frac{5}{2^5\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^9

- \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^9
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \frac{\pi}{2}\biggl[\frac{1}{2}\biggr] \biggl(\frac{c}{a}\biggr)^2 + \frac{\pi}{2}\biggl[\frac{3}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \frac{\pi}{2}\biggl[\frac{5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \frac{\pi}{2}\biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^8
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \biggl(\frac{c}{a}\biggr)
- \biggl\{
\biggl[\frac{1}{2\cdot 3}\biggr]
+ \frac{1}{2}
\biggr\}\biggl(\frac{c}{a}\biggr)^3
- \biggl\{
\biggl[\frac{3 }{2^3\cdot 5}\biggr]
+ \biggl[\frac{1}{2^2\cdot 3}\biggr]
+ \biggl[\frac{3}{2^3 }\biggr]
\biggr\}\biggl(\frac{c}{a}\biggr)^5

- \biggl\{
\biggl[ \frac{5}{2^4\cdot 7}\biggr]
+ \biggl[\frac{ 3 }{2^4 \cdot 5}\biggr]
+ \biggl[\frac{1}{2^4}\biggr]\
+ \biggl[\frac{5}{2^4 }\biggr]
\biggr\}\biggl(\frac{c}{a}\biggr)^7
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \biggl\{
\biggl[ \frac{5 \cdot 7}{2^7\cdot 3^2}\biggr]
+ \biggl[ \frac{5}{2^5 \cdot 7}\biggr]
+ \biggl[\frac{3^2 }{2^6\cdot 5}\biggr]
+ \biggl[\frac{5}{2^5\cdot 3}\biggr]
+ \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]
\biggr\} \biggl(\frac{c}{a}\biggr)^9
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} + \frac{\pi}{2}\biggl[\frac{1}{2}\biggr] \biggl(\frac{c}{a}\biggr)^2 + \frac{\pi}{2}\biggl[\frac{3}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \frac{\pi}{2}\biggl[\frac{5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \frac{\pi}{2}\biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^8
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- \biggl(\frac{c}{a}\biggr)
- \biggl[\frac{2}{3}\biggr] \biggl(\frac{c}{a}\biggr)^3

- \biggl[\frac{2^3 }{3\cdot 5}\biggr] \biggl(\frac{c}{a}\biggr)^5

- \biggl[ \frac{2^4}{5 \cdot 7}\biggr] \biggl(\frac{c}{a}\biggr)^7

- \biggl[ \frac{2^7}{3^2 \cdot 5 \cdot 7}\biggr] \biggl(\frac{c}{a}\biggr)^9
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr) \, .
</math>
</td>
</tr>
</table>

</td></tr></table>

Referring again to the [[#Binomial|above binomial-theorem expression]], we find for <math>(c^2/a^2 < 1)</math>,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ 1 - \frac{c^2}{a^2} \biggr]^{-3 / 2}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + \frac{3}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{-\tfrac{3}{2}(-\tfrac{3}{2}-1)}{2!}\biggr]\biggl(\frac{c}{a}\biggr)^4
- \biggl[\frac{-\tfrac{3}{2}(-\tfrac{3}{2}-1)(-\tfrac{3}{2}-2)}{3!}\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{-\tfrac{3}{2}(-\tfrac{3}{2}-1)(-\tfrac{3}{2}-2)(-\tfrac{3}{2}-3)}{4!}\biggr]\biggl(\frac{c}{a}\biggr)^8
- \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + \frac{3}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{\tfrac{3}{2}(\tfrac{3}{2}+1)}{2!}\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[\frac{\tfrac{3}{2}(\tfrac{3}{2}+1)(\tfrac{3}{2}+2)}{3!}\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{\tfrac{3}{2}(\tfrac{3}{2}+1)(\tfrac{3}{2}+2)(\tfrac{3}{2}+3)}{4!}\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + \frac{3}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{3\cdot 5}{2^2 \cdot 2!}\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[\frac{3\cdot 5 \cdot 7}{2^3\cdot 3!}\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{3\cdot 5 \cdot 7 \cdot 9}{2^4 \cdot 4!}\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1 + \frac{3}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{3\cdot 5}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[\frac{5 \cdot 7}{2^4}\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{5 \cdot 7 \cdot 9}{2^7}\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
</math>
</td>
</tr>
</table>

We therefore can write,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\mathcal{F}_2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
- \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^4
- \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^6
- \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^8
- \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]\biggl(\frac{c}{a}\biggr)^{10}
- \cdots
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
\times \biggl\{
1 + \frac{3}{2}\biggl(\frac{c}{a}\biggr)^2 + \biggl[\frac{3\cdot 5}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
+ \biggl[\frac{5 \cdot 7}{2^4}\biggr]\biggl(\frac{c}{a}\biggr)^6
+ \biggl[\frac{5 \cdot 7 \cdot 9}{2^7}\biggr]\biggl(\frac{c}{a}\biggr)^8
+ \cdots
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
- \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^4
- \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^6
- \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^8
- \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]\biggl(\frac{c}{a}\biggr)^{10}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ \frac{3}{2}\biggl(\frac{c}{a}\biggr)^2
\biggl\{
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
- \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^4
- \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^6
- \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^8
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ \biggl[\frac{3\cdot 5}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^4
\biggl\{
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
- \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^4
- \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^6
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+ \biggl[\frac{5 \cdot 7}{2^4}\biggr]\biggl(\frac{c}{a}\biggr)^6
\biggl\{
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
- \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^4
\biggr\}
+ \biggl[\frac{5 \cdot 7 \cdot 9}{2^7}\biggr]\biggl(\frac{c}{a}\biggr)^8
\biggl\{
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
\biggr\}
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
- \biggl[\frac{1}{2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^4
- \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^6
- \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^8
+
\pi \biggl[\frac{3}{2^2}\biggr] \biggl(\frac{c}{a}\biggr)^3
- \biggl[\frac{3}{2}\biggr]\biggl(\frac{c}{a}\biggr)^4
- \biggl[\frac{1}{2^2}\biggr] \biggl(\frac{c}{a}\biggr)^6
- \biggl[\frac{3^2 }{2^4 \cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^8
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\pi \biggl[\frac{3\cdot 5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[\frac{3\cdot 5}{2^3 }\biggr]\biggl(\frac{c}{a}\biggr)^6
- \biggl[\frac{5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^8
+
\pi\biggl[\frac{5 \cdot 7}{2^5}\biggr] \biggl(\frac{c}{a}\biggr)^7
- \biggl[\frac{5 \cdot 7}{2^4}\biggr]\biggl(\frac{c}{a}\biggr)^8
+
\pi \biggl[\frac{5 \cdot 7 \cdot 9}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
+ \pi \biggl[\frac{3}{2^2}\biggr] \biggl(\frac{c}{a}\biggr)^3
- \biggl[\frac{1}{2\cdot 3}
+ \frac{3}{2}\biggr]\biggl(\frac{c}{a}\biggr)^4
+
\pi \biggl[\frac{3\cdot 5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5} + \frac{1}{2^2} + \frac{3\cdot 5}{2^3 }\biggr] \biggl(\frac{c}{a}\biggr)^6
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
+
\pi\biggl[\frac{5 \cdot 7}{2^5}\biggr] \biggl(\frac{c}{a}\biggr)^7
- \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}
+ \frac{3^2 }{2^4 \cdot 5}
+ \frac{5}{2^4 }
+ \frac{5 \cdot 7}{2^4}\biggr]\biggl(\frac{c}{a}\biggr)^8
+
\pi \biggl[\frac{5 \cdot 7 \cdot 9}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
+ \pi \biggl[\frac{3}{2^2}\biggr] \biggl(\frac{c}{a}\biggr)^3
- \biggl[\frac{5}{3} \biggr]\biggl(\frac{c}{a}\biggr)^4
+
\pi \biggl[\frac{3\cdot 5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[\frac{11}{ 5} \biggr] \biggl(\frac{c}{a}\biggr)^6
+
\pi\biggl[\frac{5 \cdot 7}{2^5}\biggr] \biggl(\frac{c}{a}\biggr)^7
- \biggl[ \frac{ 93}{5 \cdot 7}
\biggr]\biggl(\frac{c}{a}\biggr)^8
+
\pi \biggl[\frac{5 \cdot 7 \cdot 9}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr) \, .
</math>
</td>
</tr>
</table>
Once again from the binomial theorem,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl(1 - \frac{c^2}{a^2} \biggr)^{-1}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>1 + \biggl(\frac{c}{a}\biggr)^2 + \biggl(\frac{c}{a}\biggr)^4 + \biggl(\frac{c}{a}\biggr)^6 + \biggl(\frac{c}{a}\biggr)^8 + \biggl(\frac{c}{a}\biggr)^{10} + \cdots </math>
</td>
</tr>
</table>
which gives us,
<table align="center" border=0 cellpadding="3">

<tr>
<td align="right">
<math>A_1 = \mathcal{F}_2 - \biggl(\frac{c}{a}\biggr)^2 \biggl(1 - \frac{c^2}{a^2}\biggr)^{-1}</math>
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
+ \pi \biggl[\frac{3}{2^2}\biggr] \biggl(\frac{c}{a}\biggr)^3
- \biggl[\frac{5}{3} \biggr]\biggl(\frac{c}{a}\biggr)^4
+
\pi \biggl[\frac{3\cdot 5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[\frac{11}{ 5} \biggr] \biggl(\frac{c}{a}\biggr)^6
+
\pi\biggl[\frac{5 \cdot 7}{2^5}\biggr] \biggl(\frac{c}{a}\biggr)^7
- \biggl[ \frac{ 93}{5 \cdot 7}
\biggr]\biggl(\frac{c}{a}\biggr)^8
+
\pi \biggl[\frac{3^2\cdot 5 \cdot 7}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
- ~\biggl\{
\biggl(\frac{c}{a}\biggr)^2 + \biggl(\frac{c}{a}\biggr)^4 + \biggl(\frac{c}{a}\biggr)^6 + \biggl(\frac{c}{a}\biggr)^8
\biggr\}
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
=
</math>
</td>
<td align="left">
<math>
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- 2\biggl(\frac{c}{a}\biggr)^2
+ \pi \biggl[\frac{3}{2^2}\biggr] \biggl(\frac{c}{a}\biggr)^3
- \biggl[\frac{8}{3} \biggr]\biggl(\frac{c}{a}\biggr)^4
+
\pi \biggl[\frac{3\cdot 5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[\frac{2^4}{ 5} \biggr] \biggl(\frac{c}{a}\biggr)^6
+
\pi\biggl[\frac{5 \cdot 7}{2^5}\biggr] \biggl(\frac{c}{a}\biggr)^7
- \biggl[\frac{ 2^7}{5 \cdot 7}
\biggr]\biggl(\frac{c}{a}\biggr)^8
+
\pi \biggl[\frac{3^2\cdot 5 \cdot 7}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr) \, .
</math>
</td>
</tr>
</table>

And,
<table align="center" border=0 cellpadding="3">

<tr>
<td align="right">
<math>\frac{A_3}{2} = \biggl(1 - \frac{c^2}{a^2}\biggr)^{-1} - \mathcal{F}_2 </math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl\{
1 + \biggl(\frac{c}{a}\biggr)^2 + \biggl(\frac{c}{a}\biggr)^4 + \biggl(\frac{c}{a}\biggr)^6 + \biggl(\frac{c}{a}\biggr)^8 + \biggl(\frac{c}{a}\biggr)^{10} + \cdots
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>
-~ \biggl\{
\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
- \biggl(\frac{c}{a}\biggr)^2
+ \pi \biggl[\frac{3}{2^2}\biggr] \biggl(\frac{c}{a}\biggr)^3
- \biggl[\frac{5}{3} \biggr]\biggl(\frac{c}{a}\biggr)^4
+
\pi \biggl[\frac{3\cdot 5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^5
- \biggl[\frac{11}{ 5} \biggr] \biggl(\frac{c}{a}\biggr)^6
+
\pi\biggl[\frac{5 \cdot 7}{2^5}\biggr] \biggl(\frac{c}{a}\biggr)^7
- \biggl[ \frac{ 93}{5 \cdot 7}
\biggr]\biggl(\frac{c}{a}\biggr)^8
+
\pi \biggl[\frac{3^2\cdot 5 \cdot 7}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
\biggr\}
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
1
-\frac{\pi}{2} \biggl(\frac{c}{a}\biggr)
+ 2\biggl(\frac{c}{a}\biggr)^2
- \pi \biggl[\frac{3}{2^2}\biggr] \biggl(\frac{c}{a}\biggr)^3
+ \biggl[\frac{8}{3} \biggr]\biggl(\frac{c}{a}\biggr)^4
-
\pi \biggl[\frac{3\cdot 5}{2^4 }\biggr]\biggl(\frac{c}{a}\biggr)^5
+ \biggl[\frac{2^4}{ 5} \biggr] \biggl(\frac{c}{a}\biggr)^6
-
\pi\biggl[\frac{5 \cdot 7}{2^5}\biggr] \biggl(\frac{c}{a}\biggr)^7
+ \biggl[ \frac{ 2^7}{5 \cdot 7}
\biggr]\biggl(\frac{c}{a}\biggr)^8
-
\pi \biggl[\frac{3^2\cdot 5 \cdot 7}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
\biggr\}
+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)\, .
</math>
</td>
</tr>
</table>

Notice that, to the highest order retained in these expressions, we find as expected that, <math>(A_1 + A_3/2) = 1</math>.

====Frequency (temporary)====

<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\omega_0^2
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2\pi G \rho \biggl[ A_1 - A_3 (1-e^2) \biggr]
</math>
</td>
</tr>
<tr>
<td align="right">
<math>
\Rightarrow ~~~ \Omega^2_\mathrm{Mc} \equiv \frac{\omega_0^2}{\pi G \rho }
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
2(3-2e^2)(1-e^2)^{1 / 2} \cdot \frac{\sin^{-1}e}{e^3}
- \frac{6(1-e^2)}{e^2}
\, ,
</math>
</td>
</tr>
</table>

==Taylor Series (Hunter77)==

===First (Unsuccessful) Try===

First:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- 3\Delta) f_3^' + \frac{1}{2} (- 3\Delta)^2 f^{''}_3 + \frac{1}{6} (- 3\Delta)^3 f_3^{'''} + \frac{1}{24}(- 3\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 - (3\Delta) f_3^' + \frac{3^2}{2} (\Delta)^2 f^{''}_3 - \frac{3^2}{2} (\Delta)^3 f_3^{'''} + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~
- \frac{3^2}{2} (\Delta)^2 f^{''}_3
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 - f_0 - (3\Delta) f_3^' - \frac{3^2}{2} (\Delta)^3 f_3^{'''} + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

Note that, replacing the <math>~(\Delta)^3 f_3^{'''}</math> term with the expression derived in the ''Second'' step, below, gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
- \frac{3^2}{2} (\Delta)^2 f^{''}_3
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 - f_0 - (3\Delta) f_3^' + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \frac{3^2}{2} \biggl\{
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr]
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
+ \biggl[- \frac{5}{6} \biggr] (\Delta)^4 f_3^{iv}
\biggr\}\biggl[ -\frac{3}{2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 - f_0 - 3 (\Delta) f_3^' + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
3 f_0
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{15}{2^2} \biggr] f_3
+ \biggl[- \frac{3}{2}\biggr] (\Delta) f_3^'
+ \biggl[- \frac{3^2\cdot 5}{2^3} \biggr] (\Delta)^4 f_3^{iv}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2f_0
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[1 + \frac{15}{2^2} \biggr] f_3 + \biggl[-3 - \frac{3}{2}\biggr] (\Delta) f_3^' + \biggl[\frac{3^3}{2^3}- \frac{3^2\cdot 5}{2^3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2f_0
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{19}{2^2} \biggr] f_3 + \biggl[- \frac{9}{2}\biggr] (\Delta) f_3^' + \biggl[- \frac{9}{4} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

Then, replacing the <math>~(\Delta)^4 f_3^{iv}</math> term with the expression derived in the ''Third'' step, below, gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
- \frac{3^2}{2} (\Delta)^2 f^{''}_3
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2f_0
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{19}{2^2} \biggr] f_3 + \biggl[- \frac{9}{2}\biggr] (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[- \frac{9}{4} \biggr] \biggl\{
\biggl[-\frac{1}{3^2} \biggr]f_0
+ \biggl[\frac{1}{2} \biggr] f_1
- f_2
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
\biggr\}\biggl[- 2^2\cdot 3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2f_0
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{19}{2^2} \biggr] f_3 + \biggl[- \frac{9}{2}\biggr] (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[-3 \biggr]f_0
+ \biggl[\frac{3^3 }{2} \biggr] f_1
- 3^3 f_2
+ \biggl[ \frac{3 \cdot 11}{2} \biggr] f_3
+ \biggl[- 2\cdot 3^2 \biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- f_0
+ \biggl[\frac{3^3 }{2} - \frac{3^3}{2^2}\biggr] f_1
- 3^3 f_2
+ \biggl[\frac{3 \cdot 11}{2} + \frac{19}{2^2} \biggr] f_3 + \biggl[- 2\cdot 3^2- \frac{9}{2}\biggr] (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- f_0
+ \biggl[\frac{3^3}{2^2}\biggr] f_1
- 3^3 f_2
+ \biggl[\frac{5\cdot 17}{2^2} \biggr] f_3 + \biggl[- \frac{3^2\cdot 5}{2}\biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

Second:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- 2\Delta) f_3^' + \frac{1}{2} (- 2\Delta)^2 f^{''}_3 + \frac{1}{6} (- 2\Delta)^3 f_3^{'''} + \frac{1}{24}(- 2\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 - 2(\Delta) f_3^' + 2 (\Delta)^2 f^{''}_3 - \frac{2^2}{3} (\Delta)^3 f_3^{'''} + \frac{2}{3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 - 2(\Delta) f_3^'
- \frac{2^2}{3} (\Delta)^3 f_3^{'''} + \frac{2}{3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \biggl[\frac{2^2}{3^2} \biggr] \biggl[ f_3 - f_0 - (3\Delta) f_3^' - \frac{3^2}{2} (\Delta)^3 f_3^{'''} + \frac{3^3}{2^3}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
+ f_3 \biggl[1-\frac{2^2}{3^2} \biggr]
+ \biggl[ \frac{2^2}{3^2} (3\Delta) - 2(\Delta) \biggr] f_3^'
+ \biggl[ \biggl(\frac{2^2}{3^2} \biggr) \frac{3^2}{2} (\Delta)^3 - \frac{2^2}{3} (\Delta)^3 \biggr] f_3^{'''}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ \frac{2}{3}(\Delta)^4 - \biggl( \frac{2^2}{3^2} \biggr) \frac{3^3}{2^3}(\Delta)^4 \biggr] f_3^{iv}
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
+ f_3 \biggl[\frac{5}{3^2} \biggr]
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
+ \biggl[ \frac{2}{3} \biggr] (\Delta)^3f_3^{'''}
+ \biggl[- \frac{5}{6} \biggr] (\Delta)^4 f_3^{iv}
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~
- \biggl[ \frac{2}{3} \biggr] (\Delta)^3f_3^{'''}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr]
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
+ \biggl[- \frac{5}{6} \biggr] (\Delta)^4 f_3^{iv}
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

Now, replacing the <math>~(\Delta)^4 f_3^{iv}</math> term with the expression derived in the ''Third'' step, below, gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
- \biggl[ \frac{2}{3} \biggr] (\Delta)^3f_3^{'''}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr]
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[- \frac{5}{6} \biggr] \biggl\{
\biggl[-\frac{1}{3^2} \biggr]f_0
+ \biggl[\frac{1}{2} \biggr] f_1
- f_2
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
\biggr\} \biggl[ -2^2\cdot 3\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr]
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[-\frac{2\cdot 5 }{3^2} \biggr]f_0
+ \biggl[5\biggr] f_1
+ \biggl[- 2\cdot 5 \biggr] f_2
+ \biggl[ \frac{5\cdot 11}{3^2} \biggr] f_3
+ \biggl[- \frac{2^2\cdot 5}{3}\biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{2^2}{3^2} -\frac{2\cdot 5 }{3^2} \biggr] f_0
+ \biggl[4\biggr] f_1
+ \biggl[- 2\cdot 5 \biggr] f_2
+ \biggl[\frac{5}{3^2} + \frac{5\cdot 11}{3^2}\biggr] f_3
+ \biggl[- \frac{2^2\cdot 5}{3} - \frac{2}{3}\biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[-\frac{2}{3} \biggr] f_0
+ \biggl[4\biggr] f_1
+ \biggl[- 2\cdot 5 \biggr] f_2
+ \biggl[\frac{2^2\cdot 5}{3}\biggr] f_3
+ \biggl[- \frac{2\cdot 11}{3}\biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

Third:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- \Delta) f_3^' + \frac{1}{2} (- \Delta)^2 f^{''}_3 + \frac{1}{6} (- \Delta)^3 f_3^{'''} + \frac{1}{24}(- \Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + \biggl[ -1 \biggr](\Delta) f_3^' + \biggl[ \frac{1}{2} \biggr] (\Delta)^2 f^{''}_3 + \biggl[ - \frac{1}{2\cdot 3} \biggr] (\Delta)^3 f_3^{'''} + \biggl[ \frac{1}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + \biggl[ -1 \biggr](\Delta) f_3^' + \biggl[ \frac{1}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ \frac{1}{2} \biggr] \biggl\{
2f_0
+ \biggl[- \frac{3^3}{2^2}\biggr] f_1
+ \biggl[\frac{19}{2^2} \biggr] f_3 + \biggl[- \frac{9}{2}\biggr] (\Delta) f_3^' + \biggl[- \frac{9}{4} \biggr] (\Delta)^4 f_3^{iv}
\biggr\} \biggl[-\frac{2}{3^2}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ - \frac{1}{2\cdot 3} \biggr] \biggl\{
\biggl[\frac{2^2}{3^2} \biggr] f_0
- f_1
+ f_3 \biggl[\frac{5}{3^2} \biggr]
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
+ \biggl[- \frac{5}{6} \biggr] (\Delta)^4 f_3^{iv}
\biggr\} \biggl[-\frac{3}{2}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + \biggl[ -1 \biggr](\Delta) f_3^' + \biggl[ \frac{1}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[ -\frac{2}{3^2} \biggr]f_0
+ \biggl[\frac{3}{2^2}\biggr] f_1
+ \biggl[-\frac{19}{2^2\cdot 3^2} \biggr] f_3 + \biggl[\frac{1}{2}\biggr] (\Delta) f_3^' + \biggl[\frac{1}{4} \biggr] (\Delta)^4 f_3^{iv}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{3^2} \biggr] f_0
+ \biggl[- \frac{1}{2^2} \biggr] f_1
+ f_3 \biggl[\frac{5}{2^2\cdot 3^2} \biggr]
+ \biggl[- \frac{1}{2\cdot 3}\biggr] (\Delta) f_3^'
+ \biggl[- \frac{5}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + \biggl[ -1 \biggr](\Delta) f_3^' + \biggl[ \frac{1}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{3^2} -\frac{2}{3^2} \biggr]f_0
+ \biggl[\frac{3}{2^2}- \frac{1}{2^2} \biggr] f_1
+ \biggl[\frac{5}{2^2\cdot 3^2} -\frac{19}{2^2\cdot 3^2} \biggr] f_3 + \biggl[\frac{1}{2}- \frac{1}{2\cdot 3}\biggr] (\Delta) f_3^'
+ \biggl[\frac{1}{4} - \frac{5}{2^3\cdot 3} \biggr] (\Delta)^4 f_3^{iv}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[-\frac{1}{3^2} \biggr]f_0
+ \biggl[\frac{1}{2} \biggr] f_1
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
+ \biggl[\frac{1}{2^2\cdot 3} \biggr] (\Delta)^4 f_3^{iv}
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~
- \biggl[\frac{1}{2^2\cdot 3} \biggr] (\Delta)^4 f_3^{iv}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[-\frac{1}{3^2} \biggr]f_0
+ \biggl[\frac{1}{2} \biggr] f_1
- f_2
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

And, finally:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_4</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' + \frac{1}{2} ( \Delta)^2 f^{''}_3 + \frac{1}{6} (\Delta)^3 f_3^{'''} + \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{2} \biggl\{
- f_0
+ \biggl[\frac{3^3}{2^2}\biggr] f_1
- 3^3 f_2
+ \biggl[\frac{5\cdot 17}{2^2} \biggr] f_3 + \biggl[- \frac{3^2\cdot 5}{2}\biggr] (\Delta) f_3^'
\biggr\} \biggl[ - \frac{2}{3^2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{6} \biggl\{
\biggl[-\frac{2}{3} \biggr] f_0
+ \biggl[4\biggr] f_1
+ \biggl[- 2\cdot 5 \biggr] f_2
+ \biggl[\frac{2^2\cdot 5}{3}\biggr] f_3
+ \biggl[- \frac{2\cdot 11}{3}\biggr] (\Delta) f_3^'
\biggr\} \biggl[ -\frac{3}{2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{24}\biggl\{
\biggl[-\frac{1}{3^2} \biggr]f_0
+ \biggl[\frac{1}{2} \biggr] f_1
- f_2
+ \biggl[ \frac{11}{2\cdot 3^2} \biggr] f_3
+ \biggl[- \frac{2}{3}\biggr] (\Delta) f_3^'
\biggr\} \biggl[ -2^2\cdot 3 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[ \frac{1}{3^2} \biggr] f_0
+ \biggl[- \frac{3}{2^2}\biggr] f_1
+\biggl[ 3 \biggr] f_2
+ \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3
+ \biggl[\frac{5}{2}\biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{2\cdot 3} \biggr] f_0
+ \biggl[-1 \biggr] f_1
+ \biggl[ \frac{5}{2} \biggr] f_2
+ \biggl[- \frac{5}{3}\biggr] f_3
+ \biggl[\frac{11}{2\cdot 3}\biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{2\cdot 3^2} \biggr]f_0
+ \biggl[- \frac{1}{2^2} \biggr] f_1
+ \biggl[ \frac{1}{2} \biggr] f_2
+ \biggl[ -\frac{11}{2^2 \cdot 3^2} \biggr] f_3
+ \biggl[\frac{1}{3}\biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{1}{3^2} + \frac{1}{2\cdot 3} + \frac{1}{2\cdot 3^2} \biggr] f_0 +
\biggl[- \frac{3}{2^2} - 1 - \frac{1}{2^2} \biggr] f_1
+\biggl[ 3 + \frac{5}{2} + \frac{1}{2} \biggr] f_2
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[1 - \frac{5\cdot 17}{2^2\cdot 3^2} - \frac{5}{3} - \frac{11}{2^2 \cdot 3^2} \biggr] f_3
+ \biggl[1 + \frac{5}{2} + \frac{11}{2\cdot 3} + \frac{1}{3} \biggr] (\Delta) f_3^'
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{1}{3} \biggr] f_0 +
\biggl[- 2\biggr] f_1
+\biggl[ 6 \biggr] f_2
+ \biggl[- \frac{10}{3} \biggr] f_3
+ \biggl[\frac{17}{3} \biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

Result:

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
<th align="center">
Definitely WRONG!
</th>
</tr>
<tr><td>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_4</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{3} ~ f_0 - 2 f_1 + 6 f_2
- \frac{10}{3} ~ f_3
+ \frac{17}{3} (\Delta) f_3^'
+ \mathcal{O}(\Delta^5) \, .
</math>
</td>
</tr>
</table>
</td></tr>
</table>
</div>

When I used an Excel spreadsheet to test this out against a parabola, the integration quickly became wildly unstable, strongly suggesting that there is an error in the derivation. My first attempt to uncover this error produced a new coefficient on the <math>~(\Delta) f_3^'</math>, namely,

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
<th align="center">
Somewhat Improved
</th>
</tr>
<tr><td>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_4</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{3} ~ f_0 - 2 f_1 + 6 f_2
- \frac{10}{3} ~ f_3
+ 4 (\Delta) f_3^'
+ \mathcal{O}(\Delta^5) \, .
</math>
</td>
</tr>
</table>
</td></tr>
</table>
</div>
Although it showed improvement, this expression still blows up. So I have not bothered to revise the original (definitely WRONG!) derivation. Instead, let's start all over and approach it with a more gradual derivation.

===Second Try===

We will work from the following foundation expression in which <math>~f_4</math> is the variable that we desire to evaluate, and the "known" quantities are:   <math>~f_3</math>, <math>~f_3^'</math>, <math>~f_2</math>, <math>~f_1</math>, and <math>~f_0</math>.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_4</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' + \frac{1}{2} ( \Delta)^2 f^{''}_3 + \frac{1}{6} (\Delta)^3 f_3^{'''} + \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

Let's use similar Taylor-series expansions for <math>~f_2</math>, <math>~f_3</math>, etc. in order to eliminate the <math>~f_3^{''}</math> term, the <math>~f_3^{'''}</math> term, etc.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- \Delta) f_3^' + \frac{1}{2} (- \Delta)^2 f^{''}_3 + \frac{1}{6} (- \Delta)^3 f_3^{'''} + \frac{1}{24}(- \Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~f_1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- 2\Delta) f_3^' + \frac{1}{2} (- 2\Delta)^2 f^{''}_3 + \frac{1}{6} (- 2\Delta)^3 f_3^{'''} + \frac{1}{24}(- 2\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~f_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- 3\Delta) f_3^' + \frac{1}{2} (- 3\Delta)^2 f^{''}_3 + \frac{1}{6} (- 3\Delta)^3 f_3^{'''} + \frac{1}{24}(- 3\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

----

First:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~-\frac{1}{2} (- \Delta)^2 f^{''}_3 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- \Delta) f_3^' - f_2+ \frac{1}{6} (- \Delta)^3 f_3^{'''} + \frac{1}{24}(- \Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~
\frac{1}{2} (\Delta)^2 f^{''}_3
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- f_3 + (\Delta) f_3^' + f_2+ \frac{1}{6} (\Delta)^3 f_3^{'''} - \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~
f_4
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' - f_3 + (\Delta) f_3^' + f_2 + \mathcal{O}(\Delta^3)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_2 + 2(\Delta) f_3^' + \mathcal{O}(\Delta^3)
</math>
</td>
</tr>
</table>
</div>

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center"><math>~\mathcal{O}(\Delta^3)</math></td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
f_4
</math>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_2 + 2(\Delta) f_3^' + \mathcal{O}(\Delta^3)
</math>
</td>
</tr>
</table>
</td></tr></table>
</div>

This expression works very well for a parabola.

----

Second:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- 2) \Delta f_3^' + 2 (\Delta)^2 f^{''}_3 + \biggl[- \frac{2^3}{6}\biggr] \Delta^3 f_3^{'''} + \biggl[ \frac{2^4}{2^3\cdot 3} \biggr] \Delta^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- 2) \Delta f_3^' + \biggl[- \frac{2^3}{6}\biggr] \Delta^3 f_3^{'''} + \biggl[ \frac{2^4}{2^3\cdot 3} \biggr] \Delta^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 2 \biggl\{
- f_3 + (\Delta) f_3^' + f_2+ \frac{1}{2\cdot 3} (\Delta)^3 f_3^{'''} - \frac{1}{2^3\cdot 3}(\Delta)^4 f_3^{iv}
\biggr\} \biggl[ 2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3\biggl[ 1 - 2^2\biggr] + (2^2 - 2) \Delta f_3^' + 2^2f_2
+ \biggl[\frac{2}{3} - \frac{2^3}{6}\biggr] \Delta^3 f_3^{'''}
+ \biggl[ \frac{2^4}{2^3\cdot 3} - \frac{1}{2\cdot 3}\biggr] \Delta^4 f_3^{iv}
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3\biggl[ -3\biggr] + (2) \Delta f_3^' + 2^2f_2
+ \biggl[- \frac{2}{3}\biggr] \Delta^3 f_3^{'''}
+ \biggl[ \frac{1}{2} \biggr] \Delta^4 f_3^{iv}
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~
\biggl[\frac{2}{3}\biggr] \Delta^3 f_3^{'''}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^'
+ \biggl[ \frac{1}{2} \biggr] \Delta^4 f_3^{iv}
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

This also allows us to improve the expression for the <math>~f_3^{''}</math> term, as initially derived in the "First" subsection, above. Namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{1}{2} (\Delta)^2 f^{''}_3
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_2 - f_3 + (\Delta) f_3^' - \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{6} \biggl\{
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^'
+ \biggl[ \frac{1}{2} \biggr] \Delta^4 f_3^{iv}
\biggr\} \biggl[ \frac{3}{2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^'
+ \biggl[\frac{1}{2^2\cdot 3} \biggr](\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

Hence, an improved expression for <math>~f_4</math> is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_4</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^4)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{6} \biggl\{
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^'
\biggr\} \biggl[ \frac{3}{2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{2} f_1 + 3f_2 - \frac{3}{2} f_3 + 3(\Delta) f_3^' + \mathcal{O}(\Delta^4)
</math>
</td>
</tr>
</table>
</div>

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center"><math>~\mathcal{O}(\Delta^4)</math></td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
f_4
</math>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{2} f_1 + 3f_2 - \frac{3}{2} f_3 + 3(\Delta) f_3^' + \mathcal{O}(\Delta^4)
</math>
</td>
</tr>
</table>
</td></tr></table>
</div>

----

Third:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (- 3\Delta) f_3^' + \frac{1}{2} (- 3\Delta)^2 f^{''}_3 + \frac{1}{6} (- 3\Delta)^3 f_3^{'''} + \frac{1}{24}(- 3\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + \biggl[ - 3 \biggr] (\Delta) f_3^' + \biggl[ \frac{3^3}{2^3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 3^2 \biggl\{
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^'
+ \biggl[\frac{1}{2^2\cdot 3} \biggr](\Delta)^4 f_3^{iv}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[-\frac{3^3}{2^2} \biggr] \biggl\{
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^'
+ \biggl[ \frac{1}{2} \biggr] \Delta^4 f_3^{iv}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + \biggl[ - 3 \biggr] (\Delta) f_3^' + \biggl[ \frac{3^3}{2^3} \biggr] (\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[- \frac{3^2 }{4} \biggr] f_1 + \biggl[ 2\cdot 3^2 \biggr]f_2 + \biggl[ - \frac{3^2 \cdot 7}{4} \biggr] f_3 + \biggl[ \frac{3^3}{2} \biggr] (\Delta) f_3^'
+ \biggl[\frac{3}{2^2} \biggr](\Delta)^4 f_3^{iv}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[\frac{3^3}{2^2} \biggr] f_1
+ \biggl[-3^3 \biggr] f_2
+ \biggl[\frac{3^4}{2^2} \biggr]f_3
+ \biggl[- \frac{3^3}{2} \biggr] \Delta f_3^'
+ \biggl[- \frac{3^3}{2^3} \biggr] \Delta^4 f_3^{iv}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{3^3}{2^2} - \frac{3^2 }{4} \biggr] f_1 + \biggl[ 2\cdot 3^2 -3^3\biggr]f_2 + \biggl[ 1+ \frac{3^4}{2^2} - \frac{3^2 \cdot 7}{4} \biggr] f_3
+ \biggl[ \frac{3^3}{2} - \frac{3^3}{2} -3\biggr] (\Delta) f_3^'
+ \biggl[\frac{3^3}{2^3} + \frac{3}{2^2} - \frac{3^3}{2^3}\biggr](\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{3^2}{2} \biggr] f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3
+ \biggl[ -3\biggr] (\Delta) f_3^'
+ \biggl[\frac{3}{2^2} \biggr](\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
- \biggl[\frac{3}{2^2} \biggr](\Delta)^4 f_3^{iv}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- f_0 + \biggl[\frac{3^2}{2} \biggr] f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3
+ \biggl[ -3\biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\frac{1}{2} (\Delta)^2 f^{''}_3
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[\frac{1}{2^2\cdot 3} \biggr]\biggl\{
- f_0 + \biggl[\frac{3^2}{2}\biggr] f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3
+ \biggl[ -3\biggr] (\Delta) f_3^'
\biggr\} \biggl[ - \frac{2^2}{3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{4} f_1 + 2f_2 + \biggl[ - \frac{7}{4} \biggr] f_3 + \biggl[ \frac{3}{2} \biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{1}{2} \biggr] f_1
+ f_2 + \biggl[- \frac{11}{2 \cdot 3^2} \biggr] f_3
+ \biggl[\frac{1}{3} \biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{1}{2} - \frac{1}{4} \biggr] f_1
+ 3 f_2 + \biggl[- \frac{11}{2 \cdot 3^2} - \frac{7}{4} \biggr] f_3
+ \biggl[\frac{1}{3} + \frac{3}{2} \biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{3}{4} \biggr] f_1
+ 3 f_2 + \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3
+ \biggl[\frac{11}{2\cdot 3} \biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

And,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\biggl[\frac{2}{3}\biggr] \Delta^3 f_3^{'''}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ \frac{1}{2} \biggr] \biggl\{
- f_0 + \biggl[\frac{3^2}{2} \biggr] f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3
+ \biggl[ -3\biggr] (\Delta) f_3^'
\biggr\} \biggl[ - \frac{2^2}{3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- f_1 + 2^2f_2 -3 f_3 + 2 \Delta f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[ \frac{2}{3} \biggr] f_0 + \biggl[- 3 \biggr] f_1
+ \biggl[ 2\cdot 3 \biggr] f_2 + \biggl[- \frac{11}{3} \biggr] f_3
+ \biggl[ 2 \biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2}{3} \biggr] f_0 + \biggl[- 4 \biggr] f_1
+ \biggl[ 2\cdot 5 \biggr] f_2 + \biggl[- \frac{2^2 \cdot 5}{3} \biggr] f_3
+ \biggl[ 4 \biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

----

Finally, then:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~f_4</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' + \frac{1}{2} ( \Delta)^2 f^{''}_3 + \frac{1}{6} (\Delta)^3 f_3^{'''} + \frac{1}{24}(\Delta)^4 f_3^{iv} + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{2} \biggl\{
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{3}{4} \biggr] f_1
+ 3 f_2 + \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3
+ \biggl[\frac{11}{2\cdot 3} \biggr] (\Delta) f_3^'
\biggr\}\biggl[ 2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{2\cdot 3} \biggl\{
\biggl[ \frac{2}{3} \biggr] f_0 + \biggl[- 4 \biggr] f_1
+ \biggl[ 2\cdot 5 \biggr] f_2 + \biggl[- \frac{2^2 \cdot 5}{3} \biggr] f_3
+ \biggl[ 4 \biggr] (\Delta) f_3^'
\biggr\}\biggl[ \frac{3}{2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{2^3\cdot 3} \biggl\{
- f_0 + \biggl[\frac{3^2}{2} \biggr] f_1 + \biggl[ - 3^2 \biggr]f_2 + \biggl[ \frac{11}{2} \biggr] f_3
+ \biggl[ -3\biggr] (\Delta) f_3^'
\biggr\}\biggl[- \frac{2^2}{3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{3}{4} \biggr] f_1
+ 3 f_2 + \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3
+ \biggl[\frac{11}{2\cdot 3} \biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{2^2} \biggl\{
\biggl[ \frac{2}{3} \biggr] f_0 + \biggl[- 4 \biggr] f_1
+ \biggl[ 2\cdot 5 \biggr] f_2 + \biggl[- \frac{2^2 \cdot 5}{3} \biggr] f_3
+ \biggl[ 4 \biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{2\cdot 3^2} \biggl\{
f_0 + \biggl[ - \frac{3^2}{2} \biggr] f_1 + \biggl[ 3^2 \biggr]f_2 + \biggl[- \frac{11}{2} \biggr] f_3
+ \biggl[ 3\biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
f_3 + (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[\frac{1}{3^2} \biggr] f_0 + \biggl[- \frac{3}{4} \biggr] f_1
+ 3 f_2 + \biggl[- \frac{5\cdot 17}{2^2\cdot 3^2} \biggr] f_3
+ \biggl[\frac{11}{2\cdot 3} \biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl\{
\biggl[ \frac{1}{2\cdot 3} \biggr] f_0 + \biggl[- 1 \biggr] f_1
+ \biggl[ \frac{5}{2} \biggr] f_2 + \biggl[- \frac{5}{3} \biggr] f_3
+ \biggl[ 1 \biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\biggl\{
\biggl[ \frac{1}{2\cdot 3^2} \biggr] f_0 + \biggl[ - \frac{1}{2^2} \biggr] f_1 + \biggl[ \frac{1}{2} \biggr]f_2 + \biggl[- \frac{11}{2^2\cdot 3^2} \biggr] f_3
+ \biggl[ \frac{1}{2\cdot 3} \biggr] (\Delta) f_3^'
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{1}{3^2} + \frac{1}{2\cdot 3} + \frac{1}{2\cdot 3^2} \biggr] f_0
+ \biggl[- 1 - \frac{3}{4} - \frac{1}{2^2} \biggr] f_1
+ \biggl[ 3 + \frac{5}{2} + \frac{1}{2} \biggr] f_2
+ \biggl[1 - \frac{5\cdot 17}{2^2\cdot 3^2} - \frac{5}{3} - \frac{11}{2^2\cdot 3^2} \biggr] f_3
+ \biggl[2 + \frac{11}{2\cdot 3} + \frac{1}{2\cdot 3} \biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{1}{3} \biggr] f_0
+ \biggl[- 2\biggr] f_1
+ \biggl[ 6 \biggr] f_2
+ \biggl[- \frac{2\cdot 5}{3} \biggr] f_3
+ \biggl[4 \biggr] (\Delta) f_3^'
+ \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</div>

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center"><math>~\mathcal{O}(\Delta^5)</math></td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
f_4
</math>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{3} f_0 - 2 f_1 + 6 f_2 - \frac{2\cdot 5}{3} f_3 + 4 (\Delta) f_3^' + \mathcal{O}(\Delta^5)
</math>
</td>
</tr>
</table>
</td></tr></table>
</div>

{{ SGFfooter }}

File:WolframAlphaResult.png

2024-06-28T23:29:05Z

Xxsrm:

File:Ostriker64PaperIIFig1.png

2024-06-28T23:26:31Z

Xxsrm:

Apps/Ostriker64

2024-06-28T23:21:43Z

Xxsrm: Created page with "__FORCETOC__  =Polytropic & Isothermal Tori= {| class="Ostriker64" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" |<b>Ostriker<br />(1964)</b> |} Here we will focus on the analysis of the structure self-gravitating tori that are composed of compressible — speci..."

__FORCETOC__

=Polytropic & Isothermal Tori=
{| class="Ostriker64" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like|<b>Ostriker<br />(1964)</b>]]
|}
Here we will focus on the analysis of the structure self-gravitating tori that are composed of compressible — specifically, polytropic and isothermal — fluids as presented in a series of papers by Jeremiah P. Ostriker:
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. Ostriker (1964, ApJ, 140, 1056)] — ''The Equilibrium of Polytropic and Isothermal Cylinders''
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O J. Ostriker (1964, ApJ, 140, 1067)] — ''The Equilibrium of Self-Gravitating Rings''
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1529O J. Ostriker (1964, ApJ, 140, 1529)] — ''On the Oscillations and the Stability of a Homogeneous Compressible Cylinder''
* [http://adsabs.harvard.edu/abs/1965ApJS...11..167O J. Ostriker (1965, ApJ Supplements, 11, 167)] — ''Cylindrical Emden and Associated Functions''
I believe that much, if not all, of this material was drawn from Ostriker's doctoral dissertation research at the University of Chicago (and Yerkes Observatory) under the guidance of [https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar S. Chandrasekhar].

==Coordinate System==

===Basics===
In §IIa of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker defines a set of orthogonal coordinates, <math>~(r,\phi,\theta)</math>, that is related to the traditional Cartesian coordinate system, <math>~(x,y,z)</math>, via the relations,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(R+r\cos\phi)\cos\theta \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~y</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(R+r\cos\phi)\sin\theta \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~z</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r\sin\phi \, .</math>
</td>
</tr>
</table>
As Ostriker states, <font color="darkgreen">"The coordinate <math>~r</math> is the distance from a reference circle of radius <math>~R</math> (later chosen to be the major radius of the ring) …"</font> The angle, <math>~\theta</math>, plays the role of the azimuthal angle, as is familiar in both cylindrical and spherical coordinates, while, here, <math>~\phi</math> is a meridional-plane polar angle measured counterclockwise from the equatorial plane. For axisymmetric systems, there will be no dependence on the azimuthal angle, so the pair of relevant coordinates in the meridional plane are,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\varpi \equiv (x^2+y^2)^{1 / 2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~R+r\cos\phi \, ,</math>
</td>
<td align="center">    and,    
<td align="right">
<math>~z</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r\sin\phi \, .</math>
</td>
</tr>
</table>

<div align="center" id="THH12Figure4">
<table border="1" cellpadding="8">

<tr><td align="center">
Figure 1 extracted without modification from p. 1077 of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O J. P. Ostriker (1964; Paper II)]<p></p>
"''The Equilibrium of Self-Gravitating Rings''"<p></p>
ApJ, vol. 140, pp. 1067-1087 © American Astronomical Society
</td>
</tr>

<tr>
<td align="center">
[[File:Ostriker64PaperIIFig1.png|600px|Figure 1 from Ostriker (1964) Paper II]]
</td>
</tr>
</table>
</div>
For later reference, we note that (see eq. 3 of Paper II) the corresponding line element is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\delta s^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\delta r^2 + r^2 \delta\phi^2 + (R+r\cos\phi)^2\delta\theta^2 \, ,
</math>
</td>
</tr>
</table>
which means that the relevant scale factors for the adopted coordinate system, <math>~(r,\phi,\theta)</math>, are
<div align="center">
<math>~h_1 = 1 \, ,</math>       <math>~h_2 = r \, ,</math>       <math>~h_3 = (R+r\cos\phi) \, ,</math>
</div>
and the relevant differential volume element is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~d^3 x</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~h_1 h_2 h_3 dr d\phi d\theta
= r(R+r\cos\phi) dr d\phi d\theta\, .
</math>
</td>
</tr>
</table>

===Relationship to Toroidal Coordinate===
Referring back to our separate discussion of the [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|basic elements of a toroidal coordinate system]], we know that, the meridional-plane toroidal coordinates <math>~(\eta,\theta)</math> are related to traditional meridional-plane cylindrical coordinate pair <math>~(\varpi,z)</math> via the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\varpi}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sinh\eta}{\cosh\eta - \cos\theta} \, ,</math>
</td>
<td align="center">      and,      
<td align="right">
<math>~\frac{z}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sin\theta}{\cosh\eta - \cos\theta} \, ,</math>
</td>
</tr>
</table>

assuming that the cylindrical-coordinate location of the ''anchor ring'' is <math>~(\varpi,z) = (R,0)</math>. Let's determine how to transform between these two sets of coordinate pairs.

====Independent Exploration====
First, eliminating reference to Ostriker's "polar angle" <math>~\phi</math>, we see that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{r^2}{R^2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{\varpi}{R} - 1 \biggr)^2 + \biggl(\frac{z}{R}\biggr)^2</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{\sinh\eta}{\cosh\eta - \cos\theta} - 1 \biggr]^2 + \biggl[ \frac{\sin\theta}{\cosh\eta - \cos\theta} \biggr]^2</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{(\sinh\eta - \cosh\eta + \cos\theta)^2 + \sin^2\theta}{(\cosh\eta - \cos\theta)^2} \biggr] \, .</math>
</td>
</tr>
</table>
Then, eliminating reference to Ostriker's radial coordinate <math>~r</math>, we find,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\cot\phi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\varpi/R - 1}{z/R}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sinh\eta - \cosh\eta + \cos\theta}{\sin\theta}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\cot\theta + \frac{\sinh\eta - \cosh\eta }{\sin\theta} \, .</math>
</td>
</tr>
</table>
Now let's try to derive the alternate transformation. We'll start by eliminating the "polar angle" in toroidal coordinates.
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\cosh\eta - \cos\theta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\sinh\eta}{\varpi/R}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos\theta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\cosh\eta - \frac{\sinh\eta}{\varpi/R} \, .</math>
</td>
</tr>
</table>
The same relation also implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{z}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{\varpi}{R}\biggr) \frac{\sin\theta}{\sinh\eta}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \sin\theta </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{z}{R}\biggl( \frac{\varpi}{R}\biggr)^{-1} \sinh\eta \, .</math>
</td>
</tr>
</table>
Together, then, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 = \sin^2\theta + \cos^2\theta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{z}{R}\biggl( \frac{\varpi}{R}\biggr)^{-1} \sinh\eta \biggr]^2
+ \biggl[ \cosh\eta - \frac{\sinh\eta}{\varpi/R} \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl( \frac{\varpi}{R}\biggr)^{-2} \biggl[ \frac{z}{R} \cdot \sinh\eta \biggr]^2
+ \biggl(\frac{\varpi}{R}\biggr)^{-2} \biggl[ \frac{\varpi}{R}\cdot \cosh\eta - \sinh\eta \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{\varpi}{R}\biggr)^{2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{z}{R} \cdot \sinh\eta \biggr]^2
+ \biggl[ \frac{\varpi}{R}\cdot \cosh\eta - \sinh\eta \biggr]^2 \, .
</math>
</td>
</tr>
</table>
Alternatively, in an attempt to eliminate <math>~\eta</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\sinh\eta </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\varpi}{R}\biggl( \frac{z}{R}\biggr)^{-1} \sin\theta </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cosh\eta = \biggl[ 1 + \sinh^2\eta\biggr]^{1 / 2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ 1 + \biggl(\frac{\varpi}{R}\biggr)^2 \biggl( \frac{z}{R}\biggr)^{-2} \sin^2\theta \biggr]^{1 / 2} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{z}{R}\biggr)^{-1} \biggl[ \biggl( \frac{z}{R}\biggr)^{2} + \biggl(\frac{\varpi}{R}\biggr)^2 \sin^2\theta \biggr]^{1 / 2} \, .</math>
</td>
</tr>
</table>
But, also,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\cosh\eta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{z}{R} \biggr)^{-1} \sin\theta + \cos\theta</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\biggl[ \biggl( \frac{z}{R}\biggr)^{2} + \biggl(\frac{\varpi}{R}\biggr)^2 \sin^2\theta \biggr]^{1 / 2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sin\theta + \biggl( \frac{z}{R}\biggr) \cos\theta</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\biggl( \frac{z}{R}\biggr)^{2} + \biggl(\frac{\varpi}{R}\biggr)^2 \sin^2\theta </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sin^2\theta + 2\biggl( \frac{z}{R}\biggr)\sin\theta \cos\theta + \biggl( \frac{z}{R}\biggr)^2 \cos^2\theta</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\biggl( \frac{z}{R}\biggr)^{2}\biggl[1-\cos^2\theta \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sin^2\theta\biggl[1 - \biggl(\frac{\varpi}{R}\biggr)^2 \biggr] + 2\biggl( \frac{z}{R}\biggr)\sin\theta \cos\theta </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~0 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\sin^2\theta\biggl[1 - \biggl(\frac{\varpi}{R}\biggr)^2 - \biggl( \frac{z}{R}\biggr)^{2}\biggr] + 2\biggl( \frac{z}{R}\biggr)\sin\theta \cos\theta </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\biggl[1 - \biggl(\frac{\varpi}{R}\biggr)^2 - \biggl( \frac{z}{R}\biggr)^{2}\biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ - 2\biggl( \frac{z}{R}\biggr) \cot\theta </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cot\theta </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{1}{2}\biggl( \frac{z}{R}\biggr)^{-1} \biggl[1 - \biggl(\frac{\varpi}{R}\biggr)^2 - \biggl( \frac{z}{R}\biggr)^{2}\biggr] \, .</math>
</td>
</tr>
</table>
Now that I think about it, this is all a bit silly because from the [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|basic elements of a toroidal coordinate system]] we already know how to shift from cylindrical to toroidal coordinates.

====Back to Basics====

Mapping the other direction [see equations 2.13 - 2.15 of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] ], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\eta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\cos\theta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{(r_1^2 + r_2^2 - 4R^2)}{2r_1 r_2} \, ,</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_1^2 </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~[\varpi + R]^2 + z^2 \, ,</math>
</td>
<td align="center">      and      
<td align="right">
<math>~r_2^2 </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~[\varpi - R]^2 + z^2 \, ,</math>
</td>
</tr>
</table>
</div>
and <math>~\theta</math> has the same sign as <math>~z</math>. Now, given that Ostriker's <math>~(r,\phi)</math> coordinates are related to cylindrical coordinates via the expressions,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\varpi </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~R+r\cos\phi \, ,</math>
</td>
<td align="center">    and    
<td align="right">
<math>~z</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r\sin\phi \, ,</math>
</td>
</tr>
</table>
we can write,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_1^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[2R + r\cos\phi]^2 + r^2\sin^2\phi </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~4R^2 + 4Rr\cos\phi + r^2 \, ;</math>
</td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_2^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r^2 \, .</math>
</td>
</tr>
</table>
Hence,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\cos\theta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2r[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} \biggl[ 4Rr\cos\phi + 2r^2 \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2R\cos\phi + r }{[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~e^{2\eta}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{ 4R^2 + 4Rr\cos\phi + r^2 }{r^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1 \, .
</math>
</td>
</tr>
</table>

===Summary===

<table border="1" cellpadding="10" align="center" width="85%">
<tr>
<td align="left" width="50%">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{r^2}{R^2} </math>
</td>
<td align="center">
<math>~=</math>
<td align="left">
<math>~\biggl[ \frac{(\sinh\eta - \cosh\eta + \cos\theta)^2 + \sin^2\theta}{(\cosh\eta - \cos\theta)^2} \biggr] </math>
</td>
</tr>
<tr>
<td align="right">
<math>~\cot\phi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\cot\theta + \frac{\sinh\eta - \cosh\eta }{\sin\theta} </math>
</td>
</tr>
</table>

</td>
<td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~e^{2\eta}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\cos\theta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2R\cos\phi + r }{[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}}
</math>
</td>
</tr>
</table>

</td>
</tr>
</table>

===Second Attempt===
====Single Offset Circle====
Now an [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle|off-center circle]] whose major and minor radii are, respectively, <math>~(\varpi_0,d)</math>, will be described by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~d^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(\varpi - \varpi_0)^2 + z^2 \, .
</math>
</td>
</tr>
</table>

<span id="Dsquared">where both <math>~d</math> and <math>~\varpi_0</math> are held constant while mapping out the variation of <math>~z</math> with <math>~\varpi</math>. If we acknowledge that, in general, <math>~\varpi_0 \ne R_\mathrm{JPO}</math>, then we know how <math>~r</math> varies with <math>~\phi</math> via the relation,</span>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~d^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ R_\mathrm{JPO} + r\cos\phi - \varpi_0\biggr]^2 + r^2\sin^2\phi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(R_\mathrm{JPO}-\varpi_0)^2
+ 2\biggl[ (R_\mathrm{JPO}-\varpi_0) r\cos\phi \biggr]
+r^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ 0 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
r^2 + 2r\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr] + \biggl[(R_\mathrm{JPO}-\varpi_0)^2 - d^2\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ r </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2}\biggl\{
- 2\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr] \pm
\sqrt{ 4\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr]^2 - 4\biggl[(R_\mathrm{JPO}-\varpi_0)^2 - d^2\biggr] }
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2}\biggl\{
2\biggl[ (\varpi_0 - R_\mathrm{JPO}) \cos\phi \biggr] \pm
\sqrt{ 4\biggl[ (\varpi_0 - R_\mathrm{JPO}) \cos\phi \biggr]^2 - 4\biggl[(\varpi_0 - R_\mathrm{JPO})^2 - d^2\biggr] }
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \frac{r}{ (\varpi_0 - R_\mathrm{JPO}) }</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\cos\phi \pm
\sqrt{ \cos^2\phi - 1 + d^2 (\varpi_0 - R_\mathrm{JPO})^{-2} }
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\cos\phi \pm \sqrt{ d^2 (\varpi_0 - R_\mathrm{JPO})^{-2}-\sin^2\phi }
</math>
</td>
</tr>
</table>

In order to align this expression with the terminology (and variable labels) that we use in the context of a toroidal coordinate system, we associate the radius of the ''anchor ring'' as <math>~R_\mathrm{JPO}\leftrightarrow a</math>, and we associate the major radius of each circular torus as <math>~\varpi_0 \leftrightarrow R_0</math>. We therefore have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{r}{ (R_0-a) }</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\cos\phi \pm \sqrt{ d^2 (R_0-a)^{-2}-\sin^2\phi }
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \frac{r}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{R_0}{a}-1 \biggr)
\biggl[ \cos\phi \pm \sqrt{ \biggl(\frac{d}{a}\biggr)^2 \biggl(\frac{R_0}{a}-1 \biggr)^{-2}-\sin^2\phi } \biggr]
</math>
</td>
</tr>
</table>

and, the coordinates of points along the surface of the torus <math>~(\varpi,z)</math> are provided by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\varpi</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a + (R_0 - a)\cos\phi \biggl[
\cos\phi \pm \sqrt{ d^2 (R_0 - a)^{-2}-\sin^2\phi }
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~z</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(R_0 - a)\sin\phi \biggl[
\cos\phi \pm \sqrt{ d^2 (R_0 - a)^{-2}-\sin^2\phi }
\biggr]
</math>
</td>
</tr>
</table>
We have tested this pair of expressions using Excel and have successfully demonstrated that they do, indeed, trace out a circle of radius, <math>~d</math>, whose center is offset from the symmetry axis by a distance, <math>~R_0</math>.

====Set of Circles Whose Offset Increases With Circle Diameter====

A set of nested off-center circles will be described by allowing <math>~R_0 = R_0(d)</math>, that is, by having the off-set distance, <math>~R_0</math>, vary with the size of the circle, <math>~d</math>. The above prescription for the normalized "coordinate" <math>~r/a</math> will work for ''any'' prescribed <math>~R_0(d)</math> function.

But a ''particular'' <math>~R_0(d)</math> function is demanded if we want this derived prescription to represent the behavior of toroidal coordinates. In a [[Apps/DysonWongTori#Introducing_Toroidal_Coordinates|toroidal coordinate system]], a specification of the value of the "radial" coordinate, <math>~\eta</math>, automatically dictates the ratio <math>~R_0/d</math>; but we are not at liberty to separately define the value of the ''difference,'' <math>~(R_0 - d)</math>. Instead, we must enforce the toroidal-coordinate relation,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~a^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~R_0^2 - d^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \frac{R_0}{a}-1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ 1 + \delta^2\biggr]^{1 / 2} -1 \, ,</math>
</td>
</tr>
</table>
where we have adopted the shorthand notation, <math>~\delta\equiv d/a</math>. Hence,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{r}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[ \sqrt{1+\delta^2} -1 ]
\{ \cos\phi \pm [\delta^2 ( \sqrt{1+\delta^2} -1 )^{-2}-\sin^2\phi ]^{1 / 2} \}
</math>
</td>
</tr>
</table>

Now, in a [[Apps/DysonWongTori#Introducing_Toroidal_Coordinates|toroidal coordinate system]], there is a similar "radial" coordinate, <math>~\eta</math>, whose value varies with distance from the ''anchor ring'' of radius, <math>~a</math>. Its value depends on both <math>~R_0</math> and <math>~d</math> via the relation,
<div align="center">
<math>~R_0 = d\cosh\eta \, .</math>
</div>
This means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\cosh\eta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\delta}\biggl(\frac{R_0}{a}\biggr) = \frac{\sqrt{1+\delta^2}}{\delta} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \delta^2 \cosh^2\eta</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~1 + \delta^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \delta^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\cosh^2\eta - 1} = \frac{1}{\sinh^2\eta} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~ \sqrt{1 + \delta^2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[1 + \frac{1}{\sinh^2\eta} \biggr]^{1 / 2}
= \coth\eta
\, ,</math>
</td>
</tr>
</table>
which also means that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{r}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[ \coth\eta -1 ]
\biggl\{ \cos\phi \pm \biggl[ ( \cosh\eta -\sinh\eta )^{-2} -\sin^2\phi \biggr]^{1 / 2}
\biggr\} \, .
</math>
</td>
</tr>
</table>

====Case of Small Offset====

Another way to look at this issue is to go [[#Dsquared|back to the expression]],
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~d^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
(R_\mathrm{JPO}-\varpi_0)^2
+ 2\biggl[ (R_\mathrm{JPO}-\varpi_0) r\cos\phi \biggr]
+r^2
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \delta^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{r}{a}\biggr)^2
+ \frac{r}{a}\biggl[ 2\biggl(1 - \frac{R_0}{a}\biggr)\biggr] \cos\phi
+ \biggl(1 - \frac{R_0}{a}\biggr)^2
</math>
</td>
</tr>
</table>
and assume that, while still dependent on the radial coordinate, the dimensionless offset is small. That is, assume that,
<div align="center">
<math>~\Delta(\delta) \equiv 1 - \frac{R_0(\delta)}{a} \ll 1 \, .</math>
</div>
In this case, we can write,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \delta^2</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\biggl(\frac{r}{a}\biggr)^2
+ 2\Delta(\delta) \biggl( \frac{r}{a} \biggr) \cos\phi
+\cancelto{0}{\Delta^2(\delta)} \, .
</math>
</td>
</tr>
</table>
And differentiating both sides of the expression with respect to <math>~r/a</math> gives,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0 </math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~2\biggl(\frac{r}{a}\biggr) + 2\Delta(\delta) \cos\phi</math>
</td>
</tr>
</table>

<font color="red">'''COMMENT by Tohline'''</font> (15 August 2018): I'm not sure that this is leading where I had hoped. I am gearing up to draw a comparison between these last expressions and eq. (74) in [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Ostriker's (1964) Paper II].



==Gravitational Potential==

===Potential of a Thin Hoop===
In §IIb of his [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker (1964) derives an expression for the gravitational potential of a torus in the ''Thin Ring'' approximation, beginning specifically with the [[SR/PoissonOrigin#Step_1|integral form of the Poisson equation]] that is widely referred to in the astrophysics community as an expression for the,
<div align="center" id="GravitationalPotential">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Scalar Gravitational Potential'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>~ \Phi(\vec{x})</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~ -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], §10, p. 17, Eq. (11)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], §4.2, p. 77, Eq. (12)
</td>
</tr>
</table>
</div>

(Note:   Consistent with the usage favored by his doctoral dissertation advisor in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], throughout his collection of 1964 papers Ostriker adopts a ''different sign convention'' as well as a different variable name to represent the gravitational potential.) Employing [[#Coordinate_System|Ostriker's adopted coordinate system]], and recognizing that, <font color="darkgreen">"the distance between the point of integration <math>~(0,0,\theta^')</math> and the point of observation <math>~(r,\phi,0)</math>"</font> is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~|\vec{x}^{~'} - \vec{x}|</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} \, ,</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1070, Eq. (21)
</td>
</tr>
</table>

this expression for the gravitational potential becomes,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~ \Phi(r,\phi)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left" colspan="2">
<math>~ -G \int \int \rho(r^',\phi^') r^' (R+r^'\cos\phi^') dr^' d\phi^' \int \frac{d\theta^'}{[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} } </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -G (2\sigma R) \int_0^\pi \frac{d\theta^'}{[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} } </math>
</td>
<td align="right" rowspan="4">[[File:WolframAlphaResult.png|300px|WolframAlpha result]]</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -\frac{4G \sigma R}{r} \int_0^\pi \frac{\tfrac{1}{2}d\theta^'}{[1 +n^2\sin^2(\tfrac{1}{2}\theta^')]^{1 / 2} } </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -\frac{4G \sigma R}{r} \biggl[ \frac{K(k)}{\sqrt{n^2+1}} \biggr] \, ,</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1070, Eq. (22)
</td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~n^2 \equiv \frac{4R(R+r\cos\phi)}{r^2}</math>
</td>
<td align="center">
    and    
</td>
<td align="left">
<math>~k \equiv \biggl[ \frac{n^2}{n^2+1} \biggr]^{1 / 2} \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1070, Eq. (23)
</td>
</tr>
</table>

<table border="1" cellpadding="10" align="center" width="85%"><tr><td align="left">
Mapping back to cylindrical coordinates, for the moment, we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(\varpi - R)^2 + z^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ n^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4R\varpi}{(\varpi - R)^2 + z^2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ n^2 + 1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4R\varpi + (\varpi - R)^2 + z^2}{(\varpi - R)^2 + z^2} = \frac{(\varpi + R)^2 + z^2}{(\varpi - R)^2 + z^2} \, .</math>
</td>
</tr>
</table>
Acknowledging as well that the mass of Ostriker's "thin hoop" is, <math>~M = 2\pi \sigma R</math>, his expression for the potential becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi(\varpi,z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -\frac{2G M}{\pi} \biggl[ \frac{K(k)}{\sqrt{(\varpi + R)^2 + z^2}} \biggr] \, ,</math>
</td>
</tr>
</table>
where,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~k</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4R\varpi}{(\varpi + R)^2 + z^2} \biggr]^{1 / 2} \, .</math>
</td>
</tr>
</table>

After adopting the variable association, <math>~R \leftrightarrow a</math>, it is clear that Ostriker's derived expression is identical to the Key Equation that we have [[Apps/DysonWongTori#Thin_Ring_Approximation|identified elsewhere]] as providing the,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="1"><font color="#770000">'''Gravitational Potential in the Thin Ring (TR) Approximation'''</font></td>
<td align="center" colspan="1" rowspan="2">[[File:FlatColorContoursCropped.png|220px|Contours for Thin Ring Approximation]]</td>
</tr>
<tr>
<td align="center">
{{ Math/EQ TRApproximation }}
</td>
</tr>
</table>

</td></tr></table>

===Series Expansion===
In the context of Ostriker's expression for the potential, we see that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(k')^{-2} \equiv \biggl[ \frac{1}{1-k^2}\biggr]= n^2 + 1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4R(R+r\cos\phi)}{r^2} + 1
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{2R}{r}\biggr)^2 \biggl[ 1 + \frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2\biggr] \, .
</math>
</td>
</tr>
</table>
Hence, in the vicinity of the ring where <math>~r/R \ll 1</math> and <math>~k'</math> is a "small parameter," we can draw on the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] and write,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(k')^m</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2^m}\biggl( \frac{r}{R}\biggr)^{m} \biggl[ 1 + \frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2\biggr]^{-m / 2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2^m}\biggl( \frac{r}{R}\biggr)^{m} \biggl\{ 1 -\frac{m}{2} \biggl[\frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2 \biggr]
+ \frac{1}{2}\biggl[ -\frac{m}{2}\biggl( -\frac{m}{2}-1\biggr) \biggr]\biggl[\frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2 \biggr]^2
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2^m}\biggl( \frac{r}{R}\biggr)^{m} \biggl\{ 1 - \biggl(\frac{m}{2}\biggr) \frac{r}{R}\cos\phi - \biggl(\frac{m}{2^3}\biggr) \biggl(\frac{r}{R}\biggr)^2
+ \frac{m}{4}\biggl( \frac{m}{2} + 1\biggr) \biggl[\frac{r}{R}\cos\phi \biggr]^2
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\} \, .
</math>
</td>
</tr>
</table>
Note, in particular, that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{1}{k'}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2\biggl( \frac{R}{r}\biggr) \biggl\{ 1 + \biggl(\frac{1}{2}\biggr) \frac{r}{R}\cos\phi + \biggl(\frac{1}{2^3}\biggr) \biggl(\frac{r}{R}\biggr)^2
- \frac{1}{2^3} \biggl[\frac{r}{R}\cos\phi \biggr]^2
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2R}{r} \biggl\{ 1 + \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\} \, ;
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~k'</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{r}{2R} \biggl[ 1 - \frac{r}{2R}\cos\phi
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi -1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
\, ;
</math>       and,
</td>
</tr>

<tr>
<td align="right">
<math>~(k')^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2^2}\biggl( \frac{r}{R}\biggr)^{2} \biggl[ 1 - \frac{r}{R}\cos\phi + \frac{1}{2^2} \biggl(\frac{r}{R}\biggr)^2 (4\cos^2\phi - 1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
\, .
</math>
</td>
</tr>
</table>

Next we recognize that the following series expansion for the ''complete elliptic integral of the first kind'' — written in terms of the small parameter, <math>~k'</math> — appears, for example, as eq. (8.113.3) in the Fourth Edition of [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)]:
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~K(k)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} \biggr){k'}^2
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} \biggr){k'}^4
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl( \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} - \frac{2}{5\cdot 6} \biggr){k'}^6 + \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2
+ \frac{3^2}{2^6} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4
+ \frac{5^2}{2^8} \biggl( \ln\frac{4}{k^'} - \frac{37}{30} \biggr){k'}^6 + \cdots
</math>
</td>
</tr>
</table>
[This series expansion — up through the term <math>~\mathcal{O}(k'^4)</math> — appears as equation 24 in Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II].] Put together, then, Ostriker's expression for the gravitational potential in the ''thin ring'' approximation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ -\frac{2GM}{\pi r} k' K(k) </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2GM}{\pi r} \biggl[
k' \ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^3
+ \cdots
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2GM}{\pi r} \biggl\{ \ln \frac{4}{k^'} \biggl[ k' + \frac{k'^3}{2^2} \biggr] - \frac{1}{4} k'^3 + \mathcal{O}\biggl( \frac{r^5}{R^5}\biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2GM}{\pi r} \biggl\{ \ln \frac{4}{k^'} \biggl[ k' + \frac{k'^3}{2^2} \biggr]
- \frac{1}{2^5}\biggl( \frac{r}{R}\biggr)^{3} \biggl[ 1 - \biggl(\frac{3}{2}\biggr) \frac{r}{R}\cos\phi - \biggl(\frac{3}{2^3}\biggr) \biggl(\frac{r}{R}\biggr)^2
+ \frac{3}{4}\biggl( \frac{3}{2} + 1\biggr) \biggl(\frac{r}{R}\cos\phi \biggr)^2 + \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr) \biggr]
+ \mathcal{O}\biggl( \frac{r^5}{R^5}\biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{2GM}{\pi R} \biggl\{ \ln \frac{4}{k^'} \biggl[ k' + \frac{k'^3}{2^2} \biggr] \frac{R}{r}
- \frac{1}{2^5}\biggl( \frac{r}{R}\biggr)^{2} \biggl[ 1 - \biggl(\frac{3}{2}\biggr) \frac{r}{R}\cos\phi + \mathcal{O}\biggl( \frac{r^2}{R^2}\biggr) \biggr]
+ \mathcal{O}\biggl( \frac{r^5}{R^5}\biggr)
\biggr\}\, ,
</math>
</td>
</tr>
</table>
where, again, we have recognized that the mass of the thin hoop is, <math>~M = 2\pi\sigma R</math>. Now,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ k' \biggl[ 1 + \frac{k'^2}{2^2} \biggr] \frac{R}{r}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2}\biggl[ 1 - \frac{r}{2R}\cos\phi
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi -1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]\biggl\{1 +
\frac{1}{2^4} \biggl[ \biggl( \frac{r}{R}\biggr)^{2} - \biggl(\frac{r}{R}\biggr)^3\cos\phi
+ \mathcal{O}\biggl(\frac{r^4}{R^4} \biggr) \biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[ 1 - \frac{r}{2R}\cos\phi
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi -1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]\biggl\{\frac{1}{2} +
\frac{1}{2^5} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2}\biggl[ 1 - \frac{r}{2R}\cos\phi
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr] \, ;
</math>
</td>
</tr>
</table>
and, given that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ln[a (1+x)] = \ln a + \ln(1+x)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\ln a + x - \tfrac{1}{2}x^2 + \tfrac{1}{3}x^3 - \tfrac{1}{4}x^4 + \cdots
</math>
</td>
</tr>
</table>

we also have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ln \frac{4}{k'}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \ln\biggl[ 1 + \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
- \frac{1}{2} \biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{3}\biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]^3 + \cdots
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi \biggr]
- \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2\cos^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 (1 - 2\cos^2\phi ) + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \, .
</math>
</td>
</tr>
</table>

So our series expansion for Ostriker's "thin ring" potential becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{GM}{\pi R} \biggl\{ \ln \frac{4}{k^'} \biggl[ 1 - \frac{r}{2R}\cos\phi
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
- \frac{1}{2^4}\biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{GM}{\pi R} \biggl\{ \ln \frac{8R}{r} \biggl[ 1 - \frac{r}{2R}\cos\phi
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \biggl[ \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 (1 - 2\cos^2\phi ) + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]\biggl[ 1 - \frac{r}{2R}\cos\phi
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- \frac{1}{2^4}\biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr)
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{GM}{\pi R} \biggl\{ \ln \frac{8R}{r} \biggl[ 1 - \frac{r}{2R}\cos\phi
+ \frac{1}{2^4} \biggl(\frac{r}{R}\biggr)^2 ( 6 \cos^2\phi - 1)
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 (1 - 2\cos^2\phi )
- \frac{1}{2^2} \biggl(\frac{r}{R}\biggr)^2\cos^2\phi
- \frac{1}{2^4}\biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr)
\biggr\} \, .
</math>
</td>
</tr>
</table>
Finally, dropping the explicit mention of all terms <math>~\mathcal{O}(r^3/R^3)</math> and smaller gives the series expansion formulation presented by Ostriker, namely,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{GM}{\pi R} \biggl\{\ln \frac{8R}{r}
- \frac{r}{2R}\biggl[ \ln \frac{8R}{r} - 1\biggr]\cos\phi
~+~ \frac{r^2}{2^4R^2} \biggl[ \ln \frac{8R}{r} ( 6 \cos^2\phi - 1) + (1 - 8\cos^2\phi )
\biggr] ~+ ~\cdots \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{GM}{\pi R} \biggl\{\ln \frac{8R}{r}
- \frac{r}{2R}\biggl[ \ln \frac{8R}{r} - 1\biggr]\cos\phi
~+~ \frac{r^2}{2^4R^2} \biggl[ \biggl(2\ln\frac{8R}{r} - 3 \biggr) + \biggl( 3\ln\frac{8R}{r} - 4 \biggr)\cos 2\phi
\biggr] ~+ ~\cdots \biggr\} \, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1071, Eq. (25)
</td>
</tr>
</table>

===The Dimensionless Radial Coordinate, ξ, and Smallness Parameter, β===

As we have [[SSC/Structure/Polytropes#Polytropic_Spheres|reviewed separately]], when researchers in the astrophysics community discuss the structure of ''spherical'' polytropes, the
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{Math/EQ_SSLaneEmden01}}
</div>
invariably arises, as it is the governing 2<sup>nd</sup>-order ODE whose solution, <math>~\Theta_H(\xi)</math>, defines the internal structure of spherically symmetric equlibrium configurations. Traditionally, as well, the dimensionless radial coordinate,
<div align="center">
<math>~\xi \equiv \frac{r}{a_n} \, ,</math>
</div>
is defined in terms of <math>~a_n</math>, which is a natural length scale of the (spherical) problem. Equation (42) of Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II] provides the traditional definition of <math>~a_n</math>. It is therefore not surprising that, even though Ostriker's set of 1964 papers deal largely with the equilibrium and stability of ''ring-like'' configurations, he adopts a similar definition for the dimensionless radial coordinate; specifically, eq. (5) of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II] states that,
<div align="center">
<math>~\alpha \xi \equiv r \, .</math>
</div>
But, of course, in the context of Ostriker's presentation, <math>~r</math> is not a spherical radial coordinate but is, rather, as [[#Coordinate_System|defined above]]; and <math>~\alpha</math> is of the same order as the minor, cross-sectional radius of the torus.

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 17 August 2018: There appears to be a typographical error in the definition of β that is provided by equation (6) in §IIa of Ostriker's ''Paper II''. The published equation defines β as the ratio of α to ''r'' rather than, as we have indicated here, as the ratio of α to ''R''. Equation (77) on p. 1078 of Paper II confirms this suspicion.]]In eq. (6) of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker also defines the dimensionless parameter,
<div align="center">
<math>~\beta \equiv \frac{\alpha}{R} \, ,</math>
</div>
where <math>~R</math> is associated with the major radius of the ring. Then he states that, <font color="darkgreen">"… since <math>~\alpha \ll R</math> (by hypothesis), we may be sure that <math>~\beta \ll 1</math> …"</font>

With the definitions of these two dimensionless parameters in hand — and, more specifically, after appreciating that,
<div align="center">
<math>~\frac{R}{r} = \frac{1}{\beta\xi} ~~~\Rightarrow ~~~ \ln\frac{8R}{r} = \biggl[ \ln\frac{8}{\beta} - \ln\xi \biggr] </math>
</div>
— we can follow Ostriker's lead and rewrite his derived expression for <math>~\Phi_\mathrm{TR}</math> in the form,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-\frac{GM}{\pi R} \biggl\{ \ln\frac{8}{\beta} - \ln\xi
+ \frac{\beta\xi}{2}\biggl[ - \biggl( \ln\frac{8}{\beta} -1 \biggr) + \ln\xi \biggr]\cos\phi
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+~ \frac{\beta^2\xi^2}{2^4} \biggl[ \biggl(2 \ln\frac{8}{\beta} - 3 - 2\ln\xi \biggr) + \biggl( 3 \ln\frac{8}{\beta} - 4 - 3\ln\xi \biggr)\cos 2\phi
\biggr] ~+ ~\cdots \biggr\} \, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1071, Eq. (26)
</td>
</tr>
</table>

=See Also=

* [http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016, AJ, 152, article id. 35, 31 pp.)] — ''Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects''
* [http://adsabs.harvard.edu/abs/2016ApJ...829...45K W.-T. Kim & S. Moon (2016, ApJ, 829, article id. 45, 22 pp.)] — ''Equilibrium Sequences and Gravitational Instability of Rotating Isothermal Rings''
* [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B E. Y. Bannikova, V. G. Vakulik & V. M. Shulga (2011, MNRAS, 411, 557 - 564)] — ''Gravitational Potential of a Homogeneous Circular Torus: a New Approach''
* [http://adsabs.harvard.edu/abs/2008MNRAS.389..156P D. Petroff & S. Horatschek (2008, MNRAS, 389,156 - 172)] — ''Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity''
<table border="0" cellpadding="8" align="center" width="90%">
<tr><td align="left">
The following quotes have been taken from [http://adsabs.harvard.edu/abs/2008MNRAS.389..156P Petroff & Horatschek (2008)]:<p></p>
<b>§1:</b>   "The problem of the self-gravitating ring captured the interest of such renowned scientists as Kowalewsky (1885), Poincaré (1885a,b,c) and Dyson (1892, 1893). Each of them tackled the problem of an axially symmetric, homogeneous ring in equilibrium by expanding it about the thin ring limit. In particular, Dyson provided a solution to fourth order in the parameter <math>~\sigma = a/b</math>, where <math>~a = r_t</math> provides a measure for the radius of the cross-section of the ring and <math>~b = \varpi_t</math> the distance of the cross-section's centre of mass from the axis of rotation."<p></p>
<b>§7:</b>   "In their work on homogeneous rings, Poincaré and Kowalewsky, whose results disagreed to first order, both had made mistakes as Dyson has shown. His result to fourth order is also erroneous as we point out in Appendix B."
</td></tr>
</table>
* [http://adsabs.harvard.edu/abs/2006IJMPB..20.3113C P. H. Chavanis (2006, International Journal of Modern Physics B, 20, 3113 - 3198)] — ''Phase Transitions in Self-Gravitating Systems''
* [http://adsabs.harvard.edu/abs/2003MNRAS.339..515A M. Ansorg, A. Kleinwächter & R. Meinel (2003, MNRAS, 339, 515)] — ''Uniformly Rotating Axisymmetric Fluid Configurations Bifurcating from Highly Flattened Maclaurin Spheroids''
* [http://adsabs.harvard.edu/abs/2001A%26A...375.1091L M. Lombardi & G. Bertin (2001, Astronomy & Astrophysics, 375, 1091 - 1099)] — ''Boyle's Law and Gravitational Instability''
* [http://adsabs.harvard.edu/abs/1996MNRAS.282..234K W. Kley (1996, MNRAS, 282, 234)] — ''Maclurin Discs and Bifurcations to Rings''
* [http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline, & I. Hachisu (1994, ApJ, 420, 247 - 267)] — ''The Stability of Thick, Self-Gravitating Disks in Protostellar Systems''
* [http://adsabs.harvard.edu/abs/1991ApJ...374..610B I. Bonnell & P. Bastien (1991, ApJ, 374, 610 - 622)] — ''The Collapse of Cylindrical Isothermal and Polytropic Clouds with Rotation''
* [http://adsabs.harvard.edu/abs/1990ApJ...361..394T J. E. Tohline & I. Hachisu (1990, ApJ, 361, 394 - 407)] — ''The Breakup of Self-Gravitating Rings, Tori, and Thick Accretion Disks''
* [http://adsabs.harvard.edu/abs/1988A%26A...200..127S F. Schmitz (1988, Astronomy & Astrophysics, 200, 127 - 134)] — ''Equilibrium Structures of Differentially Rotating Self-Gravitating Gases''
* [http://adsabs.harvard.edu/abs/1985Ap%26SS.109...45V P. Veugelen (1985, Astrophysics & Space Science, 109, 45 - 55)] — ''Equilibrium Models of Differentially Rotating Polytropic Cylinders''
* [http://adsabs.harvard.edu/abs/1984MNRAS.208..279A M. A. Abramowicz, A. Curir, A. Schwarzenberg-Czerny, & R. E. Wilson (1984, MNRAS, 208, 279 - 291)] — ''Self-Gravity and the Global Structure of Accretion Discs''
* [http://adsabs.harvard.edu/abs/1983A%26A...119..109B P. Bastien (1983, Astronomy & Astrophysics, 119, 109 - 116)] — ''Gravitational Collapse and Fragmentation of Isothermal, Non-Rotating, Cylindrical Clouds''
* [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] — ''Another Equilibrium Sequence of Self-Gravitating and Rotating Incompressible Fluid''
* [http://adsabs.harvard.edu/abs/1980ApJ...236..160T J. E. Tohline (1980, ApJ, 236, 160 - 171)] — ''Ring Formation in Rotating Protostellar Clouds''
* [http://adsabs.harvard.edu/abs/1980PThPh..63.1957F T. Fukushima, Y. Eriguchi, D. Sugimoto, & G. S. Bisnovatyi-Kogan (1980, Progress of Theoretical Physics, 63, 1957 - 1970)] — ''Concave Hamburger Equilibrium of Rotating Bodies''
* [http://adsabs.harvard.edu/abs/1978MNRAS.184..709K J. Katz & D. Lynden-Bell (1978, MNRAS, 184, 709 - 712)] — ''The Gravothermal Instability in Two Dimensions''
* [http://adsabs.harvard.edu/abs/1977ApJ...214..584M P. S. Marcus, W. H. Press, & S. A. Teukolsky (1977, ApJ, 214, 584- 597)] — ''Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid'' (includes torus with non-uniform rotation)
<ol type="a"><li>Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an ''equilibrium'' incompressible configuration must rotate uniformly on cylinders (the famous "Poincaré-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"</li></ol>
* [http://adsabs.harvard.edu/abs/1976ApJ...207..736H C. J. Hansen, M. L. Aizenman, & R. L. Ross (1976, ApJ, 207, 736 - 744)] — ''The Equilibrium and Stability of Uniformly Rotating, Isothermal Gas Cylinders''
* [http://adsabs.harvard.edu/abs/1974ApJ...190..675W C.-Y. Wong (1974, ApJ, 190, 675 - 694)] — ''Toroidal Figures of Equilibrium''
* [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279 - 353)] — ''Toroidal and Spherical Bubble Nuclei''
* [http://adsabs.harvard.edu/abs/1942ApJ....95...88R Gunnar Randers (1942, ApJ, 95, 88)] — ''The Equilibrium and Stability of Ring-Shaped 'barred SPIRALS'.''
* [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential William Duncan MacMillan (1958)], ''The Theory of the Potential'', New York: Dover Publications
* [https://archive.org/details/foundationsofpot033485mbp Oliver Dimon Kellogg (1929)], ''Foundations of Potential Theory'', Berlin: Verlag Von Julius Springer
* [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] — ''On the Dynamics of Revolving Fluids''
* [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106)] — ''The Potential of an Anchor Ring. Part II.'' <ol type="a"><li>In this paper, Dyson derives the gravitational potential ''inside'' the ring mass distribution</li></ol>
* [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43 - 95)] — ''The Potential of an Anchor Ring. Part I.''<ol type="a"><li>In this paper, Dyson derives the gravitational potential ''exterior to'' the ring mass distribution</li></ol>
* [http://adsabs.harvard.edu/abs/1885AN....111...37K S. Kowalewsky (1885, Astronomische Nachrichten, 111, 37)] — ''Zusätze und Bemerkungen zu Laplace's Untersuchung über die Gestalt der Saturnsringe''
* Poincaré (1885a, C. R. Acad. Sci., 100, 346), (1885b, Bull. Astr., 2, 109), (1885c, Bull. Astr. 2, 405). — references copied from paper by [http://adsabs.harvard.edu/abs/1974ApJ...190..675W Wong (1974)]

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Apps/MaclaurinToroid

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Xxsrm: Created page with "__FORCETOC__   =Overview= <table border="0" align="center" width="80%" cellpadding="3"><tr><td align="left"> <font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ EH85full }}, … Since the <math>P_4(\eta)</math> bifurc..."

__FORCETOC__ 

=Overview=

<table border="0" align="center" width="80%" cellpadding="3"><tr><td align="left">
<font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ EH85full }}, … Since the <math>P_4(\eta)</math> bifurcation point occurs along the spheroidal sequence at <math>T/|W| = 0.4574</math> … this is synonymous with saying that Maclaurin spheroids that have <math>T/|W| < 0.4574</math> are dynamically stable against linear, axisymmetric perturbations. This conclusion is based, in large part, on the fact that</font> the linear perturbation analysis performed by {{ Bardeen71full }}… <font color="darkgreen">found that the ring mode instability against a <math>P_4(\eta)</math> mode perturbation sets in at exactly the <math>P_4(\eta)</math> bifurcation point."</font>

<font color="darkgreen">"{{ Bardeen71 }}, however, did not examine the behavior of axisymmetric modes higher than <math>P_4.</math> {{ EH85 }} have tabulated bifurcation points along the Maclaurin spheroidal sequence for a number of axisymmetric modes higher than <math>P_4</math> and have found the point that occurs at the lowest value of <math>T/|W|</math> on the sequence to be the <math>P_6(\eta)</math> and not at the <math>P_4(\eta)</math> bifurcation point. We suspect, therefore that when a general linear perturbation analysis is performed, the first dynamical axisymmetric (ring) mode instability will be found to set in at <math>T/|W| = 0.4512</math> and that the unstable mode will be of a <math>P_6(\eta)</math> geometric form."</font>
</td></tr>
<tr><td align="right">
— Drawn from (p. 598 of) {{ HTE87full }}
</td></tr>
</table>

<table border="1" align="center" cellpadding="5" width="90%">
<tr>
<td align="center>
Figure 10 extracted from p. 602 of<br />{{ HTE87figure }}
</td>
</tr>
<tr>
<td align="center">
[[File:HTE78Fig10.png|700px|HTE78Fig10]]
</td>
</tr>

<tr>
<td align="left">
NOTE:
<ol type="A">
<li>In constructing a variety of different ''incompressible'' equilibrium model sequences, {{ HTE87 }} adopted a specific angular momentum distribution given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\dot\varphi\varpi^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
(1 + q)\biggl(\frac{L}{M} \biggr)\biggl\{ 1 - [1 - m(\varpi) ]^{1 / q} \biggr\} \, ,
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
{{ HTE87 }}, Eq. (8)
</td>
</tr>
</table>
When <math>q = 1.5</math>, this matches the distribution found in Maclaurin spheroids and in models along the so-called Maclaurin Toroid sequence — the curve labeled "1.5" in Fig. 10 of {{ HTE87hereafter }}.
</li>
<li>
By combining Eqs. (4) and (7) in {{ HTE87 }}, we see that their Fig. 10 ordinate parameter, <math>F</math>, is related to the parameter, <math>L_*</math>, adopted a decade earlier by {{ MPT77 }} — and heavily used below — via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>F</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\pi \biggl(\frac{5^3}{6^2}\biggr)^2 \biggl[\biggl(\frac{5}{2}\biggr)^2 \biggl(\frac{L_*^2}{3}\biggr) \biggr]^{-3}
=
\biggl(\frac{4\pi}{3}\biggr)L_*^{-6} \, .
</math>
</td>
</tr>
</table>
</li>
</ol>

</td>
</tr>
</table>

=Maclaurin Toroid (MPT77)=
{| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px;" |[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like|<b>Maclaurin<br />Toroid Sequence</b>]]<br />{{ MPT77hereafter }}
|}

In a [[#/Apps/DysonPotential|separate chapter]], we focused on the pioneering work of {{ Dyson1893full }}, {{ Dyson1893Part2full }} and, more recently, {{ Wong74full }}, who determined the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. We will refer to these ''uniformly rotating'' configurations as "Dyson-Wong tori."

Here, we summarize the work of {{ MPT77full }} — hereafter, {{ MPT77hereafter }} — who constructed a sequence of toroidal-shaped, self-gravitating, incompressible configurations that are not uniformly rotating but, rather, have a distribution of angular momentum that is identical to the distribution found in a uniformly rotating, uniform-density sphere. As we have pointed out in our [[AxisymmetricConfigurations/SolutionStrategies#Uniform-Density_Initially_(n'_=_0)|associated overview of "simple rotation curves"]], this chosen (cylindrical) radial distribution of specific angular momentum is given by the expression,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\dot\varphi\varpi^2</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5L}{2M}\biggl\{ 1 - [1 - m(\varpi) ]^{2 / 3} \biggr\} \, ,
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
{{ Stoeckly65 }}, §II.c, Eq. (12)<br />
{{ OM68 }}, Eq. (45)<br />
{{ BO70 }}, Eq. (12)<br />
{{ BO73 }}, Eq. (3)<br />
{{ EH85 }}, Eq. (1)<br />
{{ HTE87 }}, Eq. (6)
</td>
</tr>
</table>
where, <math>L</math> is the total angular momentum, <math>M</math> is the total mass, the mass fraction,
<div align="center">
<math>m(\varpi) \equiv \frac{M_\varpi(\varpi)}{M} \, ,</math>
</div>
and <math>M_\varpi(\varpi)</math> is the mass enclosed within a cylinder of radius, <math>\varpi</math>. Such equilibrium models are often referred to as <math>n' = 0</math> configurations, although {{ MPT77hereafter }} do not use this terminology. Following the lead of {{ MPT77hereafter }}, we will refer to each of their equilibrium configurations as a "Maclaurin Toroid."

==Maclaurin Spheroid Reminder==

<span id="L*">As has been demonstrated</span> in our [[Apps/MaclaurinSpheroidSequence#Corresponding_Total_Angular_Momentum|accompanying discussion of the Maclaurin spheroid sequence]], the (square of the) normalized angular momentum that is associated with a spheroid of eccentricity, <math>e \equiv (1 - c^2/a^2)^{1 / 2}</math>, is,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>L_*^2 \equiv \frac{L^2}{(GM^3\bar{a})}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{6}{5^2} \biggl[ (3-2e^2)(1-e^2)^{1 / 2} \cdot \frac{\sin^{-1}e}{e^3} - \frac{3(1-e^2)}{e^2}\biggr](1 - e^2)^{-2 / 3} \, .</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
{{ MPT77 }}, §IVa, p. 591, Eq. (4.2)
</td>
</tr>
</table>
In that [[Apps/MaclaurinSpheroidSequence#tau|same discussion]], we have demonstrated that the corresponding ratio of rotational to gravitational potential energy is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\tau \equiv \frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2e^2\sin^{-1} e}\biggl[ (3-2e^2)\sin^{-1} e - 3e(1-e^2)^{1 / 2}\biggr]
\, .
</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
{{ MPT77 }}, §IVc, p. 594, Eq. (4.4)
</td>
</tr>
</table>

[[Apps/MaclaurinSpheroidSequence#Figs3and4|Figure 4 from this accompanying discussion]] shows how <math>L_*</math> varies with <math>\tau</math> along the Maclaurin Spheroid sequence. In an effort to conform to {{ MPT77hereafter }}'s presentation, our Figure 1 (immediately below) displays the same information as displayed in Figure 4 of this separate chapter, but the axes have been swapped and the maximum displayed value of <math>L_*</math> has been extended from 1 to 1.5.

<table border="1" align="center"><tr><td align="center">
<table border="0" align="center" cellpadding="0">
<tr>
<td align="center" rowspan="2">
<b>EFE Diagram</b><br />
[[File:OurEFEannotated.png|300px|OurEFE]]
</td>
<td align="center">
 <br /><b>Figure 1</b><br />
[[File:MPT77fiveModified.png|300px|MPT77five]]
</td>
<td align="center">
 <br /><b>Figure 2</b><br />
[[File:MPT77sixModified.png|300px|MPT77six]]
</td>
</tr>
<tr>
<td align="center" colspan="2">
The multicolor curve that appears here in Figures 1 and 2 also appears as a solid black curve in, respectively,
<br />Fig. 5 (p. 594) and Fig. 4 (p. 593) of {{ MPT77 }}
</td>
</tr>
</table>

</td></tr></table>

{{ MPT77hereafter }} also evaluate the normalized total energy, <math>E_\mathrm{tot}/|E_0|</math>, of each of their constructed equilibrium configurations, where
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>E_\mathrm{tot}</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
T_\mathrm{rot} + W_\mathrm{grav} \, ,
</math>
</td>
</tr>
</table>
and, according to the caption of their Figure 4, <math>E_0</math> is <font color="darkgreen">"… the energy of a nonrotating sphere of equal mass and volume."</font> Drawing from our [[Apps/MaclaurinSpheroidSequence#EnergyNorm|separate discussion of the Maclaurin spheroid sequence]], it would be reasonable to assume that the energy normalization adopted by {{ MPT77hereafter }} is the same as the normalization used by [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], namely,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>E_\mathrm{T78}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\biggl(\frac{4\pi}{3}\biggr)^{1 / 3}G (M^5 \rho)^{1 / 3} \, .
</math>
</td>
</tr>
</table>
For models along the Maclaurin spheroid sequence, this normalization leads to expressions for the two key energy terms of the form,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{T_\mathrm{rot}}{E_\mathrm{T78}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{2\cdot 5}
\biggl[ (3 - 2e^2)\frac{\sin^{-1}e}{e} - 3(1-e^2)^{1 / 2} \biggr] \frac{(1-e^2)^{1 / 6}}{e^2}
\, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{T78}}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5}(1-e^2)^{1 / 6} \cdot \frac{\sin^{-1}e }{e} \, ;</math>
</td>
</tr>
</table>
in which case, in the limit of a nonrotating sphere,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\lim_{e\rightarrow 0}\biggl[ \frac{T_\mathrm{rot} + W_\mathrm{grav}}{E_\mathrm{T78}}\biggr]</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{3}{5} \, .
</math>
</td>
</tr>
</table>
But in Figure 4 of {{ MPT77hereafter }}, the point along the Maclaurin spheroid sequence — the solid, black curve — that represents a nonrotating <math>(L_* = 0)</math> sphere has a normalized energy, <math>(E_\mathrm{tot}/E_0) = -1.</math> We conclude, therefore, that the normalization adopted by {{ MPT77hereafter }} is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>E_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\tfrac{3}{5}E_\mathrm{T78} \, .
</math>
</td>
</tr>
</table>
Our Figure 2 (immediately above) attempts to quantitatively replicate the behavior of the Maclaurin spheroid sequence that is shown in Figure 4 (p. 213) of {{ MPT77hereafter }}; the ordinate depicts, on a base-10 logarithmic scale, how the total energy varies with the spheroid's angular momentum over the range, <math>0 \le L_* \le 1.50</math>. More specifically, for eccentricities over the range, <math>0 \le e \le 0.99998967881</math>, the corresponding value of the spheroid's normalized angular momentum is obtained from the [[#L*|above expression for]] <math>L_*</math>, and the normalized energy is given by the relation,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\frac{E_\mathrm{tot}}{E_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{3}\biggl[ \frac{T_\mathrm{rot}}{E_\mathrm{T78}} + \frac{W_\mathrm{grav}}{E_\mathrm{T78}}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2}
\biggl[ (3 - 2e^2)\frac{\sin^{-1}e}{e} - 3(1-e^2)^{1 / 2} \biggr] \frac{(1-e^2)^{1 / 6}}{e^2}
-
(1-e^2)^{1 / 6} \cdot \frac{\sin^{-1}e }{e}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(1-e^2)^{1 / 6}}{2e^2}\biggl\{
\biggl[ (3 - 2e^2)\frac{\sin^{-1}e}{e} - 3(1-e^2)^{1 / 2} \biggr]
-
\frac{2e^2\sin^{-1}e }{e}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{(1-e^2)^{1 / 6}}{2e^2}
\biggl[ (3 - 4e^2)\frac{\sin^{-1}e}{e}
- 3(1-e^2)^{1 / 2} \biggr]
\, .
</math>
</td>
</tr>
</table>

==Constructed Maclaurin Toroid Models==

{{ MPT77 }} did not create a tabulated description of the models that they constructed along their so-called "Maclaurin Toroid" sequence. Throughout their paper, however, they highlighted some properties of a selected group of equilibrium models. Column (2) of Table 1, immediately below, provides a list of the values of the normalized angular momentum, <math>L_*</math>, that corresponds to the Maclaurin Toroid models that have been explicitly referenced in their discussion.

<table border="1" align="center" cellpadding="5" width="75%">
<tr>
<td align="center" colspan="6">'''Table 1'''</td>
</tr>

<tr>
<td align="center" rowspan="2">Model</td>
<td align="center" rowspan="2"><math>L_*</math></td>
<td align="center" rowspan="1" colspan="3">Spheroid Equivalent</td>
<td align="left" rowspan="3">Notes …</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>e</math></td>
<td align="center" rowspan="1" colspan="1"><math>\tau</math></td>
<td align="center" rowspan="1" colspan="1"><math>\frac{E_\mathrm{tot}}{E_0} = \tfrac{5}{3}(T+W)/E_\mathrm{T78}</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">(1)</td>
<td align="center" rowspan="1" colspan="1">(2)</td>
<td align="center" rowspan="1" colspan="1">(3)</td>
<td align="center" rowspan="1" colspan="1">(4)</td>
<td align="center" rowspan="1" colspan="1">(5)</td>
</tr>

<tr>
<td align="center"><math>L^-</math></td>
<td align="center"><math>\sim 0.775</math></td>
<td align="center"><math>\sim 0.99409</math></td>
<td align="center"><math>\sim 0.40585</math></td>
<td align="center"><math>-0.41685</math></td>
<td align="center">(a)</td>
</tr>

<tr>
<td align="center"><math>L_0</math></td>
<td align="center"><math>0.792 \pm 0.002</math></td>
<td align="center"><math>0.9949 \pm 0.0001</math></td>
<td align="center"><math>0.41195</math></td>
<td align="center"><math>-0.40439</math></td>
<td align="center">(b)</td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>0.8732</math></td>
<td align="center"><math>0.9975</math></td>
<td align="center"><math>0.4367</math></td>
<td align="center"><math>-0.35023</math></td>
<td align="center">(c)</td>
</tr>

<tr>
<td align="center"><math>L^+</math></td>
<td align="center"><math>0.965175</math></td>
<td align="center"><math>0.99892</math></td>
<td align="center"><math>0.45747</math></td>
<td align="center"><math>-0.29762</math></td>
<td align="center">(d)</td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>0.9852</math></td>
<td align="center"><math>0.99910</math></td>
<td align="center"><math>0.46104</math></td>
<td align="center"><math>-0.28754</math></td>
<td align="center" rowspan="4">(e)</td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>1.0731</math></td>
<td align="center"><math>0.99960</math></td>
<td align="center"><math>0.47369</math></td>
<td align="center"><math>-0.24745</math></td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>1.1489</math></td>
<td align="center"><math>0.99980</math></td>
<td align="center"><math>0.48125</math></td>
<td align="center"><math>-0.21841</math></td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>1.2262</math></td>
<td align="center"><math>0.9998999</math></td>
<td align="center"><math>0.486665</math></td>
<td align="center"><math>-0.19329</math></td>
</tr>

<tr>
<td align="left" colspan="6">
'''Notes:'''
<ol type="a">
<li>Toroid does not exist.</li>
<li>Total energy of toroid is same as the total energy of Maclaurin spheroid with same <math>L_*</math>.</li>
<li>Marginally stable Maclaurin spheroid and associated toroid; see {{ MPT77hereafter}}'s Figure 2 (p. 592). Also, one (of five) meridional cross-sections displayed in {{ MPT77hereafter}}'s Figure 3 (p. 593).</li>
<li>''Analytically known'' (!) onset of dynamical instability along Maclaurin spheroid sequence; see § 33 of EFE and the last row of Table B.1 from {{ Bardeen71 }}.</li>
<li>Four (of five) meridional cross-sections displayed in {{ MPT77hereafter}}'s Figure 3 (p. 593).</li>
</ol>
</td>
</tr>
</table>

Using a simple iterative technique, we have determined the value of <math>e</math> that corresponds to each tabulated <math>L_*</math> if we assume that the referenced model is a ''Maclaurin spheroid'', not a toroid; the value that corresponds to each spheroid's eccentricity is listed in column (3) of the table. In turn, columns (4) and (5) list the values of <math>\tau</math> and <math>E_\mathrm{tot}/E_0</math> that is associated with a Maclaurin spheroid that has the stated eccentricity. If — using the coordinate pair, <math>(L_*, \tau)</math> — we were to position any one of these models into our Figure 1, above, the model would fall on the solid black portion of the "Maclaurin spheroid sequence." Similarly, if — using the coordinate pair, <math>(L_*, E_\mathrm{tot}/E_0)</math> — we were to position any one of these models into our Figure 2, above, the model would fall on the solid black portion of the "Maclaurin spheroid sequence."

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="6">'''Table 2'''<br /><hr />Data extracted via pencil & ruler measurement from Figs. 4 and 5 of …<br />{{ MPT77figure }}

</td>
</tr>

<tr>
<td align="center" rowspan="2">Model</td>
<td align="center" rowspan="2"><math>L_*</math></td>
<td align="center" rowspan="1" colspan="2">"Measured" <font color="red">Spheroid</font> Properties</td>
<td align="center" rowspan="1" colspan="2">"Measured" <font color="red">Toroid</font> Properties</td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1"><math>\tau</math></td>
<td align="center" rowspan="1" colspan="1"><math>\frac{E_\mathrm{tot}}{E_0}</math></td>
<td align="center" rowspan="1" colspan="1"><math>\tau</math></td>
<td align="center" rowspan="1" colspan="1"><math>\frac{E_\mathrm{tot}}{E_0}</math></td>
</tr>

<tr>
<td align="center" rowspan="1" colspan="1">(1)</td>
<td align="center" rowspan="1" colspan="1">(2)</td>
<td align="center" rowspan="1" colspan="1">(3)</td>
<td align="center" rowspan="1" colspan="1">(4)</td>
<td align="center" rowspan="1" colspan="1">(5)</td>
<td align="center" rowspan="1" colspan="1">(6)</td>
</tr>

<tr>
<td align="center"><math>L^-</math></td>
<td align="center"><math>\sim 0.775</math></td>
<td align="center"><math>\sim 0.4098</math></td>
<td align="center">---</td>
<td align="center"><math>0.3784</math></td>
<td align="center">---</td>
</tr>

<tr>
<td align="center"><math>L_0</math></td>
<td align="center"><math>0.792 \pm 0.002</math></td>
<td align="center"><math>0.4196</math></td>
<td align="center"><math>-0.4464</math></td>
<td align="center"><math>0.3745</math></td>
<td align="center"><math>- 0.4464</math></td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>0.8732</math></td>
<td align="center"><math>0.4431</math></td>
<td align="center"><math>-0.3670</math></td>
<td align="center"><math>0.3667</math></td>
<td align="center"><math>-0.3819</math></td>
</tr>

<tr>
<td align="center"><math>L^+</math></td>
<td align="center"><math>0.965175</math></td>
<td align="center"><math>0.4620</math></td>
<td align="center"><math>-0.3069</math></td>
<td align="center"><math>0.3627</math></td>
<td align="center"><math>-0.3441</math></td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>0.9852</math></td>
<td align="center">---</td>
<td align="center"><math>-0.2944</math></td>
<td align="center"><math>0.3623</math></td>
<td align="center"><math>-0.3388</math></td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>1.0731</math></td>
<td align="center"><math>0.4725</math></td>
<td align="center"><math>-0.2565</math></td>
<td align="center"><math>0.3608</math></td>
<td align="center"><math>-0.3061</math></td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>1.1489</math></td>
<td align="center"><math>0.4804</math></td>
<td align="center"><math>-0.2287</math></td>
<td align="center"><math>0.3588</math></td>
<td align="center"><math>-0.2795</math></td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>1.2262</math></td>
<td align="center"><math>0.4882</math></td>
<td align="center"><math>-0.2019</math></td>
<td align="center"><math>0.3573</math></td>
<td align="center"><math>-0.2585</math></td>
</tr>

<tr>
<td align="center"> </td>
<td align="center"><math>1.5</math></td>
<td align="center">---</td>
<td align="center"><math>-0.1366</math></td>
<td align="center"><math>0.3529</math></td>
<td align="center"><math>-0.1983</math></td>
</tr>
</table>

<table border="1" align="center"><tr><td align="center">
<table border="0" align="center" cellpadding="0">
<tr>
<td align="center">
 <br /><b>Figure 3</b><br />
[[File:MPT77seven.png|400px|MPT77seven]]
</td>
<td align="center">
 <br /><b>Figure 4</b><br />
[[File:MPT77eight.png|400px|MPT77eight]]
</td>
</tr>
<tr>
<td align="center">
Compare to:<br />Fig. 5 (p. 594) of {{ MPT77 }}
</td>
<td align="center">
Compare to:<br />Fig. 4 (p. 593) of {{ MPT77 }}
</td>
</tr>
</table>

</td></tr></table>

=Maclaurin Toroid (EH85)=

{{ EH85full }} — hereafter, {{ EH85hereafter }} — have constructed a set of uniform-density, axisymmetric configurations that show how the Maclaurin toroid sequence is connected to the Maclaurin spheroid sequence. The following table displays the structural characteristics of these configurations; the numbers in the first four columns have been drawn directly from Table 1 of {{ EH85hereafter }}. The quantity, <math>E_\mathrm{EH85}</math>, that has been used to normalize the total energy in, for example, the fourth column of this table, is given by the expression,
<table border="0" align="center" cellpadding="3">
<tr>
<td align="right"><math>E_\mathrm{EH85}</math></td>
<td align="center"><math>\equiv</math></td>
<td align="left"><math>(4\pi G)^2 M^5/J^2 \, .</math></td>
</tr>
<tr>
<td align="center" colspan="3">
{{ EH85 }}, §2.2, p. 291, Eq. (7)
</td>
</tr>
</table>
For purposes of comparison between the separate published works of {{ MPT77hereafter }} and {{ EH85hereafter }}, here we desire to shift back to the normalization adopted by {{ MPT77hereafter }}, namely,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>E_0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{5}E_\mathrm{T78} \, .
</math>
</td>
</tr>
</table>

Building on our [[Apps/MaclaurinSpheroidSequence#EnergyNorm|separate discussion of energy normalizations]] where we showed that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{E_\mathrm{EH85}}{E_\mathrm{T78}} \biggr]^3</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3(4\pi)^2}{j^6} \, ,
</math>
</td>
</tr>
</table>
we recognize immediately that,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>\biggl[ \frac{E_\mathrm{EH85}}{E_0} \biggr] = \biggl[ \frac{E_\mathrm{EH85}}{E_\mathrm{T78}} \biggr] \cdot \frac{E_\mathrm{T78}}{E_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{5}{3} \biggl[ \frac{3(4\pi)^2}{j^6} \biggr]^{1 / 3}
=
\frac{5(4\pi/3)^{2 / 3}}{j^2}
\, .
</math>
</td>
</tr>
</table>

We have evaluated this conversion factor and the consequential normalized total energy for each of the {{ EH85hereafter }} equilibrium configurations and have presented the results in columns six and seven, respectively, of the following table.

<table border="1" align="center" cellpadding="8">

<tr>
<td align="center" colspan="4">
Data extracted from Table 1 (p. 290) of …<br />{{ EH85figure }}
</td>
<td align="center" colspan="3">
Our Determination
</td>
</tr>
<tr>
<td align="center"><math>h_0^2</math></td>
<td align="center"><math>j^2</math></td>
<td align="center"><math>\frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math></td>
<td align="center"><math>\frac{T_\mathrm{rot} + W_\mathrm{grav}}{E_\mathrm{EH85}}</math></td>
<td align="center"><math>L_* \equiv (4\pi/3)^{2 / 3} (3j^2)^{1 / 2}</math></td>
<td align="center"><math>\frac{E_\mathrm{EH85}}{E_0} = \frac{5(4\pi/3)^{2 / 3}}{j^2}</math></td>
<td align="center"><math>\frac{T_\mathrm{rot} + W_\mathrm{grav}}{E_0}</math></td>

</tr>

<tr>
<td align="center"><math>5.802\times 10^{-2}</math></td>
<td align="center"><math>3.964\times 10^{-2}</math></td>
<td align="center"><math>0.445</math></td>
<td align="center"><math>- 1.03\times 10^{-3}</math></td>
<td align="center"><math>0.8961</math></td>
<td align="center"><math>327.76</math></td>
<td align="center"><math>-0.3376</math></td>
</tr>

<tr>
<td align="center"><math>5.834\times 10^{-2}</math></td>
<td align="center"><math>3.816\times 10^{-2}</math></td>
<td align="center"><math>0.439</math></td>
<td align="center"><math>- 1.02\times 10^{-3}</math></td>
<td align="center"><math>0.8792</math></td>
<td align="center"><math>340.48</math></td>
<td align="center"><math>-0.3473</math></td>
</tr>

<tr>
<td align="center"><math>5.916\times 10^{-2}</math></td>
<td align="center"><math>3.752\times 10^{-2}</math></td>
<td align="center"><math>0.437</math></td>
<td align="center"><math>- 1.02\times 10^{-3}</math></td>
<td align="center"><math>0.8718</math></td>
<td align="center"><math>346.28</math></td>
<td align="center"><math>-0.3532</math></td>
</tr>

<tr>
<td align="center"><math>6.075\times 10^{-2}</math></td>
<td align="center"><math>3.718\times 10^{-2}</math></td>
<td align="center"><math>0.436</math></td>
<td align="center"><math>- 1.02\times 10^{-3}</math></td>
<td align="center"><math>0.8678</math></td>
<td align="center"><math>349.45</math></td>
<td align="center"><math>-0.3564</math></td>
</tr>

<tr>
<td align="center"><math>6.416\times 10^{-2}</math></td>
<td align="center"><math>3.209\times 10^{-2}</math></td>
<td align="center"><math>0.412</math></td>
<td align="center"><math>- 9.73\times 10^{-4}</math></td>
<td align="center"><math>0.8063</math></td>
<td align="center"><math>404.89</math></td>
<td align="center"><math>-0.3939</math></td>
</tr>

<tr>
<td align="center"><math>6.766\times 10^{-2}</math></td>
<td align="center"><math>3.090\times 10^{-2}</math></td>
<td align="center"><math>0.403</math></td>
<td align="center"><math>- 9.61\times 10^{-4}</math></td>
<td align="center"><math>0.7912</math></td>
<td align="center"><math>420.47</math></td>
<td align="center"><math>-0.4041</math></td>
</tr>

<tr>
<td align="center"><math>7.070\times 10^{-2}</math></td>
<td align="center"><math>3.016\times 10^{-2}</math></td>
<td align="center"><math>0.389</math></td>
<td align="center"><math>- 9.53\times 10^{-4}</math></td>
<td align="center"><math>0.7816</math></td>
<td align="center"><math>430.79</math></td>
<td align="center"><math>-0.4105</math></td>
</tr>

<tr>
<td align="center"><math>6.739\times 10^{-2}</math></td>
<td align="center"><math>3.192\times 10^{-2}</math></td>
<td align="center"><math>0.376</math></td>
<td align="center"><math>- 9.74\times 10^{-4}</math></td>
<td align="center"><math>0.8041</math></td>
<td align="center"><math>407.04</math></td>
<td align="center"><math>-0.3965</math></td>
</tr>

<tr>
<td align="center"><math>5.376\times 10^{-2}</math></td>
<td align="center"><math>3.751\times 10^{-2}</math></td>
<td align="center"><math>0.368</math></td>
<td align="center"><math>- 1.04\times 10^{-3}</math></td>
<td align="center"><math>0.8717</math></td>
<td align="center"><math>346.38</math></td>
<td align="center"><math>-0.3602</math></td>
</tr>

<tr>
<td align="center"><math>4.007\times 10^{-2}</math></td>
<td align="center"><math>4.502\times 10^{-2}</math></td>
<td align="center"><math>0.365</math></td>
<td align="center"><math>- 1.12\times 10^{-3}</math></td>
<td align="center"><math>0.9550</math></td>
<td align="center"><math>288.60</math></td>
<td align="center"><math>-0.3232</math></td>
</tr>

<tr>
<td align="center"><math>3.202\times 10^{-2}</math></td>
<td align="center"><math>5.100\times 10^{-2}</math></td>
<td align="center"><math>0.366</math></td>
<td align="center"><math>- 1.18\times 10^{-3}</math></td>
<td align="center"><math>1.0164</math></td>
<td align="center"><math>254.76</math></td>
<td align="center"><math>-0.3006</math></td>
</tr>

<tr>
<td align="center"><math>2.525\times 10^{-2}</math></td>
<td align="center"><math>5.975\times 10^{-2}</math></td>
<td align="center"><math>0.369</math></td>
<td align="center"><math>- 1.26\times 10^{-3}</math></td>
<td align="center"><math>1.1002</math></td>
<td align="center"><math>217.45</math></td>
<td align="center"><math>-0.2740</math></td>
</tr>

<tr>
<td align="center"><math>1.811\times 10^{-2}</math></td>
<td align="center"><math>7.364\times 10^{-2}</math></td>
<td align="center"><math>0.378</math></td>
<td align="center"><math>- 1.37\times 10^{-3}</math></td>
<td align="center"><math>1.2214</math></td>
<td align="center"><math>176.43</math></td>
<td align="center"><math>-0.2417</math></td>
</tr>

<tr>
<td align="center"><math>1.127\times 10^{-2}</math></td>
<td align="center"><math>9.914\times 10^{-2}</math></td>
<td align="center"><math>0.396</math></td>
<td align="center"><math>- 1.53\times 10^{-3}</math></td>
<td align="center"><math>1.4171</math></td>
<td align="center"><math>131.05</math></td>
<td align="center"><math>-0.2005</math></td>
</tr>

<tr>
<td align="center"><math>4.812\times 10^{-3}</math></td>
<td align="center"><math>1.618\times 10^{-1}</math></td>
<td align="center"><math>0.430</math></td>
<td align="center"><math>- 1.76\times 10^{-3}</math></td>
<td align="center"><math>1.8104</math></td>
<td align="center"><math>80.300</math></td>
<td align="center"><math>-0.1413</math></td>
</tr>
</table>

<table border="1" align="center"><tr><td align="center">
<table border="0" align="center" cellpadding="0">
<tr>
<td align="center">
 <br /><b>Figure 5</b><br />
[[File:EH85nine.png|400px|EH85nine]]
</td>
<td align="center">
 <br /><b>Figure 6</b><br />
[[File:EH85ten.png|400px|EH85ten]]
</td>
</tr>
</table>

</td></tr></table>

=Relationship with the Dyson-Wong One-Ring Sequence=

{{ CKST95bfull }} discuss how to interpret the physical meaning of the Maclaurin Toroid sequence, especially in the context of its relationship to the Maclaurin spheroid sequence and to the Dyson-Wong one-ring sequence. In this discussion, reference is made to the work of {{ HTE87full }}.

=See Also=

{{ SGFfooter }}

File:RoverD5over2.png

2024-06-28T21:55:31Z

Xxsrm:

File:DysonCompare01.png

2024-06-28T21:54:35Z

Xxsrm:

Appendix/Ramblings/ToroidalCoordinates

2024-06-28T21:48:52Z

Xxsrm: /* Multipole Moment in Toroidal Coordinates */

__FORCETOC__ 

=Toroidal Configurations and Related Coordinate Systems=

This rather long and rambling chapter reveals how my train of thought progressed as I effectively taught myself how to use a toroidal coordinate system to determine the gravitational potential of an axisymmetric configuration. A summary of the key results from this chapter can be found in a primary chapter of this H_Book titled, [[2DStructure/ToroidalCoordinates|Using Toroidal Coordinates to Determine the Gravitational Potential]].

==Preamble==
As I have studied the structure and analyzed the stability of (both self-gravitating and non-self-gravitating) toroidal configurations over the years, I have often wondered whether it might be useful to examine such systems mathematically using a toroidal — or at least a toroidal-like — coordinate system. Is it possible, for example, to build an equilibrium torus for which the density distribution is one-dimensional as viewed from a well-chosen toroidal-like system of coordinates?

I should begin by clarifying my terminology. In volume II (p. 666) of their treatise on ''Methods of Theoretical Physics'', Morse & Feshbach (1953; hereafter MF53) define an orthogonal toroidal coordinate system in which the Laplacian is separable.<sup>1</sup> (See details, below.) It is only this system that I will refer to as ''the'' toroidal coordinate system; all other functions that trace out toroidal surfaces but that don't conform precisely to Morse & Feshbach's coordinate system will be referred to as ''toroidal-like.''

I became particularly interested in this idea while working with Howard Cohl (when he was an LSU graduate student). Howie's dissertation research uncovered a ''Compact Cylindrical Greens Function'' technique for evaluating Newtonian potentials of rotationally flattened (especially axisymmetric) configurations.<sup>2,3</sup> The technique involves a multipole expansion in terms of half-integer-degree Legendre functions of the <math>2^\mathrm{nd}</math> kind — see [http://dlmf.nist.gov/14.19 NIST digital library discussion] — where, if I recall correctly, the argument of this special function (or its inverse) seemed to resemble the ''radial'' coordinate of Morse & Feshbach's orthogonal toroidal coordinate system — see more on this, [[#Relating_CCGF_Expansion_to_Toroidal_Coordinates|below]].

==Off-center Circle==
In what follows, it will be useful to recall the algebraic expression that defines a circle whose center is not positioned at the origin of a Cartesian coordinate system. Specifically, consider a circle of radius, <math>~r_c</math>, whose center is located a distance <math>~x_0</math> along the plus-x axis and a distance <math>~y_0</math> along the plus-y axis. The equation for this circle is,
<div align="center">
<math>
~(x-x_0)^2 + (y-y_0)^2 = r_c^2 .
</math>
</div>

==Toroidal Coordinates==
===Presentation by MF53===
The orthogonal toroidal coordinate system <math>(\xi_1,\xi_2,\xi_3=\cos\varphi)</math> discussed by MF53 has the following properties:
<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
~\frac{x}{a}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]\cos\varphi \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{y}{a}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]\sin\varphi \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{z}{a}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{\varpi}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 \biggr]^{1/2}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{r}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 + \biggl(\frac{z}{a}\biggr)^2\biggr]^{1/2}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} \, .
</math>
</td>
</tr>
</table>

<span id="ToroidalScaleFactors">According to MF53, the associated scale factors of this orthogonal coordinate system are:</span>
<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
~\frac{h_1}{a}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)(\xi_1^2 - 1)^{1/2}} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{h_2}{a}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)(1-\xi_2^2)^{1/2}} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{h_3}{a}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]\frac{1}{\sin\varphi} \, .
</math>
</td>
</tr>
</table>

That means that, in the meridional plane, an area element should be,
<div align="center">
<math>
~d\sigma = (h_1 d\xi_1)(h_2 d\xi_2) = a^2 \biggl[ \frac{d\xi_1}{(\xi_1 - \xi_2)(\xi_1^2 - 1)^{1/2}} \biggr]
\biggl[ \frac{d\xi_2}{(\xi_1 - \xi_2)(1-\xi_2^2)^{1/2}} \biggr] \, .
</math>
</div>

===Tohline's Ramblings===
My inversion of these coordinate definitions has led to the following expressions:
<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
~\xi_1
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
\frac{r(r^2 + 1)} {[\chi^2(r^2-1)^2 + \zeta^2(r^2+1)^2]^{1/2}} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\xi_2
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
\frac{r(r^2 - 1)}{[\chi^2(r^2-1)^2 + \zeta^2(r^2+1)^2]^{1/2}} \, ,
</math>
</td>
</tr>
</table>
where,
<div align="center">
<math>
\chi \equiv \frac{\varpi}{a} ~~~;~~~\zeta\equiv\frac{z}{a} ~~~\mathrm{and} ~~~ r=(\chi^2 + \zeta^2)^{1/2} .
</math>
</div>

Apparently the allowed ranges of the two meridional-plane coordinates are:
<div align="center">
<math>
+1 \leq \xi_1 \leq \infty ~~~\mathrm{and} ~~~ -1 \leq \xi_2 \leq +1 .
</math>
</div>

===Example Toroidal Surfaces===
In the accompanying figure labeled "Toroidal Coordinate System," we've outlined three different <math>~\xi_1 = \mathrm{constant}</math> meridional contours for the MF53 toroidal coordinate system. The illustrated values are,
<table align="center" border="0" cellpadding="4">
<tr>
<td align="right">
<math>
~\xi_1
</math>
</td>
<td align="center>
<math>~=</math>
</td>
<td align="left">
<math>~1.1</math>
</td>
<td align="left" width="25%">
 
</td>
<td align="left">
<math>~\mathrm{(blue)} \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\xi_1
</math>
</td>
<td align="center>
<math>~=</math>
</td>
<td align="left">
<math>~1.2</math>
</td>
<td align="left" width="25%">
 
</td>
<td align="left">
<math>~\mathrm{(red)} \, ;</math>
</td>
</tr>
<tr>
<td align="right">
<math>
~\xi_1
</math>
</td>
<td align="center>
<math>~=</math>
</td>
<td align="left">
<math>~1.5</math>
</td>
<td align="left" width="25%">
 
</td>
<td align="left">
<math>\mathrm{(gold)} \, .</math>
</td>
</tr>
</table>

The inner and outer edges of the toroidal surface in the equatorial plane should be determined by setting <math>~\xi_2 = -1</math> (inner) and <math>~\xi_2 = +1</math> (outer). Hence,
<table align="center" border="0" cellpadding="4">
<tr>
<td align="right">
<math>
~\chi_\mathrm{inner}
</math>
</td>
<td align="center>
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 +1} = \biggl[\frac{(\xi_1 - 1)}{(\xi_1 + 1)} \biggr]^{1/2}
</math>

</td>
</tr>

<tr>
<td align="right">
<math>
~\chi_\mathrm{outer}
</math>
</td>
<td align="center>
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - 1} = \biggl[\frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2}
</math>

</td>
</tr>
</table>
The equatorial-plane location of the "center" of each torus is,
<div align="center">
<math>
\chi_0 = \frac{1}{2} (\chi_\mathrm{outer} + \chi_\mathrm{inner}) = \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} ,
</math>
</div>
and the so-called distortion parameter,
<div align="center">
<math>
\delta \equiv \frac{\chi_\mathrm{outer}-\chi_\mathrm{inner}}{\chi_0}= \frac{2}{\xi_1} .
</math>
</div>

<table align="center" border="1" cellpadding="8">
<tr>
<th align="center" colspan="6">
<font color="maroon">
Properties of <math>\xi_1 = \mathrm{constant}</math> Toroidal Surfaces
</font>
</th>
</tr>
<tr>
<td align="center">
Curve in<br />Figure
</td>
<td align="center">
<math>\xi_1</math>
</td>
<td align="center">
<math>\chi_\mathrm{inner}</math>
</td>
<td align="center">
<math>\chi_\mathrm{outer}</math>
</td>
<td align="center">
<math>\chi_0</math>
</td>
<td align="center">
<math>\delta</math>
</td>
</tr>

<tr>
<td align="center">
Blue
</td>
<td align="center">
1.1
</td>
<td align="center">
0.218
</td>
<td align="center">
4.583
</td>
<td align="center">
2.400
</td>
<td align="center">
1.818
</td>
</tr>

<tr>
<td align="center">
Red
</td>
<td align="center">
1.2
</td>
<td align="center">
0.302
</td>
<td align="center">
3.317
</td>
<td align="center">
1.809
</td>
<td align="center">
1.667
</td>
</tr>

<tr>
<td align="center">
Gold
</td>
<td align="center">
1.5
</td>
<td align="center">
0.447
</td>
<td align="center">
2.236
</td>
<td align="center">
1.342
</td>
<td align="center">
1.333
</td>
</tr>
</table>

What function <math>~\zeta(\varpi)</math> coincides with these <math>~\xi_1 = \mathrm{constant}</math> surfaces? (To be answered!)

<div align="center">
[[File:LSU_CombinedTori.jpg|none|800px|Meridional contours of constant <math>\xi_1</math>.]]
</div>

===Off-center Circle===
The curves drawn in the above figure labeled "Toroidal Coordinate System" resemble circles whose centers are positioned a distance <math>~\chi_0</math> away from the origin. Let's examine whether this is the case by drawing on the familiar expression for such a configuration, [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle|as presented above]]. If this is the case, then the circle as illustrated in the figure will have <math>~z_0 = 0</math> and a radius,

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>
~\alpha_c \equiv \frac{r_c}{a}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\chi_\mathrm{outer} - \chi_0
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\biggl[\frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2} - \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{1}{(\xi_1^2 - 1)^{1/2}} \, ,
</math>
</td>
</tr>
</table>

and the algebraic expression describing the circle will take the form,

<div align="center">
<math>
~(\chi - \chi_0)^2 + \zeta^2 = \alpha_c^2 = (\xi_1^2 - 1)^{-1} .
</math>
</div>

Let's evaluate the left-hand-side of this expression to see if it indeed reduces to <math>(\xi_1^2 - 1)^{-1}</math>.

<table align="center" border="0" cellpadding="10">
<tr>
<td align="right">
<math>~\mathrm{LHS}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\biggl\{ \biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2}
\biggr] - \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr\}^2 + \biggl[ \frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} \biggr]^2
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)} \biggl\{ (\xi_1^2 - 1) - \xi_1 (\xi_1-\xi_2) \biggr\}^2 + \frac{(1-\xi_2^2)}{(\xi_1 - \xi_2)^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{(\xi_1 \xi_2 - 1)^2}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)} + \frac{(1-\xi_2^2)}{(\xi_1 - \xi_2)^2}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)} \biggl[(\xi_1 \xi_2 - 1)^2 + (\xi_1^2 - 1)(1-\xi_2^2) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{1}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)} \biggl[ \xi_1^2 \xi_2^2 - 2\xi_1 \xi_2 + 1 ) + (\xi_1^2 - 1 -\xi_1^2 \xi_2^2 + \xi_2^2) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{\xi_1^2 - 2\xi_1 \xi_2 + \xi_2^2}{(\xi_1 - \xi_2)^2 (\xi_1^2 - 1)}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>
~\frac{1}{(\xi_1^2 - 1)} \, .
</math>
</td>
</tr>

</table>

Yes! So this means that the <math>~\xi_1 = \mathrm{constant}</math> toroidal contours can be described by the off-center circle expression,

<div align="center">
<math>
~(\chi - \chi_0)^2 + \zeta^2 = (\chi_\mathrm{outer} - \chi_0)^2 \, ,
</math>
</div>
or,

<div align="center">
<math>
~\biggl[ \chi - \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^2 + \zeta^2 = \frac{1}{(\xi_1^2 - 1)} \, .
</math>
</div>

It also means that, while <math>~\xi_1</math> is the official ''radial'' coordinate of MF53's toroidal coordinate system, the actual dimensionless radius of the relevant cross-sectional circle is,

<table align="center" border="0" cellpadding="3">
<tr>
<td align="right">
<math>
~\alpha_c
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{1}{(\xi_1^2 - 1)^{1/2}} .
</math>
</td>
</tr>
</table>

==Papaloizou-Pringle Tori==
===Summary of Structure===
As derived [[Apps/PapaloizouPringleTori|elsewhere]], the accretion tori constructed by Papaloizou & Pringle (1984; hereafter PP84) have the following surface properties. For a given choice of the dimensionless Bernoulli constant, <math>C_\mathrm{B}^'</math>,

<div align="center">
<table border="0" cellpadding="3" align="center">
<tr>
<td align="right">
<math>
~\chi_\mathrm{inner}
</math>
</td>
<td align="center">
<math>~ =</math>
</td>
<td align="left"><math>
~\frac{1}{1 + \sqrt{1-2C_\mathrm{B}^'}} \, ;
</math>
</td>
</tr>
<tr>
<td align="right">
<math>
~\chi_\mathrm{outer}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left"><math>
~\frac{1}{1 - \sqrt{1-2C_\mathrm{B}^'}} \, ;
</math>
</td>
</tr>
<tr>
<td align="right">
<math>
~\chi_0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left"><math>
~\frac{1}{2} (\chi_\mathrm{outer} + \chi_\mathrm{inner}) = \frac{1}{2C_\mathrm{B}^'} \, ;
</math>
</td>
</tr>
<tr>
<td align="right">
<math>
~\delta
</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td algin="left">
<math>
~\frac{\chi_\mathrm{outer} - \chi_\mathrm{inner}}{\chi_0} = 2\sqrt{1-2C_\mathrm{B}^'} \, .
</math>
</td>
</tr>
</table>
</div>

So if I want to construct PP84 tori that are approximately the same size/shape as the MF53 tori illustrated above, I should choose values of the dimensionless Bernoulli constant as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
<math>
~{\chi_0}\biggr|_\mathrm{PP84}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left"><math>
~{\chi_0}\biggr|_\mathrm{MF53}
</math>
</td>
</tr>
<tr>
<td align="right">
<math>
\Rightarrow~~~~ \frac{1}{2C_\mathrm{B}^'}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\xi_1}{(\xi_1^2-1)^{1/2}}
</math>
</td>
</tr>
<tr>
<td align="right">
<math>
\Rightarrow~~~~ C_\mathrm{B}^'
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{(\xi_1^2-1)^{1/2}}{2\xi_1} \, .
</math>
</td>
</tr>
</table>
</div>

In the accompanying figure labeled "Papaloizou-Pringle Tori," we've drawn three different <math>~C_\mathrm{B}^' = \mathrm{constant}</math> meridional contours for the PP84 tori where the values of the dimensionless Bernoulli constants have been chosen to produce values of <math>~\chi_0</math> that are identical to the values displayed by the three MF53 tori shown above. The following table details properties of these three PP84 tori that have been constructed in an effort to facilitate comparison with the table shown above for MF53 tori.

<table align="center" border="1" cellpadding="8">
<tr>
<th align="center" colspan="6">
<font color="maroon">
Properties of <math>C_\mathrm{B}^' = \mathrm{constant}</math> PP84 Toroidal Surfaces
</font>
</th>
</tr>
<tr>
<td align="center">
Curve in<br />Figure
</td>
<td align="center">
<math>C_\mathrm{B}^'</math>
</td>
<td align="center">
<math>\chi_\mathrm{inner}</math>
</td>
<td align="center">
<math>\chi_\mathrm{outer}</math>
</td>
<td align="center">
<math>\chi_0</math>
</td>
<td align="center">
<math>\delta</math>
</td>
</tr>

<tr>
<td align="center">
Red
</td>
<td align="center">
0.208
</td>
<td align="center">
0.567
</td>
<td align="center">
4.234
</td>
<td align="center">
2.400
</td>
<td align="center">
1.528
</td>
</tr>

<tr>
<td align="center">
Blue
</td>
<td align="center">
0.276
</td>
<td align="center">
0.599
</td>
<td align="center">
3.019
</td>
<td align="center">
1.809
</td>
<td align="center">
1.338
</td>
</tr>

<tr>
<td align="center">
Gold
</td>
<td align="center">
0.373
</td>
<td align="center">
0.665
</td>
<td align="center">
2.019
</td>
<td align="center">
1.342
</td>
<td align="center">
1.009
</td>
</tr>
</table>

===Advantageous Coordinate System===

According to [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] review of the [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] discussion — see his equation (14) — surfaces of constant density can be defined by the coordinate, <math>~\chi_{PP}</math>, where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\tan\chi_{PP}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{z}{ \frac{z^2}{\sqrt{\varpi^2 + z^2}} + \sqrt{\varpi^2 + z^2} - 1}
</math>
</td>
</tr>
</table>
</div>

Indeed, equation (6.6) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] defines the coordinate, <math>~\chi_{PP}</math>, via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\tan\chi_{PP}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{\cos\theta}{ \cos^2\theta + 1 - \varpi_0/r} \, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~z = r\cos\theta</math>
</td>
<td align="center">
        and        
</td>
<td align="left">
<math>~\varpi = r\sin\theta \, .</math>
</td>
</tr>
</table>
</div>

Let's see if these match. Starting from the Kojima expression, we have,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\tan\chi_{PP}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{r\cos\theta}{ \frac{r^2\cos^2\theta}{r} + r - 1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\cos\theta}{ \cos^2\theta + 1 - 1/r} \, .
</math>
</td>
</tr>
</table>
</div>
Hence, they are the same, as long as we appreciate that Kojima assumes all length scales are normalized to <math>~\varpi_0</math>. Let's express this coordinate in terms of the <math>~(\xi_1, \xi_2)</math> toroidal coordinates as defined by MF53, namely,

<table align="center" border="0" cellpadding="5">

<tr>
<td align="right">
<math>
~\frac{z}{a}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{\varpi}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 \biggr]^{1/2}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{r}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 + \biggl(\frac{z}{a}\biggr)^2\biggr]^{1/2}
</math>
</td>
<td align="center">
<math>
~=
</math>
</td>
<td align="left">
<math>
~\biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} \, .
</math>
</td>
</tr>
</table>

Kojima's expression becomes:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\tan\chi_{PP}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~z \biggl[ \frac{z^2}{\sqrt{\varpi^2 + z^2}} + \sqrt{\varpi^2 + z^2} - 1 \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} \biggl\{ \biggl[\frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)^2} \biggr] \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}
+ \biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} - 1 \biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(1-\xi_2^2)^{1/2} \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)} \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}
+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - ( \xi_1 - \xi_2 ) \biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(1-\xi_2^2)^{1/2} \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)^{1/2} (\xi_1 + \xi_2)^{1/2}}
+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - ( \xi_1 - \xi_2 ) \biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(1-\xi_2^2)^{1/2} (\xi_1 - \xi_2)^{1/2} (\xi_1 + \xi_2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1 - \xi_2)(\xi_1 + \xi_2)
- ( \xi_1 - \xi_2 )(\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} \biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(1-\xi_2^2)^{1/2} (\xi_1^2 - \xi_2^2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1^2 - \xi_2^2)
- ( \xi_1 - \xi_2 )(\xi_1^2 - \xi_2^2)^{1/2} \biggr\}^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{(1-\xi_2^2)}{ (\xi_1^2 - \xi_2^2) } \biggr]^{1/2} \biggl\{ \biggl[ \frac{(1-\xi_2^2)}{(\xi_1^2 - \xi_2^2)}\biggr] + 1
- \biggl[ \frac{ ( \xi_1 - \xi_2 )}{(\xi_1 + \xi_2)} \biggr]^{1/2}\biggr\}^{-1} \, .
</math>
</td>
</tr>
</table>
</div>
This does not appear to be very useful or productive!

==CCGF Expansion==
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999; hereafter CT99)] derive an expression for the Newtonian gravitational potential in terms of a ''Compact Cylindrical Green's Function'' expansion. They show, for example, that when expressed in terms of cylindrical coordinates, the axisymmetric potential is,
<div align="center">
<math>
\Phi(R,z) = - \frac{2G}{R^{1/2}} q_0 ,
</math>
</div>
where,
<div align="center">
<math>
q_0 = \int\int (R')^{1/2} \rho(R',z') Q_{-1/2}(\Chi) dR' dz',
</math>
</div>
and the dimensionless argument (the modulus) of the special function, <math>~Q_{-1/2}</math>, is,
<div align="center">
<math>
\Chi \equiv \frac{R^2 + {R'}^2 + (z - z')^2}{2R R'} .
</math>
</div>
Note: Here we are using <math>~\Chi</math> instead of <math>~\chi</math> (as used by CT99) to represent this dimensionless parameter in order to avoid confusion with our use of <math>~\chi</math>, above. Next, following the lead of CT99, we note that according to the Abramowitz & Stegun (1965),
<div align="center">
<math>Q_{-1/2}(\Chi) = \mu K(\mu) \, ,</math>
</div>
where, the function <math>~K(\mu)</math> is the complete elliptical integral of the first kind and, for our particular problem,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mu^2</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~2(1+\Chi)^{-1}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
2\biggl[ 1+\frac{R^2 + {R'}^2 + (z - z')^2}{2R R'} \biggr]^{-1}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\biggl[\frac{4R R'}{(R + {R'})^2 + (z - z')^2} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Hence, we can write,
<div align="center">
<math>
q_0 = \int\int (R')^{1/2} \rho(R',z') \mu K(\mu) dR' dz' \, .
</math>
</div>

===Confirmation Provided by Trova, Huré and Hersant===

In their study of the potential of self-gravitating, axisymmetric discs, [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T A. Trova, J.-M. Huré and F. Hersant (2012; MNRAS, 424, 2635)] write (see their equation 1),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Phi(R,Z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
-2G \int\int \sqrt{\frac{\varpi}{R}} \rho(\varpi, z) k K(k) d\varpi dz \, ,
</math>
</td>
</tr>
</table>
</div>
where, the modulus, <math>~k</math>, of the complete elliptical integral of the first kind is (see their equation 2),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2\sqrt{\varpi R}}{\sqrt{(\varpi + R)^2 + (Z-z)^2}} \, ,
</math>
</td>
</tr>
</table>
</div>
and its relevant domain is, <math>~0 \leq k \leq 1</math>. After associating <math>~z \leftrightarrow Z</math>, <math>~z^' \leftrightarrow z</math>, and <math>~R^' \leftrightarrow \varpi</math>, we see that the modulus, <math>~k</math>, used by Trova et al. (2012), is precisely the same as the argument, <math>~\mu</math>, defined in CT99. Hence, the two expressions for the axisymmetric potential, <math>~\Phi(R,Z)</math>, are identical.

===Recognition as Circle===
If we scale all of the lengths in [http://adsabs.harvard.edu/abs/1999ApJ...527...86C CT99's] expression for <math>~\Chi</math>, to <math>~a</math> and, along the lines of what was done above, define,
<div align="center">
<math>
~\chi \equiv \frac{R'}{a} ~~~~\mathrm{and}~~~~\zeta \equiv \frac{z-z'}{a} ,
</math>
</div>
we can rewrite the expression in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\biggl(\frac{R}{a}\biggr)^2 + \chi^2 + \zeta^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2 \Chi \biggl( \frac{R}{a}\biggr) \chi \, .</math>
</td>
</tr>
</table>
</div>
Now, because,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[ \chi - \Chi\biggl(\frac{R}{a}\biggr) \biggr]^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi^2 - 2\Chi\biggl(\frac{R}{a}\biggr)\chi + \Chi^2 \biggl(\frac{R}{a}\biggr)^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ 2\Chi\biggl(\frac{R}{a}\biggr)\chi </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi^2 - \biggl[ \chi - \Chi\biggl(\frac{R}{a}\biggr) \biggr]^2 + \Chi^2 \biggl(\frac{R}{a}\biggr)^2 \, ,</math>
</td>
</tr>
</table>
</div>
we can further rewrite the expression as,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\biggl(\frac{R}{a}\biggr)^2 + \chi^2 + \zeta^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\chi^2 - \biggl[ \chi - \Chi\biggl(\frac{R}{a}\biggr) \biggr]^2 + \Chi^2 \biggl(\frac{R}{a}\biggr)^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~
\biggl[ \chi - \Chi\biggl(\frac{R}{a}\biggr) \biggr]^2 + \zeta^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{R}{a}\biggr)^2(\Chi^2 -1) \, .</math>
</td>
</tr>
</table>
</div>

Finally, if we adopt the ''specific'' scale factor, <math>~a = R</math>, we have,
<div align="center">
<math>
~(\chi - \Chi)^2 + \zeta^2 = (\Chi^2 - 1) .
</math>
</div>
So, a curve of constant <math>~\Chi</math> produces an off-center circle whose center is located at <math>~\Chi</math> and whose radius is <math>~\sqrt{\Chi^2 - 1}</math>.

===Relating CCGF Expansion to Toroidal Coordinates===
We see that curves of constant <math>\Chi</math> (as defined in CT99) are in every respect identical to curves of constant <math>\xi_1</math> (as defined in MF53). The association is straightforward:
<table align="center" border="2" cellpadding="5" bgcolor="pink">
<tr>
<th align="center">
<font color="red">EUREKA!</font>
</th>
</tr>
<tr>
<td align="center">
<math>
\Chi^2 - 1 = \frac{1}{\xi_1^2 - 1} .
</math>
</td>
</tr>
</table>

==Do It Again==

The question that I've had in the back of my mind for quite some time is, "For what astrophysically interesting problem might we effectively use the toroidal coordinate system in order to derive a much cleaner ''analytic'' description of an axisymmetric potential?" Originally, I thought that a suitable configuration might be a uniform-density torus that has a perfectly circular cross-section. After all, the surface of such a torus can be perfectly described as a <math>~\xi_1 = </math>  constant configuration. In a [[#Multipole_Moment_in_Toroidal_Coordinates|subsection presented below]], I began investigating this problem, setting up a toroidal coordinate system to appropriately conform to the surface of such a torus, then calling upon WolframAlpha's online integration tool to complete the integral over the orthogonal coordinate, <math>~\xi_2</math>, analytically. After giving the problem considerable more thought, however, I realized that, while I could legitimately move the mass-density outside of that first integral, it was not legitimate to move the <math>~\mu K(\mu)</math> factor outside of that integral. While it is true that CT99 showed that the <math>~\mu K(\mu)</math> factor only depends on the first coordinate in a toroidal coordinate system, it is a ''different'' toroidal coordinate system from the one that conveniently aligns with the physical torus! Let's set up the double integral again, but this time let's use the toroidal coordinate system that is defined within the CCGF discussion. We begin by describing geometric relationships between pairs of off-center circles and deriving algebraic expressions that define the conditions under which such circles overlap and/or simply intersect.

===Overlap Between Two Off-Center Circles===

Figure 2 displays two off-center circles. The solid pink circle represents a meridional cross-section through a uniform-density, axisymmetric torus whose center lies in the equatorial plane of a <math>~(\varpi, Z) </math>, cylindrical coordinate system; as depicted, <math>~\varpi_t</math> is the size of the major radius of this torus and its cross-sectional radius is <math>~r_t</math>. The other circle represents a single, <math>~\xi_1</math> = constant (toroidal) surface in toroidal coordinates; its major radius is, <math>~R_0</math>, and its cross-sectional radius is <math>~r_0</math>. The center of this <math>~\xi_1</math> = constant circle lies in the equatorial system of the associated toroidal coordinate system, which is parallel to but, as depicted, lies a distance, <math>~Z_0</math>, above the equatorial plane of the <math>~(\varpi, Z) </math>, cylindrical coordinate system.

As drawn, the figure does not identify the precise location of the ''origin'' of the toroidal coordinate system. But, in accordance with the properties of such coordinate systems, the origin must lie inside of the referenced circle and to the left of — that is, closer to the <math>~Z</math> (symmetry) axis than — the center of the circle, <math>~R_0</math>.

<div align="center" id="Figure2">
<table border="1" cellpadding="8">

<tr>
<th align="center"><font size="+1">Figure 2</font></th>
</tr>

<tr>
<td align="center">
[[File:DiagramToroidalCoordinates.png|350px|Diagram of Torus and Toroidal Coordinates]]
</td>
</tr>
</table>
</div>

If the size of the <math>~\xi_1</math> = constant surface is varied while all the other key parameters <math>~(R_0, Z_0, \varpi_t, r_t)</math> are held fixed, what is the range of values of <math>~r_0</math> over which the two depicted circles overlap and/or simply intersect?

====Initial Contact====
Geometrically we appreciate that, as <math>~r_0</math> is increased, the two circles will first touch at a point that lies along the (blue-dashed) line-segment that connects the centers of both circles. More specifically, the initial interception will be at the point identified in Figure 2 by the solid blue dot lying on the surface of the pink torus. The distance between the two centers — which we will denote as <math>~h</math> — is also the hypotenuse of a right triangle whose other two sides are of length (opposite the angle, <math>~\alpha</math>) <math>~\varpi_t - R_0</math> and (adjacent to the angle, <math>~\alpha</math>) <math>~Z_0</math>. We see that the initial interception will occur when <math>~r_0 + r_t = h</math>, that is, when
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_0 = r_+</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~h - r_t</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[(\varpi_t - R_0)^2 + Z_0^2 ]^{1/2} - r_t </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} - r_t \, ,</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Lambda</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{Z_0}{\varpi_t - R_0} \, .</math>
</td>
</tr>
</table>
</div>
For later reference, we note that the cylindrical coordinates associated with this initial point of contact — ''i.e.,'' the point identified in Figure 2 by the solid blue dot lying on the surface of the pink torus — are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\varpi_+</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\varpi_t - r_t \sin\alpha</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\varpi_t - \frac{r_t (\varpi_t-R_0)}{[(\varpi_t - R_0)^2 + Z_0^2 ]^{1/2} }</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\varpi_t - \frac{r_t }{[1+\Lambda^2 ]^{1/2} } \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~Z_+</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r_t \cos\alpha</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{r_t Z_0}{[(\varpi_t - R_0)^2 + Z_0^2 ]^{1/2} }</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{r_t \Lambda}{[1+\Lambda^2 ]^{1/2} } \, .</math>
</td>
</tr>
</table>
</div>

====Final Contact====
It is easy to see, geometrically, that if the (blue-dashed) line-of-centers and, in particular, if <math>~r_0</math> is increased beyond the "initial contact" length of <math>~r_+</math>, by exactly a length that equals the diameter of the pink torus, <math>~2r_t</math>, then the <math>~\xi_1</math> = constant circle will make its ''last'' contact with the circle that defines the surface of the equatorial-plane torus. Associating the subscript "-" with this point of last contact, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_-</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~r_+ + 2r_t</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} + r_t \, ,</math>
</td>
</tr>
</table>
</div>
and the associated coordinate-location of this ''last'' point of contact,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\varpi_-</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\varpi_t + \frac{r_t }{[1+\Lambda^2 ]^{1/2} } \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~Z_-</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-\frac{r_t \Lambda}{[1+\Lambda^2 ]^{1/2} } \, .</math>
</td>
</tr>
</table>
</div>

====Region of Overlap====
From the above discussion and derivations, we conclude that the <math>~\xi_1</math> = constant circle will overlap the pink torus and will, accordingly, intersect the surface of that torus in two places for all values of <math>~r_+ < r_0 < r_-</math>, that is, for,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} - r_t </math>
</td>
<td align="center">
<math>~< r_0 < </math>
</td>
<td align="left">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} + r_t \, .</math>
</td>
</tr>
</table>
</div>

====Reality Check One====
Let's see if these derived results make sense. As a first example, let's assign values of various Figure 2 parameters as follows:

<div id="Example1A">
<table align="center" border="1" cellpadding="5">
<tr><th align="center" colspan="4">Example 1A</th></tr>
<tr>
<td align="center" width="25%"><math>~\varpi_t</math></td>
<td align="center" width="25%"><math>~r_t</math></td>
<td align="center" width="25%"><math>~Z_0</math></td>
<td align="center"><math>~\alpha</math></td>
</tr>
<tr>
<td align="center"><math>~\tfrac{3}{4}</math></td>
<td align="center"><math>~\tfrac{1}{4}</math></td>
<td align="center"><math>~1</math></td>
<td align="center"><math>~\tfrac{\pi}{6}</math></td>
</tr>
</table>
</div>

(Notice that the first pair of these parameter values aligns with the properties of the pink torus that was sketched in Figure 4 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)] — as [[#THH12Figure4|reprinted immediately below]] — and that the chosen value of <math>~Z_0</math> aligns with the z-coordinate of their "Point B.")

<div align="center" id="THH12Figure4">
<table border="1" cellpadding="8">

<tr><td align="center">
Figure 4 extracted without modification from p. 2640 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)]<p></p>
"''The Potential of Discs from a 'Mean Green Function' ''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 424, pp. 2635-2645 © RAS
</td>
</tr>

<tr>
<td align="center">
[[File:Figure4THH2012.png|350px|Figure 4 from Trova, Huré & Hersant (2012)]]
</td>
</tr>
<tr>
<td align="center">
<math>~\varpi_t = \tfrac{3}{4}\, ; ~ r_t = \tfrac{1}{4}</math><p></p>
Point A: <math>~(\varpi, Z) = (\tfrac{3}{4}, 0)</math><p></p>
Point B: <math>~(\varpi, Z) = (1, 1)</math><p></p>
Point C: <math>~(\varpi, Z) = (10, 10)</math>
</td>
</tr>
</table>
</div>

Taken together, this choice for the values of <math>~\alpha</math> and <math>~Z_0</math> implies: (1) That the hypotenuse of the blue right-triangle in [[#THH12Figure4|our Figure 2]] and, hence, the distance between the centers of the two circles, is
<div align="center">
<math>~h = \frac{Z_0}{\cos\alpha} = \frac{2\sqrt{3}}{3} \, ;</math>
</div>

and, (2) that the side of the triangle that is opposite the angle, <math>~\alpha</math>, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\varpi_t - R_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~h \sin\alpha = \frac{\sqrt{3}}{3} \, ,</math>
</td>
</tr>
</table>
</div>
which, taken together with the choice of <math>~\varpi_t</math>, gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~R_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3}{4} - \frac{\sqrt{3}}{3} = \frac{9-4\sqrt{3}}{12} \approx 0.17265\, .</math>
</td>
</tr>
</table>
</div>

With this set of parameters held fixed, it is clear that, in order for the <math>~\xi_1</math> = constant circle to make first/final contact with the pink torus, it will need to have a radius,
<div align="center">
<math>~r_\pm = h \mp r_t = \frac{2\sqrt{3}}{3} \mp \frac{1}{4} \, .</math>
</div>

Let's see if this expectation matches the result obtained via the expressions derived above. Specifically, we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Lambda</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{Z_0}{\varpi_t - R_0} = \sqrt{3} \, ;</math>
</td>
</tr>
</table>
</div>
hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_\pm</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(\varpi_t - R_0)[1 + \Lambda^2 ]^{1/2} \mp r_t </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{\sqrt{3}}{3}\biggl[1 + (\sqrt{3})^2 \biggr]^{1/2} \mp \frac{1}{4} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{2\sqrt{3}}{3} \mp \frac{1}{4} \, .</math>
</td>
</tr>
</table>
</div>
This precisely matches our expectation.

===Relate to Toroidal Coordinate System===

====Determine Overall Scale Length====
In order to fully tie our "region of overlap" discussion back to MF53's system of toroidal coordinates, we must identify the specific location of the origin of that coordinate system in, for example, the Figure 2 diagram. [[#Presentation_by_MF53|As above]], we will place the origin of the coordinate system an, as yet unspecified, distance, <math>~a</math>, from the symmetry axis while, as illustrated in Figure 2, displacing it a distance, <math>~Z_0</math>, above the (cylindrical coordinate system's) equatorial plane. Referring back to the properties of toroidal coordinate systems, as [[#Example_Toroidal_Surfaces|discussed above]], we know that in the <math>~Z = Z_0</math> plane, the inner and outer edges of a <math>~\xi_1</math> = constant torus/circle have radial locations,
<table align="center" border="0" cellpadding="4">
<tr>
<td align="right">
<math>
~\frac{\varpi_\mathrm{inner}}{a} = \chi_\mathrm{inner}
</math>
</td>
<td align="center>
<math>~=</math>
</td>
<td align="left">
<math>
~\biggl[\frac{(\xi_1 - 1)}{(\xi_1 + 1)} \biggr]^{1/2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{\varpi_\mathrm{outer}}{a} = \chi_\mathrm{outer}
</math>
</td>
<td align="center>
<math>~=</math>
</td>
<td align="left">
<math>
~\biggl[\frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2} \, .
</math>
</td>
</tr>
</table>

Hence, the major radius of the <math>~\xi_1</math> = constant toroidal surface is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~R_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{2} (\varpi_\mathrm{outer} + \varpi_\mathrm{inner})</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{a}{2} \biggl\{
\biggl[\frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2}
+ \biggl[\frac{(\xi_1 - 1)}{(\xi_1 + 1)} \biggr]^{1/2}
\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{a}{2}
\biggl[\frac{(\xi_1 + 1) + (\xi_1-1)}{(\xi_1^2 - 1)^{1/2}} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}} \, ,
</math>
</td>
</tr>
</table>
</div>
and its cross-sectional radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{2} (\varpi_\mathrm{outer} - \varpi_\mathrm{inner})</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{a}{2} \biggl\{
\biggl[\frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2}
- \biggl[\frac{(\xi_1 - 1)}{(\xi_1 + 1)} \biggr]^{1/2}
\biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{a}{2}
\biggl[\frac{(\xi_1 + 1) - (\xi_1-1)}{(\xi_1^2 - 1)^{1/2}} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{a}{(\xi_1^2 - 1)^{1/2}} \, .
</math>
</td>
</tr>
</table>
</div>

This also means that, if <math>~r_0</math> and <math>~R_0</math> are specified, the associated values of <math>~\xi_1</math> and the scale length, <math>~a</math>, are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi_1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{R_0}{r_0} \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~a</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r_0 \biggl[\biggl( \frac{R_0}{r_0} \biggr)^2 - 1 \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

====Revise Overlap Discussion====
Let's reassess the conclusions drawn in our [[#Overlap_Between_Two_Off-Center_Circles|overlap discussion, above]]. Rather than varying <math>~r_0</math> while holding <math>~R_0</math> fixed, let's consider varying <math>~\xi_1</math> while fixing the coordinate location of the origin of the toroidal coordinate system, <math>~(a, Z_0)</math>. This is the approach that is appropriately aligned with integration over the (pink) toroidal mass distribution.

Re-expressed, the pair of boundaries of the "region of overlap," <math>~r_\pm</math>, give:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(r_0 \pm r_t)^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(\varpi_t - R_0)^2 + Z_0^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \biggl[ \frac{a}{(\xi_1^2 - 1)^{1/2}} \pm r_t \biggr]^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \varpi_t - \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^2 + Z_0^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ \biggl[ a \pm r_t (\xi_1^2 - 1)^{1/2}\biggr]^2 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \varpi_t (\xi_1^2 - 1)^{1/2} - a\xi_1 \biggr]^2 + Z_0^2 (\xi_1^2 - 1)</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~
a^2 \pm 2a r_t (\xi_1^2 - 1)^{1/2} + r_t^2 (\xi_1^2 - 1)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\varpi_t^2 (\xi_1^2 - 1) - 2a \varpi_t \xi_1 (\xi_1^2 - 1)^{1/2} + a^2\xi_1^2 + Z_0^2 (\xi_1^2 - 1)</math>
</td>
</tr>

<tr>
<td align="right">
<math> ~\Rightarrow~~~~(\xi_1^2 - 1)^{1/2}[2a \varpi_t \xi_1 \pm 2a r_t ]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[\varpi_t^2 +a^2 + Z_0^2](\xi_1^2 - 1) </math>
</td>
</tr>

<tr>
<td align="right">
<math> ~\Rightarrow~~~~(\xi_1^2 - 1)^{1/2}
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2a(\varpi_t \xi_1 \pm r_t )}{(\varpi_t^2 +a^2 + Z_0^2)} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{\ell} \biggl[ \xi_1 \pm \frac{r_t}{\varpi_t} \biggr] \, ,</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\ell \equiv \frac{1}{2}\biggl[ \frac{a^2 + \varpi_t^2 + Z_0^2}{a\varpi_t} \biggr] \, .</math>
</div>

After squaring both sides of this equation, we find that the values of <math>~\xi_1</math> corresponding to the limits of overlap can be obtained from the roots of the following quadratic equation:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math> ~\ell^2 (\xi_1^2 - 1)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[ \xi_1 \pm \frac{r_t}{\varpi_t} \biggr]^2</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \xi_1^2 \pm \xi_1\biggl(\frac{2r_t}{\varpi_t}\biggr) + \biggl(\frac{r_t}{\varpi_t}\biggr)^2 \, ,</math>
</td>
</tr>
</table>
</div>
that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(1-\ell^2) \xi_1^2 \pm \xi_1\biggl(\frac{2r_t}{\varpi_t}\biggr) + \biggl[ \biggl(\frac{r_t}{\varpi_t}\biggr)^2 +\ell^2\biggr] \, .</math>
</td>
</tr>
</table>
</div>



After setting up this expression, it dawned on me that the "plus or minus" generalization is not appropriate in this situation. While either result — say, the "plus" result — can be shifted from a <math>~r_0 - R_0</math> specification to a <math>~a - \xi_1</math> specification, the pair of results generally will not share the same value of the scale length, <math>~a</math>. Hence the pair of solutions will be unrelated when viewed from the perspective of the toroidal coordinate system. Instead, let's determine the value of <math>~a</math> from the "first contact" solution — the ''superior'' sign in the expression — then figure out what the "final contact" solution will be if this scale length is held fixed. The solution to the quadratic equation is:

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi_1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2(1-\ell^2)} \biggl\{
- \biggl(\frac{2r_t}{\varpi_t}\biggr) \pm \sqrt{\biggl(\frac{2r_t}{\varpi_t}\biggr)^2 -4(1-\ell^2)
\biggl[ \biggl(\frac{r_t}{\varpi_t}\biggr)^2 +\ell^2\biggr]}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{r_t}{\varpi_t(1-\ell^2)} \biggl\{
- 1 \pm \sqrt{1 -(1-\ell^2)
\biggl[ 1 +\biggl(\frac{\ell \varpi_t}{r_t}\biggr)^2\biggr]}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Given that the allowed range of values for the "radial" toroidal coordinate is, <math>~1 \leq \xi_1 \leq \infty</math>, the relevant root is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi_1\biggr|_\mathrm{first}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{r_t}{\varpi_t(1-\ell^2)} \biggl\{
\sqrt{1 -(1-\ell^2)
\biggl[ 1 +\biggl(\frac{\ell \varpi_t}{r_t}\biggr)^2\biggr]} -1
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

====Reality Check Two====
Let's examine the behavior of these expressions, given the structural parameters provided in [[#Example1A|Example 1A, as defined above]]. [[#Reality_Check_One|Earlier]], we deduced that "first contact" occurs when,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~R_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{3}{4} - \frac{\sqrt{3}}{3} = \frac{9-4\sqrt{3}}{12} \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~r_0 = r_+</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{2\sqrt{3}}{3} - \frac{1}{4} = \frac{8\sqrt{3} - 3}{12} \, .</math>
</td>
</tr>
</table>
</div>

Hence, we should find that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi_1\biggr|_\mathrm{first}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{R_0}{r_0} = \frac{9-4\sqrt{3}}{8\sqrt{3} - 3} \approx 0.19084\, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~a</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r_0 \biggl[\biggl( \frac{R_0}{r_0} \biggr)^2 - 1 \biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

===Other===
Now, the surface of an equatorial-plane torus having major radius, <math>\varpi_t</math>, and cross-sectional radius, <math>~r_t</math>, is described by the expression,

So, as the vertical coordinate varies over the range, <math>-r_t \leq z \leq + r_t</math>, the horizontal coordinate varies over the range, <math>(\varpi_t - r_t ) \leq \varpi \leq (\varpi_t + r_t)</math>. But, more importantly, for a ''given'' value of <math>~\varpi</math>, the corresponding value of the vertical coordinate is,
<div align="center">
<math>
~ z = \pm \biggl[ r_t^2 - (\varpi-\varpi_t)^2 \biggr]^{1/2}.
</math>
</div>

==Yet Again==
===Walk Through Step-By-Step===
Keep the scale length of the toroidal coordinate system, <math>~a</math>, fixed while varying the value of <math>~\xi_1</math> and, hence, the radius,
<div align="center">
<math>~r_0 = \frac{a}{(\xi_1^2 - 1)^{1/2}} \, ,</math>
</div>
of the <math>~\xi_1</math> = constant circle (hereafter, <math>\xi_1</math>-circle). The (cylindrical) coordinate location of the center of this circle will be, <math>~(R_0, Z_0)</math>, where,
<div align="center">
<math>~R_0 = \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}} \, .</math>
</div>
For the time being, we will assume that <math>~Z_0 > 0</math>, as illustrated in our [[#Figure2|Figure 2]]. Our initial aim is to determine the range of values of <math>~\xi_1</math> for which the <math>\xi_1</math>-circle touches or overlaps the equatorial-plane torus, whose position and size are as defined in our [[#Figure2|Figure 2]].

====Lowest Point on Circle====
We will identify the (cylindrical) coordinates of the lowest point on the <math>\xi_1</math>-circle as <math>~(R_0, Z_\mathrm{min})</math>, where,
<div align="center">
<math>~Z_\mathrm{min} = Z_0 - r_0 = Z_0 - \frac{a}{(\xi_1^2 - 1)^{1/2}} \, .</math>
</div>
The <math>\xi_1</math>-circle cannot possibly touch the equatorial-plane torus until <math>~\xi_1</math> drops to a value such that <math>~Z_\mathrm{min}</math> is less than or equal to the radius of the torus, <math>~r_t</math>. This means that touching/overlap cannot occur unless,
<div align="center">
<math>~\xi_1 \leq \xi_\mathrm{max} \equiv \biggl[1 + \biggl(\frac{a}{Z_0 - r_t} \biggr)^2\biggr]^{1/2} \, .</math>
</div>

====A Critical Value of the Scale Length====
Now, the two circles will come into contact at this limiting value, <math>~\xi_\mathrm{max}</math>, only if the corresponding "radial" coordinate location of the center of the <math>~\xi_1</math> circle exactly equals <math>~\varpi_t</math>, that is, only if
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{a\xi_\mathrm{max}}{(\xi_\mathrm{max}^2 - 1)^{1/2}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\varpi_t</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~
a^2\xi_\mathrm{max}^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\varpi_t^2(\xi_\mathrm{max}^2 - 1)</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~
\xi_\mathrm{max}^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\varpi_t^2}{(\varpi_t^2 - a^2)}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~
1 + \biggl(\frac{a}{Z_0 - r_t} \biggr)^2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\varpi_t^2}{(\varpi_t^2 - a^2)}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~
(\varpi_t^2 - a^2)(Z_0 - r_t)^2 + a^2(\varpi_t^2 - a^2)
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\varpi_t^2 (Z_0 - r_t)^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a^2 (Z_0 - r_t)^2 - a^2(\varpi_t^2 - a^2)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
a^2 [(Z_0 - r_t)^2 - \varpi_t^2] + a^4
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ a = a_\mathrm{crit} </math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
[\varpi_t^2 - (Z_0 - r_t)^2 ]^{1/2} \, .
</math>
</td>
</tr>
</table>
</div>

====Points of Intersection====
In all meridional planes, the surface of the equatorial-plane torus is defined by the off-center circle expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\varpi - \varpi_t)^2 + z^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r_t^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~z^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r_t^2 - (\varpi - \varpi_t)^2 \, .</math>
</td>
</tr>
</table>
</div>

Independently, we know that the surface of the off-center, <math>\xi_1</math>-circle is defined by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(\varpi - R_0)^2 + (z- Z_0)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~r_0^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~[z- Z_0]^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a^2}{(\xi_1^2 - 1)} - \biggl[ \varpi - \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^2 </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~z^2 - 2 z Z_0 + Z_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{a^2}{(\xi_1^2 - 1)} - \biggl[ \varpi^2 - \frac{2a \varpi \xi_1}{(\xi_1^2 - 1)^{1/2}} + \frac{a^2\xi_1^2}{(\xi_1^2 - 1)} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-a^2 - \varpi^2 + \frac{2a \varpi \xi_1}{(\xi_1^2 - 1)^{1/2}} \, .</math>
</td>
</tr>
</table>
</div>

When the two circles intersect, the (cylindrical) coordinates of the point(s) at which the intersection occurs, <math>~(\varpi, z)=(\varpi_i, z_i)</math> must be shared by both circles. Eliminating <math>~z</math> between these two off-center circle expressions allows us to solve for the "radial" coordinate, <math>~\varpi_i</math>, of the intersection point(s). Specifically we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~[ r_t^2 - (\varpi - \varpi_t)^2 ] - 2 [ r_t^2 - (\varpi - \varpi_t)^2 ]^{1/2} Z_0 + Z_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-a^2 - \varpi^2 + \frac{2a \varpi \xi_1}{(\xi_1^2 - 1)^{1/2}} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~2 [ r_t^2 - (\varpi - \varpi_t)^2 ]^{1/2} Z_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~Z_0^2 + a^2 + \varpi^2 + [ r_t^2 - (\varpi - \varpi_t)^2 ] - \frac{2a \varpi \xi_1}{(\xi_1^2 - 1)^{1/2}} </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~Z_0^2 + a^2 + \varpi^2 + r_t^2 - (\varpi^2 - 2 \varpi \varpi_t + \varpi_t^2) - \varpi\biggl[ \frac{2a \xi_1}{(\xi_1^2 - 1)^{1/2}}\biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Kappa + 2 \varpi \biggl[ \varpi_t - \frac{a \xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr] \, ,</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\Kappa \equiv Z_0^2 + a^2 - (\varpi_t^2 - r_t^2) \, .</math>
</div>

Squaring both sides of this expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~4Z_0^2 [ r_t^2 - (\varpi^2 - 2\varpi \varpi_t + \varpi_t^2) ] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\Kappa^2 + 4 \varpi \Kappa\biggl[ \varpi_t - \frac{a \xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]
+ 4 \varpi^2 \biggl[ \varpi_t - \frac{a \xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^2</math>
</td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\Rightarrow ~~~~ 0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~[\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)] + \varpi [ 4\Kappa \beta - 8Z_0^2 \varpi_t ]
+ 4\varpi^2 [Z_0^2 + \beta^2 ] \, ,</math>
</td>
</tr>
</table>
</div>
<span id="BetaDefinition">where,</span>
<div align="center">
<math>\beta \equiv \varpi_t - \frac{a \xi_1}{(\xi_1^2 - 1)^{1/2}} \, .</math>
</div>

<span id="IntersectionVarpi">The roots of this quadratic equation provide the sought-after coordinate(s), <math>~\varpi_i</math>, of the point(s) of intersection.</span> Specifically,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\varpi_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{8 [Z_0^2 + \beta^2 ]} \biggl\{
[ 8Z_0^2 \varpi_t- 4\Kappa \beta ] \pm \sqrt{[ 8Z_0^2 \varpi_t- 4\Kappa \beta ]^2 - 16[Z_0^2 + \beta^2 ] [\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)]}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{[ 8Z_0^2 \varpi_t- 4\Kappa \beta ]}{8 [Z_0^2 + \beta^2 ]} \biggl\{
1 \pm \sqrt{1 - \frac{16[Z_0^2 + \beta^2 ] [\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)]}{[ 8Z_0^2 \varpi_t- 4\Kappa \beta ]^2 }}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{2Z_0^2 \varpi_t-\Kappa \beta }{2 (Z_0^2 + \beta^2 )} \biggl\{
1 \pm \sqrt{1 - \ell } \biggr\}\, ,
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\ell \equiv \frac{(Z_0^2 + \beta^2 )[\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)]}{( 2Z_0^2 \varpi_t-\Kappa \beta )^2 } \, .</math>
</div>

Now, from the [[#Toroidal_Coordinates|definition of Toroidal Coordinates, as provided above]], we know that the cylindrical coordinate, <math>~\varpi</math>, is related to the pair of meridional-plane toroidal coordinates via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\varpi}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1-\xi_2} \, .</math>
</td>
</tr>
</table>
</div>

Therefore, once <math>~\varpi_i</math> has been determined for a given choice of <math>~\xi_1</math>, the corresponding value of <math>~\xi_2</math> at the intersection point is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi_2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\xi_1 - \frac{(\xi_1^2 - 1)^{1/2}}{(\varpi_i/a)} \, .</math>
</td>
</tr>
</table>
</div>
Finally, given the pair of coordinate values, <math>~(\xi_1, \xi_2)_i</math>, the value of the (cylindrical) z-coordinate at the intersection point can be obtained via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{Z_0}{a} - \frac{z}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{(1-\xi_2^2 )^{1/2}}{\xi_1-\xi_2} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~~z</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~Z_0 - \frac{a(1-\xi_2^2 )^{1/2}}{\xi_1-\xi_2} \, .</math>
</td>
</tr>
</table>
</div>

====Limiting Values====
All other parameters <math>~(a, Z_0, \varpi_t, r_t)</math> being held fixed, as the coordinate, <math>~\xi_1</math>, is varied, there will be a maximum value, <math>~\xi_1|_\mathrm{max}</math>, at which the <math>\xi_1</math>-circle will first make contact with the (pink) equatorial-plane torus, and there will be a minimum value, <math>~\xi_1|_\mathrm{min}</math>, at which it will have its final contact. At all values within the parameter range,
<div align="center">
<math>~\xi_1|_\mathrm{max} > \xi_1 > ~\xi_1|_\mathrm{min} \, ,</math>
</div>
the <math>\xi_1</math>-circle will intersect the surface of the torus in two locations, defined by two different values of the associated angular coordinate, <math>~\xi_2</math> — see, for example, the coordinates listed in the table associated with [[#Example2|example 2, below]] — but ''at'' the first and final points of contact, the two values of <math>~\xi_2</math> will be degenerate. Let's derive the mathematical relations that give the values of <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math>.

The expression [[#IntersectionVarpi|derived above]] for the "radial" coordinate of the points of intersection, <math>~\varpi_i</math>, gives two physically viable, real numbers as long as the composite parameter, <math>~1 > \ell \geq 0</math>. But only one real value is obtained when <math>~\ell = 1</math>, and that occurs when,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~( 2Z_0^2 \varpi_t-\Kappa \beta )^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(Z_0^2 + \beta^2 )[\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)] \, .</math>
</td>
</tr>
</table>
</div>
In this expression, <math>~\beta</math> is the only parameter that depends on <math>~\xi_1</math>. So, temporarily using the shorthand notation,
<div align="center">
<math>~\Lambda \equiv [\Kappa^2 +4Z_0^2 (\varpi_t^2- r_t^2)] \, </math>
</div>
let's solve for the "critical" value(s), <math>~\beta_\mathrm{crit}</math>. We have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~4 Z_0^4 \varpi_t^2 - 4Z_0^2 \varpi_t \Kappa\beta + \Kappa^2\beta^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~Z_0^2\Lambda + \Lambda \beta^2 </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~(\Kappa^2 - \Lambda)\beta^2 - (4Z_0^2 \varpi_t \Kappa)\beta + (4 Z_0^4 \varpi_t^2 -Z_0^2\Lambda)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>
</div>
The roots of this quadratic equation give,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\beta_\mathrm{crit}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{ 2Z_0^2 \varpi_t \Kappa}{ (\Kappa^2 - \Lambda)} \biggl[1\mp
\sqrt{1-\frac{(\Kappa^2 - \Lambda) (4 Z_0^4 \varpi_t^2 -Z_0^2\Lambda)}{4Z_0^4 \varpi_t^2 \Kappa^2 } } \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{ 2Z_0^2 \varpi_t \Kappa}{ 4Z_0^2 (\varpi_t^2- r_t^2)} \biggl[1\mp
\sqrt{1+\frac{4Z_0^2 (\varpi_t^2- r_t^2)(4 Z_0^4 \varpi_t^2 -Z_0^2\Lambda)}{4Z_0^4 \varpi_t^2 \Kappa^2 } } \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{ \varpi_t \Kappa}{ 2 (\varpi_t^2- r_t^2)} \biggl[1\mp
\sqrt{1+\frac{(\varpi_t^2- r_t^2)(4 Z_0^2 r_t^2 -\Kappa^2)}{\varpi_t^2 \Kappa^2 } } \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Notice that a ''single'' critical value of <math>~\ell</math> — specifically, <math>~\ell = 1</math> — translates nicely into a pair of values of <math>~\beta_\mathrm{crit}</math>; these presumably relate directly to the pair of limiting coordinate values, <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math>, that we are seeking. Via the [[#BetaDefinition|definition of <math>~\beta</math>]], we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\xi_1}{(\xi_1^2-1)^{1/2}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{\varpi_t - \beta}{a} \biggr)</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~\xi_1^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{\varpi_t - \beta}{a} \biggr)^2(\xi_1^2-1)</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~\xi_1^2\biggl[ \biggl(\frac{\varpi_t - \beta}{a} \biggr)^2-1\biggr]</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{\varpi_t - \beta}{a} \biggr)^2</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow~~~~\xi_1</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{(\varpi_t-\beta)^2}{(\varpi_t-\beta)^2-a^2}\biggr]^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

Upon evaluation of this expression in conjunction with the ''pair'' of <math>~\beta_\mathrm{crit}</math> values, the table, below, provides numerical values for the limiting values of <math>~\xi_1|_\mathrm{max}</math> and <math>~\xi_1|_\mathrm{min}</math>, along with the respective values of their accompanying (degenerate) coordinate, <math>~\xi_2</math>.

===Examples===

<div align="center" id="Figure2">
<table border="1" cellpadding="8">

<tr>
<td align="center">
[[File:DiagramToroidalCoordinates.png|350px|Diagram of Torus and Toroidal Coordinates]]
</td>
<td align="center">
[[File:TCoordsE.gif|Diagram of Torus and Toroidal Coordinates]]
</td>
</tr>
</table>
</div>

For reference purposes, Figure 2 has been displayed here, again, in the lefthand panel of Figure 4; the animation sequence presented in the righthand panel illustrates how the <math>\xi_1</math>-circle (depicted by the locus of small black dots) intersects the surface of the (pink) equatorial-plane torus as the value of <math>~\xi_1</math> is varied over the parameter range,
<div align="center">
<math>~\xi_1|_\mathrm{max} \geq \xi_1 \geq ~\xi_1|_\mathrm{min} \, ,</math>
</div>
for a toroidal coordinate system whose origin (filled, red dot) remains fixed at the (cylindrical) coordinate location, <math>~(\varpi, z) = (a, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math>. For a toroidal coordinate system with this specified origin and an equatorial-plane torus having <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math> — as recorded in the top row of numbers in the Table, below — the <math>\xi_1</math>-circle makes ''first contact'' with the torus when <math>~\xi_1 = \xi_1|_\mathrm{max} = 1.1927843</math> and it makes ''final contact'' when <math>~\xi_1 = \xi_1|_\mathrm{min} = 1.0449467</math>. The animation sequence contains ten unique frames: The value of <math>~\xi_1</math> that is associated with the <math>\xi_1</math>-circle in each case appears near the bottom-right corner of the animation frame. These parameter values have also been recorded in the first column of ten separate rows in the following table, along with other relevant parameter values. For example, in each frame of the animation, the points of intersection between the surface of the torus and the <math>\xi_1</math>-circle are identified by filled, green diamonds; the (cylindrical) coordinates associated with these points of intersection, <math>~(\varpi_i, z_i)</math>, are listed in each table row, along with the corresponding value of the toroidal coordinate system's angular, <math>~\xi_2</math> coordinate.

<div id="Example2" style="width: 85%; height: 15em; overflow: auto;">
<table align="center" border="1" cellpadding="5">
<tr><th align="center" colspan="10">Example 2</th></tr>
<tr>
<td align="center" colspan="2" width="25%"><math>~\varpi_t</math></td>
<td align="center" colspan="2" width="25%"><math>~r_t</math></td>
<td align="center" colspan="2" width="25%"><math>~Z_0</math></td>
<td align="center" colspan="2"><math>~a</math></td>
<td align="center" colspan="2"><math>~\Kappa</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{3}{4}</math></td>
<td align="center" colspan="2"><math>~\tfrac{1}{3}</math></td>
<td align="center" colspan="2"><math>~(\tfrac{5}{12})^2</math></td>
</tr>
<tr>
<th colspan="10" align="center">Torus Intersection Points</th>
</tr>
<tr>
<td align="center" colspan="2" rowspan="2"><math>~\xi_1</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\beta</math></td>
<td align="center" colspan="1" rowspan="2"><math>~\ell</math></td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #1 (''superior'' sign)</td>
<td align="center" colspan="3" bgcolor="yellow">Intersection #2 (''inferior'' sign)</td>
</tr>
<tr>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
<td align="center"><math>~\xi_2</math>
<td align="center"><math>~\varpi_i</math>
<td align="center"><math>~z_i</math>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.1927843</math></td>
<td align="center" colspan="1"><math>~+0.138485</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.704606</math></td>
<td align="center" colspan="1"><math>~0.245844</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.176</math></td>
<td align="center" colspan="1"><math>~+0.116568</math></td>
<td align="center" colspan="1"><math>~0.981258</math></td>
<td align="center" colspan="1"><math>~0.922142</math></td>
<td align="center" colspan="1"><math>~0.812595</math></td>
<td align="center" colspan="1"><math>~0.242037</math></td>
<td align="center" colspan="1"><math>~0.841611</math></td>
<td align="center" colspan="1"><math>~0.616896</math></td>
<td align="center" colspan="1"><math>~0.211621</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.160</math></td>
<td align="center" colspan="1"><math>~+0.092267</math></td>
<td align="center" colspan="1"><math>~0.962725</math></td>
<td align="center" colspan="1"><math>~0.933386</math></td>
<td align="center" colspan="1"><math>~0.864726</math></td>
<td align="center" colspan="1"><math>~0.222121</math></td>
<td align="center" colspan="1"><math>~0.824945</math></td>
<td align="center" colspan="1"><math>~0.584858</math></td>
<td align="center" colspan="1"><math>~0.187691</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.144</math></td>
<td align="center" colspan="1"><math>~+0.063705</math></td>
<td align="center" colspan="1"><math>~0.943871</math></td>
<td align="center" colspan="1"><math>~0.940238</math></td>
<td align="center" colspan="1"><math>~0.908969</math></td>
<td align="center" colspan="1"><math>~0.192948</math></td>
<td align="center" colspan="1"><math>~0.813713</math></td>
<td align="center" colspan="1"><math>~0.560766</math></td>
<td align="center" colspan="1"><math>~0.163372</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.127</math></td>
<td align="center" colspan="1"><math>~+0.027202</math></td>
<td align="center" colspan="1"><math>~0.924221</math></td>
<td align="center" colspan="1"><math>~0.944608</math></td>
<td align="center" colspan="1"><math>~0.949856</math></td>
<td align="center" colspan="1"><math>~0.150191</math></td>
<td align="center" colspan="1"><math>~0.806047</math></td>
<td align="center" colspan="1"><math>~0.539788</math></td>
<td align="center" colspan="1"><math>~0.135318</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.111</math></td>
<td align="center" colspan="1"><math>~-0.015045</math></td>
<td align="center" colspan="1"><math>~0.907444</math></td>
<td align="center" colspan="1"><math>~0.946487</math></td>
<td align="center" colspan="1"><math>~0.980806</math></td>
<td align="center" colspan="1"><math>~0.096065</math></td>
<td align="center" colspan="1"><math>~0.802617</math></td>
<td align="center" colspan="1"><math>~0.523232</math></td>
<td align="center" colspan="1"><math>~0.105244</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.094</math></td>
<td align="center" colspan="1"><math>~-0.071947</math></td>
<td align="center" colspan="1"><math>~0.894425</math></td>
<td align="center" colspan="1"><math>~0.945995</math></td>
<td align="center" colspan="1"><math>~0.999208</math></td>
<td align="center" colspan="1"><math>~0.019887</math></td>
<td align="center" colspan="1"><math>~0.803522</math></td>
<td align="center" colspan="1"><math>~0.509118</math></td>
<td align="center" colspan="1"><math>~0.066901</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.078</math></td>
<td align="center" colspan="1"><math>~-0.142539</math></td>
<td align="center" colspan="1"><math>~0.892548</math></td>
<td align="center" colspan="1"><math>~0.942353</math></td>
<td align="center" colspan="1"><math>~0.989322</math></td>
<td align="center" colspan="1"><math>~-0.072283</math></td>
<td align="center" colspan="1"><math>~0.810056</math></td>
<td align="center" colspan="1"><math>~0.500846</math></td>
<td align="center" colspan="1"><math>~0.020554</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.061</math></td>
<td align="center" colspan="1"><math>~-0.247448</math></td>
<td align="center" colspan="1"><math>~0.916366</math></td>
<td align="center" colspan="1"><math>~0.932024</math></td>
<td align="center" colspan="1"><math>~0.916375</math></td>
<td align="center" colspan="1"><math>~-0.186599</math></td>
<td align="center" colspan="1"><math>~0.827074</math></td>
<td align="center" colspan="1"><math>~0.505248</math></td>
<td align="center" colspan="1"><math>~-0.050956</math></td>
</tr>
<tr>
<td align="center" colspan="2"><math>~1.0449467</math></td>
<td align="center" colspan="1"><math>~-0.398902</math></td>
<td align="center" colspan="1"><math>~1.000000</math></td>
<td align="center" colspan="1"><math>~0.885198</math></td>
<td align="center" colspan="1"><math>~0.632605</math></td>
<td align="center" colspan="1"><math>~-0.220722</math></td>
<td align="center" colspan="3">Degenerate Coordinate Values</td>
</tr>
</table>
</div>

Notice in the animation that, while the origin of the selected toroidal coordinate system (the filled red dot) remains fixed, the ''center'' of the <math>\xi_1</math>-circle does not remain fixed. In order to highlight this behavior, the location of the center of the <math>\xi_1</math>-circle has been marked by a filled, light-blue square and, in keeping with the earlier Figure 2 sketch, a vertical, light-blue line connects this center to the equatorial plane.

{{SGFworkInProgress}}

==Confusing and Misleading Steps==

===But Not Every Circle Will Do===
It is very important to appreciate that, although surfaces of constant <math>\Chi</math> (or, equivalently, surfaces of constant <math>\xi_1</math>) are always off-center circles, it is not the case that every off-center circle will prove to be a <math>\Chi= \mathrm{constant}</math> surface in the most relevant toroidal coordinate system. To be more specific, suppose we want to evaluate the potential at some location <math>(R,0)</math> inside or outside of a uniform-density torus whose meridional cross-section is a circle of radius <math>r_c</math> and whose center is located on the <math>x</math>-axis at position <math>x_0</math>. The equation describing the cross-sectional surface of this torus is,
<div align="center">
<math>
(R' - x_0)^2 - {z'}^2 = r_c^2 .
</math>
</div>
Dividing through by the square of a (as yet unspecified) scale length, <math>a</math>, gives,
<div align="center">
<math>
\biggl[ \chi^2 - \frac{x_0}{a} \biggr]^2 - \zeta^2 = \frac{r_c^2}{a^2} .
</math>
</div>
This dimensionless expression will only describe a <math>\Chi = \mathrm{constant}</math> surface in an MF53 toroidal coordinate system if, simultaneously,
<div align="center">
<math>
\Chi = \frac{x_0}{a} ~~~~~\mathrm{and}~~~~~ \Chi^2 - 1 = \frac{r_c^2}{a^2} .
</math>
</div>
That is, only if,
<div align="center">
<math>
a = (x_0^2 - r_c^2)^{1/2} .
</math>
</div>
But in the above discussion we were only able to associate the dimensionless argument of the special function in CT99's CCGF expansion with the "radial" coordinate of the MF53 toroidal coordinate system by setting <math>a = R</math>, that is, only by setting the scale length equal to the cylindrical coordinate value <math>R</math> ''at which the potential is to be evaluated''. So the surface of our torus will only align with a <math>\xi_1 = \mathrm{constant}</math> surface in a toroidal coordinate system if,
<div align="center">
<math>
R = (x_0^2 - r_c^2)^{1/2} .
</math>
</div>
This is a very tight constraint that usually will not be satisfied.

===Multipole Moment in Toroidal Coordinates===
While it might not be interesting or useful to impose this constraint ''in general'', it will likely be instructive to evaluate the potential at the location where this constraint is satisfied. That is, we want to evaluate the potential inside a uniform density, circular-cross-section torus at the location,
<div align="center">
<math>
\Phi[(x_0^2 - r_c^2)^{1/2},0] = - \frac{2G}{(x_0^2 - r_c^2)^{1/4}} q_0 .
</math>
</div>

Since in this case the argument of <math>~Q_{-1/2}</math> can be expressed in terms of the "radial" toroidal coordinate, it is reasonable to write the relevant moment of the mass distribution, <math>~q_0</math>, entirely in terms of toroidal coordinates. Specifically,
<div align="center">
<math>
~q_0 = a^{5/2} \int\int \biggl[ \frac{({\xi_1'}^2 - 1)^{1/2}}{\xi_1' - \xi_2'} \biggr]^{1/2} \rho(\xi_1',\xi_2') Q_{-1/2}(\xi_1') \biggl[ \frac{d\xi_1'}{(\xi_1' - \xi_2')({\xi_1'}^2 - 1)^{1/2}} \biggr] \biggl[ \frac{d\xi_2'}{(\xi_1' - \xi_2')(1-{\xi_2'}^2)^{1/2}} \biggr] .
</math>
</div>

Now suppose that the density distribution is only a function of the ''radial'' coordinate, that is, suppose that <math>~\rho = \rho(\xi_1')</math>. Then the integral can be written as,
<div align="center">
<math>
~q_0 = a^{5/2} \int \rho(\xi_1') Q_{-1/2}(\xi_1')\biggl[ \frac{d\xi_1'}{({\xi_1'}^2 - 1)^{1/4}} \biggr] \int \biggl[ \frac{d\xi_2'}{(\xi_1' - \xi_2')^{5/2}(1-{\xi_2'}^2)^{1/2}} \biggr] .
</math>
</div>
Presumably the integral over <math>~d\xi_2'</math> can be completed in closed form if the density distribution fills out the entire circular cross-section, that is, if the limits on integration are <math>~-1</math> to <math>~+1</math>. Alternatively, write <math>~\xi_2'</math> in terms of <math>~\sin\theta</math> and integrate from <math>~-\tfrac{\pi}{2}</math> to <math>~\tfrac{\pi}{2}</math>. Let's do this.

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{q_0}{a^{5/2}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int\limits_0^{\xi_1^'} \rho(\xi_1') Q_{-1/2}(\xi_1')\biggl[ \frac{d\xi_1'}{({\xi_1'}^2 - 1)^{1/4}} \biggr]
\int\limits_{-1}^{1} \biggl[ \frac{d\xi_2'}{(\xi_1' - \xi_2')^{5/2}(1-{\xi_2'}^2)^{1/2}} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int\limits_0^{\xi_1^'} \rho(\xi_1') Q_{-1/2}(\xi_1')\biggl[ \frac{d\xi_1'}{({\xi_1'}^2 - 1)^{1/4}} \biggr]
\int\limits_{-\pi/2}^{\pi/2} \biggl[ \frac{d\theta}{(\xi_1' - \sin\theta)^{5/2}} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Now, using WolframAlpha's online integrator, we find …
<div align="center">
[[File:TorusIntegration.png|450px|WolframAlpha Integration]]
</div>

Hence — continuing to substitute <math>~a</math> for <math>~\xi_1^'</math> — the definite integral gives,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int\limits_{-\pi/2}^{\pi/2} \biggl[ \frac{d\theta}{(a - \sin\theta)^{5/2}} \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{ \frac{2}{3(a^2-1)^2 (a-\sin\theta)^{3/2}}
\biggl[
\cos\theta ( -5a^2 + 4a\sin\theta + 1)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ (a+1)(a-1)^2 \biggl( \frac{a-\sin\theta}{a-1} \biggr)^{3/2} F\biggl(\frac{\pi - 2\theta}{4} \biggr| \frac{-2}{a-1} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 4a(a-1)(\sin\theta-a) \biggl( \frac{a-\sin\theta}{a-1} \biggr)^{1/2} E\biggl(\frac{\pi - 2\theta}{4} \biggr| \frac{-2}{a-1} \biggr)
\biggr] \biggr\}_{-\tfrac{\pi}{2}}^{\tfrac{\pi}{2}}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{ \frac{2}{3(a^2-1)^2 (a-1)^{3/2}}
\biggl[(a+1)(a-1)^2 \biggl( \frac{a-1}{a-1} \biggr)^{3/2} F\biggl(0\biggr| \frac{-2}{a-1} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
+ 4a(a-1)(1-a) \biggl( \frac{a-1}{a-1} \biggr)^{1/2} E\biggl(0 \biggr| \frac{-2}{a-1} \biggr)
\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~-\biggl\{ \frac{2}{3(a^2-1)^2 (a+1)^{3/2}}
\biggl[(a+1)(a-1)^2 \biggl( \frac{a+1}{a-1} \biggr)^{3/2} F\biggl(\frac{\pi}{2}\biggr| \frac{-2}{a-1} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- 4a(a-1)(1+a) \biggl( \frac{a+1}{a-1} \biggr)^{1/2} E\biggl(\frac{\pi}{2} \biggr| \frac{-2}{a-1} \biggr)
\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{ \frac{2}{3(a-1)^2 (a+1)^2 (a-1)^{3/2}}\biggl[(a+1)(a-1)^2 F\biggl(0\biggr| \frac{2}{1-a} \biggr)
+ 4a(a-1)(1-a) E\biggl(0 \biggr| \frac{2}{1-a} \biggr)
\biggr] \biggr\}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~-\biggl\{ \frac{2}{3(a-1)^2 (a+1)^2 (a+1)^{3/2}}
\biggl[(a+1)(a-1)^2 \biggl( \frac{a+1}{a-1} \biggr)^{3/2} F\biggl(\frac{\pi}{2}\biggr| \frac{2}{1-a} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
- 4a(a-1)(1+a) \biggl( \frac{a+1}{a-1} \biggr)^{1/2} E\biggl(\frac{\pi}{2} \biggr| \frac{2}{1-a} \biggr)
\biggr] \biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{3 (a+1) (a-1)^{3/2}} ~F\biggl(0\biggr| \frac{2}{1-a} \biggr)
- \frac{8a }{3 (a+1)^2 (a-1)^{3/2}}~ E\biggl(0 \biggr| \frac{2}{1-a} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~-\frac{ 2 }{3 (a+1) (a-1)^{3/2}} ~ F\biggl(\frac{\pi}{2}\biggr| \frac{2}{1-a} \biggr)
+ \frac{8a }{3 (a+1)^{2} (a-1)^{3/2}} ~ E\biggl(\frac{\pi}{2} \biggr| \frac{2}{1-a} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{3 (a+1) (a-1)^{3/2}} ~\biggl\{ \biggl[ F\biggl(0\biggr| \frac{2}{1-a} \biggr) - F\biggl(\frac{\pi}{2}\biggr| \frac{2}{1-a} \biggr) \biggr]
- \frac{4a }{(a+1) }~ \biggl[ E\biggl(0 \biggr| \frac{2}{1-a} \biggr) - E\biggl(\frac{\pi}{2} \biggr| \frac{2}{1-a} \biggr) \biggr] \biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

Now, according to (for example) [http://dlmf.nist.gov/19.6#ii NIST's ''Digital Library of Mathematical Functions''],
<div align="center">
<math>~F(0,k) = 0</math>   and   <math>F(\tfrac{\pi}{2},k) = K(k) \, ,</math>
</div>
where, <math>~K(k)</math> is the complete elliptic integral of the first kind. Also, according to [http://dlmf.nist.gov/19.6#iii NIST's ''Digital Library of Mathematical Functions''],
<div align="center">
<math>~E(0,k) = 0</math>   and   <math>E(\tfrac{\pi}{2},k) = E(k) \, ,</math>
</div>
where, <math>~E(k)</math> is the complete elliptic integral of the second kind. Hence we deduce that,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\int\limits_{-\pi/2}^{\pi/2} \biggl[ \frac{d\theta}{(\xi_1' - \sin\theta)^{5/2}} \biggr] </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{3 ({\xi_2'}^2-1) (\xi_1'-1)^{1/2}} ~\biggl[
\frac{4\xi_1' }{(\xi_1'+1) } \cdot E\biggl(\frac{2}{1-\xi_1'} \biggr) - K\biggl(\frac{2}{1-\xi_1'} \biggr)
\biggr] \, ,
</math>
</td>
</tr>

</table>
</div>
which implies,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{3q_0}{2a^{5/2}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\int\limits_0^{\xi_1^'} \rho(\xi_1') Q_{-1/2}(\xi_1')\biggl[ \frac{d\xi_1'}{({\xi_1'}^2 - 1)^{5/4} (\xi_1'-1)^{1/2}} \biggr] ~\biggl[
\frac{4\xi_1' }{(\xi_1'+1) } \cdot E\biggl(\frac{2}{1-\xi_1'} \biggr) - K\biggl(\frac{2}{1-\xi_1'} \biggr)
\biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Next, we might as well also insert the [http://dlmf.nist.gov/14.5#v NIST relation],
<div align="center">
<math>Q_{-1/2}(\cos\theta) = K\biggl[\cos(\tfrac{1}{2}\theta)\biggr]\, .</math>
</div>

===Older, Apparently Irrelevant Material===
If we subtract "1" from both sides of this expression, the right-hand-side (RHS) takes on a familiar form:
<table align="center" border="0" cellpadding="5">
<tr>
<td align="right">
<math>
\chi_\mathrm{CT99}-1
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2R R'} \biggl[ R^2 + {R'}^2 + (z - z')^2 - 2 R R' \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{2R R'} \biggl[ (R - R')^2 + (z - z')^2 \biggr] .
</math>
</td>
</tr>

</table>

It appears as though the quantity, <math>[2R R'(\chi_\mathrm{CT99}-1)]^{1/2}</math>, is the radius <math>r_c</math> of a circle whose center is located at either <math>(R,z)</math> or <math>(R',z')</math>, that is, whose center is shifted off the origin of a cylindrical coordinate system. I'm not yet sure how/if we can benefit from recognizing this association.

<font color="red">'''Case A:'''</font> Suppose we associate <math>R'</math> with the center of a toroidal cross-section and, at the same time, associate <math>R</math> with the inner edge of a ''particular'' toroidal cross-section that is associated with the toroidal coordinate <math>\xi_1</math>. We know that the scale-length <math>a</math> that is associated with the chosen toroidal coordinate system must be given by the ratio,
<div align="center">
<math>
a = \frac{R'}{\chi_0} .
</math>
</div>
[NOTE: As I'm doing this, I'm realizing that it may be wiser to associate <math>a</math> directly with the coordinate location <math>R'</math>. But let's play this out first and see.]

Then it also will be true that,
<div align="center">
<math>
a = \frac{R}{\chi_\mathrm{inner}} = R \biggl[ \frac{\xi_1+1}{\xi_1 - 1} \biggr]^{1/2}.
</math>
</div>
Hence, we conclude that,
<div align="center">
<math>
\mathrm{Case~A:}~~~~~\frac{R}{R'} = \frac{\chi_\mathrm{inner}}{\chi_0} = \biggl[\frac{\xi_1 - 1}{\xi_1 + 1}\biggr]^{1/2} \frac{(\xi_1^2 - 1)^{1/2}}{\xi_1} = \frac{(\xi_1 - 1)}{\xi_1}
</math><br />

<math>
\Rightarrow~~~~~ \xi_1 = \biggl[1 - \frac{R}{R'} \biggr]^{-1} .
</math>
</div>

What, then, is the expression for the scale-length <math>a</math> in terms of just <math>R</math> and <math>R'</math>? Well ...
<div align="center">
<math>
\xi_1 + 1 = \frac{1}{1-(R/R')} +1 = \frac{2 - R/R'}{1-(R/R')} ,
</math>
</div>
and
<div align="center">
<math>
\xi_1 - 1 = \frac{1}{1-(R/R')} -1 = \frac{R/R'}{1-(R/R')} .
</math>
</div>
Hence,
<div align="center">
<math>
a = R \biggl[ \frac{2 - R/R'}{R/R'} \biggr]^{1/2} = ( 2RR' - R^2 )^{1/2}.
</math>
</div>

<font color="red">'''Case B:'''</font> On the other hand, if we associate <math>R'</math> directly with <math>a</math>, then we conclude,
<div align="center">
<math>
\mathrm{Case~B:}~~~~~\frac{R}{R'} = \chi_\mathrm{inner} = \biggl[\frac{\xi_1 - 1}{\xi_1 + 1}\biggr]^{1/2} .
</math><br />

<math>
\Rightarrow~~~~~ (\xi_1 + 1)\biggl(\frac{R}{R'}\biggr)^2 = \xi_1 - 1 .
</math><br />

<math>
\Rightarrow~~~~~ \xi_1 \biggl[ 1 - \biggl(\frac{R}{R'}\biggr)^2 \biggr] = \biggl[1 + \biggl(\frac{R}{R'}\biggr)^2 \biggr].
</math>
<br />

<math>
\Rightarrow~~~~~ \xi_1 = \biggl[1 + \biggl(\frac{R}{R'}\biggr)^2 \biggr]\biggl[ 1 - \biggl(\frac{R}{R'}\biggr)^2 \biggr] ^{-1}.
</math>

</div>

----

=References=
#Morse, P.M. & Feshmach, H. 1953, ''Methods of Theoretical Physics'' — Volumes I and II
#Cohl, H.S. & Tohline, J.E. [http://adsabs.harvard.edu/abs/1999ApJ...527...86C 1999, ApJ, 527, 86-101]
#Cohl, H.S., Rau, A.R.P., Tohline, J.E., Browne, D.A., Cazes, J.E. & Barnes, E.I. [http://adsabs.harvard.edu/abs/2001PhRvA..64e2509C 2001, Phys. Rev. A, 64, 052509]

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Xxsrm: /* Other Equations with Assigned Templates */

__FORCETOC__


=[[File:LSUkey.png|50px]]<font size="+2" color="black">Key Equations</font>=

Each of the equations displayed in the Tables, below, encapsulates a physical concept that is fundamental to our understanding of — and, hence our discussion of — the '''structure, stability, and dynamics of self-gravitating fluids.''' The pervasiveness of these physical concepts throughout astrophysics is reflected in the fact that the same equations — perhaps written in slightly different forms — appear in numerous published books and research papers. When attempting to understand the physical concept that is associated with any one of these mathematical relations, it can be helpful to read how and in what context different authors have introduced the expression in their own work. These Tables offer guides to some parallel discussions that have appeared in published texts over the past 5+ decades in connection with selected sets of key physical relations.

<font color="darkgreen">EXAMPLE:</font> Suppose you want to gain a better understanding of the origin of the ideal gas equation of state, the definition of the gas constant {{ Template:Math/C_GasConstant }}, or how to determine the value of the mean molecular weight {{Template:Math/MP_MeanMolecularWeight}} of a gas. According to the Table entitled ''Equations of State'', you will find a discussion of the ideal gas equation of state: near Eq. (1) in §II.1 of Chandrasekhar (1967); near Eq. (80.8) in §IX.80 of Landau & Lifshitz (1975); near Eq. (5.91) in Vol. I, §5.6 of Padmanabhan (2000); etc. A "note" (linked to a comment further down on this page) appears along with a table entry if the relevant equation in the cited reference contains notations or symbol names that differ significantly from the equation as displayed here.

==Principal Governing Equations==
<span id="PGE">
<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<th colspan=9 align="center">
<font size="+1" color="darkblue">Principal Governing Equations</font>
</th>
</tr>
<tr>
<td colspan=2>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
<td colspan=7 align="center">
<font color="red">Parallel References</font> <br />§ no. and (Eq. no.)
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<th colspan=1>
<font color="red">C67</font>
</th>
<th colspan=1>
<font color="red">LL75</font>
</th>
<th colspan=1>
<font color="red">H87</font>
</th>
<th colspan=1>
<font color="red">ST83</font>
</th>
<th colspan=1>
<font color="red">KW94</font>
</th>
<th colspan=1>
<font color="red">P00</font>
</th>
<th colspan=1>
<font color="red">BLRY07</font>
</th>
</tr>

<tr>
<td>
[[Template:Math/EQ_Continuity01|EQ_Continuity01]]
</td>
<td align="center">
Continuity Equation:<br />
{{ Math/EQ_Continuity01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§I.1<br /> (1.2)<br />[[#LL75note_Continuity01|Note]]
</td>
<td colspan=1 align="center">
§5.4<br /> (5.37)<br />[[#H87note_Continuity01|Note]]
</td>
<td colspan=1 align="center">
§6.1<br /> (6.1.1)<br />[[#ST83note_Continuity01|Note]]
</td>
<td colspan=1 align="center">
§2.5<br /> (2.22)<br />[[#KW94note_Continuity01|Note]]
</td>
<td colspan=1 align="center">
I: §8.5<br /> (8.45)
</td>
<td colspan=1 align="center">
§1.4<br /> (1.53)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_Euler01|EQ_Euler01]]
</td>
<td align="center">
Euler Equation:<br />
{{ Math/EQ_Euler01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§I.2<br /> (2.1)<br />[[#LL75note_Euler01|Note]]
</td>
<td colspan=1 align="center">
§5.4<br /> (5.38)<br />[[#H87note_Euler01|Note]]
</td>
<td colspan=1 align="center">
§6.1<br /> (6.1.2)<br />
</td>
<td colspan=1 align="center">
§2.5<br /> (2.20)
</td>
<td colspan=1 align="center">
I: §8.5<br /> (8.48)
</td>
<td colspan=1 align="center">
§1.4<br /> (1.55)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_FirstLaw01|EQ_FirstLaw01]]
</td>
<td align="center">
1<sup>st</sup> Law of Thermodynamics:<br />
{{ Math/EQ_FirstLaw01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§I.2<br /> (2.5)<br />[[#LL75note_FirstLaw01|Note]]
</td>
<td colspan=1 align="center">
§4.2<br /> (4.31)<br />[[#H87note_FirstLaw01|Note]]
</td>
<td colspan=1 align="center">
§6.1<br /> (6.1.8)<br />
</td>
<td colspan=1 align="center">
§4.1<br /> (4.1)<br />[[#KW94note_FirstLaw01|Note]]
</td>
<td colspan=1 align="center">
I: §8.5<br /> (8.53)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_Poisson01|EQ_Poisson01]]
</td>
<td align="center">
Poisson Equation:<br />
{{Template:Math/EQ_Poisson01}}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§I.3<br /> (3.5)<br />[[#LL75note_Poisson01|Note]]
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§6.1<br /> (6.1.4)<br />
</td>
<td colspan=1 align="center">
§1.3<br /> (1.9)
</td>
<td colspan=1 align="center">
I: §10.2<br /> (10.1)<br />[[#P00note_Poisson01|Note]]
</td>
<td colspan=1 align="center">
Chap. 7
</td>
</tr>

</table>
</div>
</span>

==Equations of State==
<span id="EOS">
<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<th colspan=9 align="center">
<font size="+1" color="darkblue">Equations of State</font>
</th>
</tr>
<tr>
<td colspan=2>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
<td colspan=7 align="center">
<font color="red">Parallel References</font> <br />§ no. and (Eq. no.)
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<th colspan=1>
<font color="red">C67</font>
</th>
<th colspan=1>
<font color="red">LL75</font>
</th>
<th colspan=1>
<font color="red">H87</font>
</th>
<th colspan=1>
<font color="red">ST83</font>
</th>
<th colspan=1>
<font color="red">KW94</font>
</th>
<th colspan=1>
<font color="red">P00</font>
</th>
<th colspan=1>
<font color="red">BLRY07</font>
</th>
</tr>

<tr>
<td>
[[Template:Math/EQ_EOSideal0A|EQ_EOSideal0A]]
</td>
<td align="center">
Ideal Gas Equation of State:<br />
{{ Math/EQ_EOSideal0A }}
</td>
<td colspan=1 align="center">
§II.1<br /> (1) <br />[[#C67note_EOSideal0A|Note]]
</td>
<td colspan=1 align="center">
§IX.80<br /> (80.8) <br />[[#LL75note_EOSideal0A|Note]]
</td>
<td colspan=1 align="center">
§1.1<br /> ("n")
</td>
<td colspan=1 align="center">
§2.3<br /> (2.3.32)<br /> or <br /> (3.2.12)
</td>
<td colspan=1 align="center">
§13.0<br /> (13.1) <br />[[#KW94note_EOSideal0A|Note]]
</td>
<td colspan=1 align="center">
I: §5.6<br /> (5.91)
</td>
<td colspan=1 align="center">
§5.4<br /> (5.34)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_ZTFG01|EQ_ZTFG01]]
</td>
<td align="center">
Degenerate Electron Pressure:<br />
{{ Math/EQ_ZTFG01 }}
———   <font color="darkgreen">NOTE:</font>   ———<br />
<math>
F(\chi) = \frac{8}{5}\chi^5 - \frac{4}{7}\chi^7 + \cdots ~~~~~~(\mathrm{for}~~ \chi\ll 1)
</math>

<math>
F(\chi) = 2\chi^4 - 2\chi^2 + \cdots ~~~~~~~(\mathrm{for}~~ \chi\gg 1)
</math>
</td>
<td colspan=1 align="center">
§X.1<br /> (19) <br /> + <br /> (20)
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§11.2<br /> (11.41)
</td>
<td colspan=1 align="center">
§2.3<br /> (2.3.5)<br /> + <br /> (2.3.6)
</td>
<td colspan=1 align="center">
§15.0<br /> (15.13)<br /> + <br /> (15.14)
</td>
<td colspan=1 align="center">
I: §5.9.2<br /> (5.156)<br /> + <br /> (5.158)
</td>
<td colspan=1 align="center">
§5.6.1<br /> (5.86)<br /> + <br /> (5.87)<br /> + <br /> (5.88)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_EOSradiation01|EQ_EOSradiation01]]
</td>
<td align="center">
Radiation Pressure:<br />
{{ Math/EQ_EOSradiation01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§12.1<br /> (12.12)<br /> +<br /> (12.15)
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§5.6.1<br /> (5.85)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_PressureTotal01|EQ_PressureTotal01]]
</td>
<td align="center">
Normalized Total Pressure:<br />
{{ Math/EQ_PressureTotal01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
</tr>

</table>
</div>
</span>

==Traditional Equations of (Spherical) Stellar Structure==

<span id="SS">
<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<th colspan=9 align="center">
<font size="+1" color="darkblue">Traditional Equations of (Spherical) Stellar Structure</font>
</th>
</tr>
<tr>
<td colspan=2>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
<td colspan=7 align="center">
<font color="red">Parallel References</font> <br />§ no. and (Eq. no.)
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<th colspan=1>
<font color="red">C67</font>
</th>
<th colspan=1>
<font color="red">LL75</font>
</th>
<th colspan=1>
<font color="red">H87</font>
</th>
<th colspan=1>
<font color="red">ST83</font>
</th>
<th colspan=1>
<font color="red">KW94</font>
</th>
<th colspan=1>
<font color="red">P00</font>
</th>
<th colspan=1>
<font color="red">BLRY07</font>
</th>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSmassConservation01|EQ_SSmassConservation01]]
</td>
<td align="center">
Mass Conservation:<br />
{{ Math/EQ_SSmassConservation01 }}
</td>
<td colspan=1 align="center">
§IV.2<br /> (6)
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§3.2<br /> (3.2.1)
</td>
<td colspan=1 align="center">
§2.1<br /> (2.4)
</td>
<td colspan=1 align="center">
II: §2.2<br /> (2.2)
</td>
<td colspan=1 align="center">
§5.1<br /> (5.2)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SShydrostaticBalance01|EQ_SShydrostaticBalance01]]
</td>
<td align="center">
Hydrostatic Balance:<br />
{{ Math/EQ_SShydrostaticBalance01 }}
</td>
<td colspan=1 align="center">
§IV.2<br /> (6)
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§3.2<br /> (3.2.2)
</td>
<td colspan=1 align="center">
§1.1<br /> (1.2) <br />[[#KW94note_SShydrostaticBalance|Note]]
</td>
<td colspan=1 align="center">
II: §2.2<br /> (2.1)
</td>
<td colspan=1 align="center">
§5.1<br /> (5.1)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSLaneEmden01|EQ_SSLaneEmden01]]
</td>
<td align="center">
Polytropic Lane-Emden Equation:<br />
{{ Math/EQ_SSLaneEmden01 }}<br />

</td>
<td colspan=1 align="center">
§IV.2<br /> (11) <br />[[#C67note_SSLaneEmden01|Note]]
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§3.3<br /> (3.3.6)
</td>
<td colspan=1 align="center">
§19.2<br /> (19.10)
</td>
<td colspan=1 align="center">
I: §10.3<br /> (10.4)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSLaneEmden02|EQ_SSLaneEmden02]]
</td>
<td align="center">
Isothermal Lane-Emden Equation:<br />
{{ Math/EQ_SSLaneEmden02 }}<br />
</td>
<td colspan=1 align="center">
§IV.22<br /> (374) <br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§19.8<br /> (19.35)
</td>
<td colspan=1 align="center">
I: §10.3.3<br /> (10.23)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSradiationTransport01|EQ_SSradiationTransport01]]
</td>
<td align="center">
Radiation Transport:<br />
{{ Math/EQ_SSradiationTransport01 }}<br />
</td>
<td colspan=1 align="center">
§IV.22<br /> (374) <br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§5.1.2<br /> (5.11)
</td>
<td colspan=1 align="center">
II: §2.2<br /> (2.8)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSenergyConservation01|EQ_SSenergyConservation01]]
</td>
<td align="center">
Energy Conservation:<br />
{{ Math/EQ_SSenergyConservation01 }}<br />
</td>
<td colspan=1 align="center">
§IV.22<br /> (374) <br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§4.2<br /> (4.22)
</td>
<td colspan=1 align="center">
II: §2.2<br /> (2.18)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

</table>
</div>
</span>

==Stability: Radial Pulsation==
<span id="RadialStability">
<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<th colspan="7" align="center">
<font size="+1" color="darkblue">Stability: Radial Pulsation</font>
</th>
</tr>
<tr>
<td colspan=2>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
<td colspan=5 align="center">
<font color="red">Parallel References</font> <br />§ no. and (Eq. no.)
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<th colspan=1>
<font color="red">C67</font>
</th>
<th colspan=1>
<font color="red">ST83</font>
</th>
<th colspan=1>
<font color="red">KW94</font>
</th>
<th colspan=1>
<font color="red">HK94</font>
</th>
<th colspan=1>
<font color="red">P00</font>
</th>
</tr>

<tr>
<td>
[[Template:Math/EQ_RadialPulsation01|EQ_RadialPulsation01]]
</td>
<td align="center">
LAWE:   Linear Adiabatic Wave (or Radial Pulsation) Equation<br />
{{ Math/EQ_RadialPulsation01 }}<br />

</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§6.5<br /> (6.5.6)
</td>
<td colspan=1 align="center">
§38.1<br /> (38.8)
</td>
<td colspan=1 align="center">
§10.1.1<br /> (10.16)
</td>
<td colspan=1 align="center">
II: §3.7.1<br /> (3.144)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_RadialPulsation04|EQ_RadialPulsation04]]
</td>
<td align="center">
Δ-Highlighted LAWE:<br />
{{ Math/EQ_RadialPulsation04 }}<br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_RadialPulsation02|EQ_RadialPulsation02]]
</td>
<td align="center">
Polytropic LAWE:<br />
{{ Math/EQ_RadialPulsation02 }}<br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_RadialPulsation03|EQ_RadialPulsation03]]
</td>
<td align="center">
Isothermal LAWE:<br />
{{ Math/EQ_RadialPulsation03 }}<br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
</tr>

</table>
</div>
</span>

=Key Parallel References (printed texts spanning 5+ decades)=

* [<b><font color="red">C67</font></b>] [http://adsabs.harvard.edu/abs/1967aits.book.....C '''Chandrasekhar, S.''' 1967] (originally, 1939), An Introduction to the Study of Stellar Structure (New York: Dover)
** <span id="C67note_EOSideal0A">EQ_EOSideal0A</span> — In C67, the ideal gas equation of state is initially written in terms of the specific volume {{Template:Math/VAR_SpecificVolume01}}, instead of the mass density {{Template:Math/VAR_Density01}}; also, it is initially assumed that {{Template:Math/MP_MeanMolecularWeight}} = 1. Both {{Template:Math/VAR_Density01}} and {{Template:Math/MP_MeanMolecularWeight}} are introduced in §III.1, Eq.(5).
** <span id="C67note_SSLaneEmden01">EQ_SSLaneEmden01</span> — At the end of his Chapter IV, C67 writes an extensive history of the earliest work on stellar structure pointing especially the origins of the so-called Lane-Emden equation. He points out, for example, that [http://gallica.bnf.fr/ark:/12148/bpt6k152556/f340.image.r=Annalen%20der%20Physic.langEN Ritter (1880)] actually published this governing differential equation prior to Emden.

* [<b><font color="red">LL75</font></b>] [http://adsabs.harvard.edu/abs/1959flme.book.....L '''Laundau, L. D. & Lifshitz, E. M.''' 1975 (originally, 1959)], Fluid Mechanics (New York: Pergamon Press)
** <span id="LL75note_Continuity01">EQ_Continuity01</span> — LL75 present the Eulerian, rather than the Lagrangian form of the Continuity equation.
** <span id="LL75note_Euler01">EQ_Euler01</span> — In the Euler equation, LL75 do not initially include a source term to account for a gradient in the Newtonian gravitational potential, {{Template:Math/VAR_NewtonianPotential01}}; a term representing acceleration due to gravity, <math>\vec{g} = -\nabla\Phi</math>, is introduced in Eq.(2.4), but in LL75 this is intended primarily to describe gravity at the surface of the Earth.
** <span id="LL75note_FirstLaw01">EQ_FirstLaw01</span> — LL75's Eq.(2.5) must be combined with their discussion of what they refer to as ''the familiar thermodynamic relation'' (between LL75 Eqs. 2.8 and 2.9) in order to appreciate the similarity with our expression.
** <span id="LL75note_Poisson01">EQ_Poisson01</span> — In LL75, the symbol <math>\Delta</math>, rather than <math>\nabla^2</math>, is used to represent the Laplacian spatial operator.
** <span id="LL75note_EOSideal0A">EQ_EOSideal0A</span> — In LL75, the ideal gas equation of state is written in terms of the specific volume {{Template:Math/VAR_SpecificVolume01}}, as well as in terms of the mass density {{Template:Math/VAR_Density01}}.

* [<b><font color="red">ST83</font></b>] [http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983], Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA
** <span id="ST83note_Euler01">EQ_Continuity01</span> — ST83 present the Eulerian, rather than the Lagrangian form of the Continuity equation.

* [<b><font color="red">H87</font></b>] [http://adsabs.harvard.edu/abs/1987stme.book.....H '''Huang, K.''' 1987] (originally 1963), Statistical Mechanics (New York: John Wiley & Sons)
** <span id="H87note_Euler01">EQ_Continuity01</span> — H87 presents the Eulerian, rather than the Lagrangian form of the Continuity equation, and the variable <math>\vec{u}</math> is used instead of {{Template:Math/VAR_VelocityVector01}} to represent the velocity.
** <span id="H87note_Continuity01">EQ_Euler01</span> — H87 presents the Eulerian, rather than the Lagrangian form of the Euler equation, and the variable <math>\vec{u}</math> is used instead of {{Template:Math/VAR_VelocityVector01}} to represent the velocity. Furthermore, to match the source term in our version of the Euler equation, we must set H87's applied acceleration, <math>\vec{F}/m = -\nabla</math>{{Template:Math/VAR_NewtonianPotential01}}.
** <span id="H87note_FirstLaw01">EQ_FirstLaw01</span> — H87 begins a discussion of the 1<sup>st</sup> Law of Thermodynamics in the first section of the first chapter, but it does not appear in the form we present (relevant for a "dilute gas") until Eq.(4.31).

* [<b><font color="red">BT87</font></b>] '''Binney, J. & Tremaine, S.''' 1987, Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">KW94</font></b>] '''Kippenhahn, R. & Weigert, A.''' 1994, Stellar Structure and Evolution (New York: Springer-Verlag)
** <span id="KW94note_Continuity01">EQ_Continuity01</span> — KW94 present the Eulerian, rather than the Lagrangian form of the Continuity equation.
** <span id="KW94note_FirstLaw01">EQ_FirstLaw01</span> — In KW94, the symbol <math>u</math> instead of {{Template:Math/VAR_SpecificInternalEnergy01}} is used to represent the specific internal energy.
** <span id="KW94note_EOSideal0A">EQ_EOSideal0A</span> — In KW94, the ideal gas equation of state is actually first introduced in §2.2, Eq.(27), but it is seriously discussed in Chapter 13. KW94 provide a particularly nice explanation of how to calculate the model parameter, {{Template:Math/MP_MeanMolecularWeight}}.
** <span id="KW94note_SShydrostaticBalance">EQ_SShydrostaticBalance01</span> — In KW94, the hydrostatic balance equation is expressed in terms of <math>dP/dM_r</math> instead of <math>dP/dr</math>; and the second term on the right-hand-side allows for a net radial acceleration.

* [<b><font color="red">HK94</font></b>] '''Hansen, C. J. & Kawaler, S. D.''' 1994, Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer)

* [<b><font color="red">P00</font></b>] '''Padmanabhan, T.''' 2000, Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)
** <span id="P00note_Poisson01">EQ_Poisson01</span> — See also Vol.I: §10.4, Eq.(10.58).

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

=Other Equations with Assigned Templates=

<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<td colspan=8>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Template:Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<td colspan=6 align="center">
<font color="red">Description</font>
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Continuity02|EQ_Continuity02]]
</td>
<td align="center">
{{Template:Math/EQ Continuity02}}
</td>
<td colspan=6>
Eulerian (and Conservative) form of the continuity equation.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Euler02|EQ_Euler02]]
</td>
<td align="center">
{{Template:Math/EQ Euler02}}
</td>
<td colspan=6>
Eulerian form of the Euler equation.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Euler03|EQ_Euler03]]
</td>
<td align="center">
{{Template:Math/EQ Euler03}}
</td>
<td colspan=6>
Conservative form of the Euler equation.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Euler04|EQ_Euler04]]
</td>
<td align="center">
{{Template:Math/EQ Euler04}}
</td>
<td colspan=6>
Euler equation in terms of vorticity.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ FirstLaw02|EQ_FirstLaw02]]
</td>
<td align="center">
{{Template:Math/EQ FirstLaw02}}
</td>
<td colspan=6>
Adiabatic form of the 1<sup>st</sup> Law of Thermodynamics.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Polytrope01|EQ_Polytrope01]]
</td>
<td align="center">
{{Template:Math/EQ Polytrope01}}
</td>
<td colspan=6>
Polytropic equation of state.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Polytrope02|EQ_Polytrope02]]
</td>
<td align="center">
{{Template:Math/EQ Polytrope02}}
</td>
<td colspan=6>
Enthalpy in a polytrope.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_Polytrope03|EQ_Polytrope03]]
</td>
<td align="center">
{{Math/EQ_Polytrope03}}
</td>
<td colspan=6>
Density in terms of enthalpy for polytrope.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_EOSideal00|EQ_EOSideal00]]
</td>
<td align="center">
{{Template:Math/EQ_EOSideal00}}
</td>
<td colspan=6>
Alternate form of the ideal gas equation of state.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ EOSideal02|EQ_EOSideal02]]
</td>
<td align="center">
{{Template:Math/EQ EOSideal02}}
</td>
<td colspan=6>
Alternate form of the ideal gas equation of state.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ TRApproximation|EQ_TRApproximation]]
</td>
<td align="center">
{{Template:Math/EQ TRApproximation}}
</td>
<td colspan=6>
Gravitational potential exterior to an axisymmetric torus,<br />in the [[Apps/DysonWongTori#TRApproximation|Thin Ring (TR) Approximation]].
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ CT99Axisymmetric|EQ_CT99Axisymmetric]]
</td>
<td align="center">
{{Template:Math/EQ CT99Axisymmetric}}
</td>
<td colspan=6>
Gravitational potential of any axisymmetric mass distribution.
</td>
</tr>

</table>
</div>

 <br />
{{SGFfooter}}

Template:Math/EQ TRApproximation

2023-12-12T20:29:15Z

Xxsrm: Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> link=Appendix/EquationTemplates#Other_Equations_with_Assigned_Templates </td> <td align="right"> <math>\Phi_\mathrm{TR}(\varpi,z)</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math>-\biggl[ \frac{2GM}{\pi } \biggr]\frac{K(k)}{\sqrt{(\varpi+a)^2 + z^2}}</math> </td> </tr> <tr> <td align="center" colspan="4"> <math>\mathrm{where:}~~~k..."

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
[[File:LSUkey.png|25px|link=Appendix/EquationTemplates#Other_Equations_with_Assigned_Templates]]
</td>
<td align="right">
<math>\Phi_\mathrm{TR}(\varpi,z)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>-\biggl[ \frac{2GM}{\pi } \biggr]\frac{K(k)}{\sqrt{(\varpi+a)^2 + z^2}}</math>
</td>
</tr>
<tr>
<td align="center" colspan="4">
<math>\mathrm{where:}~~~k \equiv \{4\varpi a/[ (\varpi+a)^2 + z^2]\}^{1 / 2}</math>
</td>
</tr>
</table>

Template:Math/EQ Polytrope03

2023-12-12T20:27:16Z

Xxsrm: Created page with "<math>\rho = \biggl[ \frac{H}{(n+1)K_\mathrm{n}} \biggr]^n </math>"

<math>\rho = \biggl[ \frac{H}{(n+1)K_\mathrm{n}} \biggr]^n </math>

Template:Math/EQ Polytrope02

2023-12-12T20:26:37Z

Xxsrm: Created page with "<math>H = (n+1)K_\mathrm{n} \rho^{1/n}</math>"

<math>H = (n+1)K_\mathrm{n} \rho^{1/n}</math>

Template:Math/EQ Polytrope01

2023-12-12T20:25:52Z

Xxsrm: Created page with "<math>P = K_\mathrm{n} \rho^{1+1/n}</math>"

<math>P = K_\mathrm{n} \rho^{1+1/n}</math>

Template:Math/EQ EOSideal02

2023-12-12T20:24:51Z

Xxsrm: Created page with "<math>P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>"

<math>P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>

Template:Math/EQ EOSideal00

2023-12-12T20:23:41Z

Xxsrm: Created page with "<math>P = n_g k T</math>"

Template:Math/EQ CT99Axisymmetric

2023-12-12T20:22:21Z

Xxsrm: Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> link=Appendix/EquationTemplates#Other_Equations_with_Assigned_Templates </td> <td align="right"> <math>\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> - \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(\varpi~ \varpi^')^{1 / 2}} \biggr] K(\mu) \rho(\varpi^', z^') 2\pi \varpi^'~ d\v..."

<table border="0" cellpadding="5" align="center">
<tr>
<td align="right">
[[File:LSUkey.png|25px|link=Appendix/EquationTemplates#Other_Equations_with_Assigned_Templates]]
</td>
<td align="right">
<math>\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(\varpi~ \varpi^')^{1 / 2}} \biggr] K(\mu) \rho(\varpi^', z^') 2\pi \varpi^'~ d\varpi^' dz^' </math>
</td>
</tr>
<tr>
<td align="center" colspan="4">
<math>\mathrm{where:}~~~\mu \equiv \{4\varpi \varpi^' /[ (\varpi+\varpi^')^2 + (z-z^')^2]\}^{1 / 2}</math>
</td>
</tr>
</table>

Template:Math/EQ RadialPulsation04

2023-12-12T20:19:35Z

Xxsrm: Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> link=Appendix/EquationTemplates#Stability:_Radial_Pulsation </td> <td align="center"> <math>0 = \frac{d^2x}{dr_0^2} + \frac{1}{r_0}\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{dx}{dr_0} + \biggl[ \biggl(3 - \frac{4}{\gamma_g}\biggr) \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{x}{r_0^2} - \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c..."

<table border=0 cellpadding=2>
<tr>
<td align="right">
[[Image:LSUkey.png|25px|link=Appendix/EquationTemplates#Stability:_Radial_Pulsation]]
</td>
<td align="center">
<math>0 =
\frac{d^2x}{dr_0^2} + \frac{1}{r_0}\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{dx}{dr_0}
+ \biggl[ \biggl(3 - \frac{4}{\gamma_g}\biggr) \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{x}{r_0^2}
- \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \frac{x}{r_0^2}
</math>
</td>
</tr>
<tr>
<td align="center" colspan="2">
where:    <math>\Delta \equiv \frac{M_r}{4\pi r_0^3 \rho_0} \, ,</math>    <math>\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} </math>
</td>
</tr>
</table>

Template:Math/EQ RadialPulsation01

2023-12-12T20:18:55Z

Xxsrm: Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> link=Appendix/EquationTemplates#Stability:_Radial_Pulsation </td> <td align="center"> <math> \frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x = 0 </math> </td> </tr> </table>"

Template:Math/EQ SSradiationTransport01

2023-12-12T19:48:32Z

Xxsrm: Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> link=Appendix/EquationTemplates#Traditional_Equations_of_.28Spherical.29_Stellar_Structure </td> <td align="center"> <math>\frac{dT}{dr} = - \frac{ 3 }{ 4a_\mathrm{rad} c} \biggl(\frac{ \kappa \rho }{ T^3 }\biggr) \frac{ L_r }{ 4\pi r^2 }</math> </td> </tr> </table>"

<table border=0 cellpadding=2>
<tr>
<td align="right">
[[Image:LSUkey.png|25px|link=Appendix/EquationTemplates#Traditional_Equations_of_.28Spherical.29_Stellar_Structure]]
</td>
<td align="center">
<math>\frac{dT}{dr} = - \frac{ 3 }{ 4a_\mathrm{rad} c}
\biggl(\frac{ \kappa \rho }{ T^3 }\biggr)
\frac{ L_r }{ 4\pi r^2 }</math>
</td>
</tr>
</table>

Template:Math/EQ SSmassConservation01

2023-12-12T19:47:38Z

Xxsrm: Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> link=Appendix/EquationTemplates#Traditional_Equations_of_.28Spherical.29_Stellar_Structure </td> <td align="center"> <math>\frac{dM_r}{dr} = 4\pi r^2 \rho</math> </td> </tr> </table>"

<table border=0 cellpadding=2>
<tr>
<td align="right">
[[Image:LSUkey.png|25px|link=Appendix/EquationTemplates#Traditional_Equations_of_.28Spherical.29_Stellar_Structure]]
</td>
<td align="center">
<math>\frac{dM_r}{dr} = 4\pi r^2 \rho</math>
</td>
</tr>
</table>

Template:Math/EQ SSenergyConservation01

2023-12-12T19:46:56Z

Xxsrm: Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> link=Appendix/EquationTemplates#Traditional_Equations_of_.28Spherical.29_Stellar_Structure </td> <td align="center"> <math>\frac{dL_r}{dr} = 4\pi r^2 \rho \epsilon_\mathrm{nuc}</math> </td> </tr> </table>"

<table border=0 cellpadding=2>
<tr>
<td align="right">
[[Image:LSUkey.png|25px|link=Appendix/EquationTemplates#Traditional_Equations_of_.28Spherical.29_Stellar_Structure]]
</td>
<td align="center">
<math>\frac{dL_r}{dr} = 4\pi r^2 \rho \epsilon_\mathrm{nuc}</math>
</td>
</tr>
</table>

Template:Math/EQ ZTFG01

2023-12-12T19:44:39Z

Xxsrm: Created page with "<table border=0 cellpadding=2> <tr> <td align="right" width="35%"> link=Appendix/EquationTemplates#Equations_of_State </td> <td align="left"> <font><math>P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math></font> </td> </tr> <tr> <td align="center" colspan="2"> where:  <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math> </td> </tr> <tr> <td align="right" colspan="1"> and:    </td> <td align="left"..."

<table border=0 cellpadding=2>
<tr>
<td align="right" width="35%">
[[File:LSUkey.png|25px|link=Appendix/EquationTemplates#Equations_of_State]]
</td>
<td align="left">
<font><math>P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math></font>
</td>
</tr>
<tr>
<td align="center" colspan="2">
where:  <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math>
</td>
</tr>
<tr>
<td align="right" colspan="1">
and:   
</td>
<td align="left" colspan="1">
<math>\chi \equiv (\rho/B_\mathrm{F})^{1/3}</math>
</td>
</tr>
</table>

Template:Math/EQ PressureTotal01

2023-12-12T19:43:54Z

Xxsrm: Created page with "<table border=0 cellpadding=2> <tr> <td align="right"> link=Appendix/EquationTemplates#Equations_of_State </td> <td align="center"> <math>p_\mathrm{total} = \biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) 8 \chi^3 \frac{T}{T_e} + F(\chi) + \frac{8\pi^4}{15} \biggl( \frac{T}{T_e} \biggr)^4</math> </td> </tr> </table>"

<table border=0 cellpadding=2>
<tr>
<td align="right">
[[File:LSUkey.png|25px|link=Appendix/EquationTemplates#Equations_of_State]]
</td>
<td align="center">
<math>p_\mathrm{total} = \biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) 8 \chi^3 \frac{T}{T_e} + F(\chi) +
\frac{8\pi^4}{15} \biggl( \frac{T}{T_e} \biggr)^4</math>
</td>
</tr>
</table>

Template:Math/EQ EOSradiation01

2023-12-12T19:43:01Z

Template:Math/EQ EOSideal0A

2023-12-12T19:42:25Z

<table border=0 cellpadding=2>
<tr>
<td align="right">
[[Image:LSUkey.png|25px|link=Appendix/EquationTemplates#Equations_of_State]]
</td>
<td align="center">
<math>P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>
</td>
</tr>
</table>

Appendix/EquationTemplates

2023-12-12T19:39:27Z

Xxsrm: Created page with "__FORCETOC__  =50px<font size="+2" color="black">Key Equations</font>= Each of the equations displayed in the Tables, below, encapsulates a physical concept that is fundamental to our understanding of — and, hence our discussion of — the '''structure, stability, and dynamics of self-gravitating fluids.''' The pervasiveness of these physical concepts throughout astrophysics is reflected in the fact t..."

__FORCETOC__


=[[File:LSUkey.png|50px]]<font size="+2" color="black">Key Equations</font>=

Each of the equations displayed in the Tables, below, encapsulates a physical concept that is fundamental to our understanding of — and, hence our discussion of — the '''structure, stability, and dynamics of self-gravitating fluids.''' The pervasiveness of these physical concepts throughout astrophysics is reflected in the fact that the same equations — perhaps written in slightly different forms — appear in numerous published books and research papers. When attempting to understand the physical concept that is associated with any one of these mathematical relations, it can be helpful to read how and in what context different authors have introduced the expression in their own work. These Tables offer guides to some parallel discussions that have appeared in published texts over the past 5+ decades in connection with selected sets of key physical relations.

<font color="darkgreen">EXAMPLE:</font> Suppose you want to gain a better understanding of the origin of the ideal gas equation of state, the definition of the gas constant {{ Template:Math/C_GasConstant }}, or how to determine the value of the mean molecular weight {{Template:Math/MP_MeanMolecularWeight}} of a gas. According to the Table entitled ''Equations of State'', you will find a discussion of the ideal gas equation of state: near Eq. (1) in §II.1 of Chandrasekhar (1967); near Eq. (80.8) in §IX.80 of Landau & Lifshitz (1975); near Eq. (5.91) in Vol. I, §5.6 of Padmanabhan (2000); etc. A "note" (linked to a comment further down on this page) appears along with a table entry if the relevant equation in the cited reference contains notations or symbol names that differ significantly from the equation as displayed here.

==Principal Governing Equations==
<span id="PGE">
<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<th colspan=9 align="center">
<font size="+1" color="darkblue">Principal Governing Equations</font>
</th>
</tr>
<tr>
<td colspan=2>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
<td colspan=7 align="center">
<font color="red">Parallel References</font> <br />§ no. and (Eq. no.)
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<th colspan=1>
<font color="red">C67</font>
</th>
<th colspan=1>
<font color="red">LL75</font>
</th>
<th colspan=1>
<font color="red">H87</font>
</th>
<th colspan=1>
<font color="red">ST83</font>
</th>
<th colspan=1>
<font color="red">KW94</font>
</th>
<th colspan=1>
<font color="red">P00</font>
</th>
<th colspan=1>
<font color="red">BLRY07</font>
</th>
</tr>

<tr>
<td>
[[Template:Math/EQ_Continuity01|EQ_Continuity01]]
</td>
<td align="center">
Continuity Equation:<br />
{{ Math/EQ_Continuity01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§I.1<br /> (1.2)<br />[[#LL75note_Continuity01|Note]]
</td>
<td colspan=1 align="center">
§5.4<br /> (5.37)<br />[[#H87note_Continuity01|Note]]
</td>
<td colspan=1 align="center">
§6.1<br /> (6.1.1)<br />[[#ST83note_Continuity01|Note]]
</td>
<td colspan=1 align="center">
§2.5<br /> (2.22)<br />[[#KW94note_Continuity01|Note]]
</td>
<td colspan=1 align="center">
I: §8.5<br /> (8.45)
</td>
<td colspan=1 align="center">
§1.4<br /> (1.53)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_Euler01|EQ_Euler01]]
</td>
<td align="center">
Euler Equation:<br />
{{ Math/EQ_Euler01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§I.2<br /> (2.1)<br />[[#LL75note_Euler01|Note]]
</td>
<td colspan=1 align="center">
§5.4<br /> (5.38)<br />[[#H87note_Euler01|Note]]
</td>
<td colspan=1 align="center">
§6.1<br /> (6.1.2)<br />
</td>
<td colspan=1 align="center">
§2.5<br /> (2.20)
</td>
<td colspan=1 align="center">
I: §8.5<br /> (8.48)
</td>
<td colspan=1 align="center">
§1.4<br /> (1.55)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_FirstLaw01|EQ_FirstLaw01]]
</td>
<td align="center">
1<sup>st</sup> Law of Thermodynamics:<br />
{{ Math/EQ_FirstLaw01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§I.2<br /> (2.5)<br />[[#LL75note_FirstLaw01|Note]]
</td>
<td colspan=1 align="center">
§4.2<br /> (4.31)<br />[[#H87note_FirstLaw01|Note]]
</td>
<td colspan=1 align="center">
§6.1<br /> (6.1.8)<br />
</td>
<td colspan=1 align="center">
§4.1<br /> (4.1)<br />[[#KW94note_FirstLaw01|Note]]
</td>
<td colspan=1 align="center">
I: §8.5<br /> (8.53)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_Poisson01|EQ_Poisson01]]
</td>
<td align="center">
Poisson Equation:<br />
{{Template:Math/EQ_Poisson01}}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§I.3<br /> (3.5)<br />[[#LL75note_Poisson01|Note]]
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§6.1<br /> (6.1.4)<br />
</td>
<td colspan=1 align="center">
§1.3<br /> (1.9)
</td>
<td colspan=1 align="center">
I: §10.2<br /> (10.1)<br />[[#P00note_Poisson01|Note]]
</td>
<td colspan=1 align="center">
Chap. 7
</td>
</tr>

</table>
</div>
</span>

==Equations of State==
<span id="EOS">
<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<th colspan=9 align="center">
<font size="+1" color="darkblue">Equations of State</font>
</th>
</tr>
<tr>
<td colspan=2>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
<td colspan=7 align="center">
<font color="red">Parallel References</font> <br />§ no. and (Eq. no.)
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<th colspan=1>
<font color="red">C67</font>
</th>
<th colspan=1>
<font color="red">LL75</font>
</th>
<th colspan=1>
<font color="red">H87</font>
</th>
<th colspan=1>
<font color="red">ST83</font>
</th>
<th colspan=1>
<font color="red">KW94</font>
</th>
<th colspan=1>
<font color="red">P00</font>
</th>
<th colspan=1>
<font color="red">BLRY07</font>
</th>
</tr>

<tr>
<td>
[[Template:Math/EQ_EOSideal0A|EQ_EOSideal0A]]
</td>
<td align="center">
Ideal Gas Equation of State:<br />
{{ Math/EQ_EOSideal0A }}
</td>
<td colspan=1 align="center">
§II.1<br /> (1) <br />[[#C67note_EOSideal0A|Note]]
</td>
<td colspan=1 align="center">
§IX.80<br /> (80.8) <br />[[#LL75note_EOSideal0A|Note]]
</td>
<td colspan=1 align="center">
§1.1<br /> ("n")
</td>
<td colspan=1 align="center">
§2.3<br /> (2.3.32)<br /> or <br /> (3.2.12)
</td>
<td colspan=1 align="center">
§13.0<br /> (13.1) <br />[[#KW94note_EOSideal0A|Note]]
</td>
<td colspan=1 align="center">
I: §5.6<br /> (5.91)
</td>
<td colspan=1 align="center">
§5.4<br /> (5.34)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_ZTFG01|EQ_ZTFG01]]
</td>
<td align="center">
Degenerate Electron Pressure:<br />
{{ Math/EQ_ZTFG01 }}
———   <font color="darkgreen">NOTE:</font>   ———<br />
<math>
F(\chi) = \frac{8}{5}\chi^5 - \frac{4}{7}\chi^7 + \cdots ~~~~~~(\mathrm{for}~~ \chi\ll 1)
</math>

<math>
F(\chi) = 2\chi^4 - 2\chi^2 + \cdots ~~~~~~~(\mathrm{for}~~ \chi\gg 1)
</math>
</td>
<td colspan=1 align="center">
§X.1<br /> (19) <br /> + <br /> (20)
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§11.2<br /> (11.41)
</td>
<td colspan=1 align="center">
§2.3<br /> (2.3.5)<br /> + <br /> (2.3.6)
</td>
<td colspan=1 align="center">
§15.0<br /> (15.13)<br /> + <br /> (15.14)
</td>
<td colspan=1 align="center">
I: §5.9.2<br /> (5.156)<br /> + <br /> (5.158)
</td>
<td colspan=1 align="center">
§5.6.1<br /> (5.86)<br /> + <br /> (5.87)<br /> + <br /> (5.88)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_EOSradiation01|EQ_EOSradiation01]]
</td>
<td align="center">
Radiation Pressure:<br />
{{ Math/EQ_EOSradiation01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§12.1<br /> (12.12)<br /> +<br /> (12.15)
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§5.6.1<br /> (5.85)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_PressureTotal01|EQ_PressureTotal01]]
</td>
<td align="center">
Normalized Total Pressure:<br />
{{ Math/EQ_PressureTotal01 }}
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
</tr>

</table>
</div>
</span>

==Traditional Equations of (Spherical) Stellar Structure==

<span id="SS">
<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<th colspan=9 align="center">
<font size="+1" color="darkblue">Traditional Equations of (Spherical) Stellar Structure</font>
</th>
</tr>
<tr>
<td colspan=2>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
<td colspan=7 align="center">
<font color="red">Parallel References</font> <br />§ no. and (Eq. no.)
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<th colspan=1>
<font color="red">C67</font>
</th>
<th colspan=1>
<font color="red">LL75</font>
</th>
<th colspan=1>
<font color="red">H87</font>
</th>
<th colspan=1>
<font color="red">ST83</font>
</th>
<th colspan=1>
<font color="red">KW94</font>
</th>
<th colspan=1>
<font color="red">P00</font>
</th>
<th colspan=1>
<font color="red">BLRY07</font>
</th>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSmassConservation01|EQ_SSmassConservation01]]
</td>
<td align="center">
Mass Conservation:<br />
{{ Math/EQ_SSmassConservation01 }}
</td>
<td colspan=1 align="center">
§IV.2<br /> (6)
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§3.2<br /> (3.2.1)
</td>
<td colspan=1 align="center">
§2.1<br /> (2.4)
</td>
<td colspan=1 align="center">
II: §2.2<br /> (2.2)
</td>
<td colspan=1 align="center">
§5.1<br /> (5.2)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SShydrostaticBalance01|EQ_SShydrostaticBalance01]]
</td>
<td align="center">
Hydrostatic Balance:<br />
{{ Math/EQ_SShydrostaticBalance01 }}
</td>
<td colspan=1 align="center">
§IV.2<br /> (6)
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§3.2<br /> (3.2.2)
</td>
<td colspan=1 align="center">
§1.1<br /> (1.2) <br />[[#KW94note_SShydrostaticBalance|Note]]
</td>
<td colspan=1 align="center">
II: §2.2<br /> (2.1)
</td>
<td colspan=1 align="center">
§5.1<br /> (5.1)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSLaneEmden01|EQ_SSLaneEmden01]]
</td>
<td align="center">
Polytropic Lane-Emden Equation:<br />
{{ Math/EQ_SSLaneEmden01 }}<br />

</td>
<td colspan=1 align="center">
§IV.2<br /> (11) <br />[[#C67note_SSLaneEmden01|Note]]
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§3.3<br /> (3.3.6)
</td>
<td colspan=1 align="center">
§19.2<br /> (19.10)
</td>
<td colspan=1 align="center">
I: §10.3<br /> (10.4)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSLaneEmden02|EQ_SSLaneEmden02]]
</td>
<td align="center">
Isothermal Lane-Emden Equation:<br />
{{ Math/EQ_SSLaneEmden02 }}<br />
</td>
<td colspan=1 align="center">
§IV.22<br /> (374) <br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§19.8<br /> (19.35)
</td>
<td colspan=1 align="center">
I: §10.3.3<br /> (10.23)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSradiationTransport01|EQ_SSradiationTransport01]]
</td>
<td align="center">
Radiation Transport:<br />
{{ Math/EQ_SSradiationTransport01 }}<br />
</td>
<td colspan=1 align="center">
§IV.22<br /> (374) <br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§5.1.2<br /> (5.11)
</td>
<td colspan=1 align="center">
II: §2.2<br /> (2.8)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_SSenergyConservation01|EQ_SSenergyConservation01]]
</td>
<td align="center">
Energy Conservation:<br />
{{ Math/EQ_SSenergyConservation01 }}<br />
</td>
<td colspan=1 align="center">
§IV.22<br /> (374) <br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§4.2<br /> (4.22)
</td>
<td colspan=1 align="center">
II: §2.2<br /> (2.18)
</td>
<td colspan=1 align="center">
 
</td>
</tr>

</table>
</div>
</span>

==Stability: Radial Pulsation==
<span id="RadialStability">
<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<th colspan="7" align="center">
<font size="+1" color="darkblue">Stability: Radial Pulsation</font>
</th>
</tr>
<tr>
<td colspan=2>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
<td colspan=5 align="center">
<font color="red">Parallel References</font> <br />§ no. and (Eq. no.)
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<th colspan=1>
<font color="red">C67</font>
</th>
<th colspan=1>
<font color="red">ST83</font>
</th>
<th colspan=1>
<font color="red">KW94</font>
</th>
<th colspan=1>
<font color="red">HK94</font>
</th>
<th colspan=1>
<font color="red">P00</font>
</th>
</tr>

<tr>
<td>
[[Template:Math/EQ_RadialPulsation01|EQ_RadialPulsation01]]
</td>
<td align="center">
LAWE:   Linear Adiabatic Wave (or Radial Pulsation) Equation<br />
{{ Math/EQ_RadialPulsation01 }}<br />

</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
§6.5<br /> (6.5.6)
</td>
<td colspan=1 align="center">
§38.1<br /> (38.8)
</td>
<td colspan=1 align="center">
§10.1.1<br /> (10.16)
</td>
<td colspan=1 align="center">
II: §3.7.1<br /> (3.144)
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_RadialPulsation04|EQ_RadialPulsation04]]
</td>
<td align="center">
Δ-Highlighted LAWE:<br />
{{ Math/EQ_RadialPulsation04 }}<br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_RadialPulsation02|EQ_RadialPulsation02]]
</td>
<td align="center">
Polytropic LAWE:<br />
{{ Math/EQ_RadialPulsation02 }}<br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_RadialPulsation03|EQ_RadialPulsation03]]
</td>
<td align="center">
Isothermal LAWE:<br />
{{ Math/EQ_RadialPulsation03 }}<br />
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
<td colspan=1 align="center">
 
</td>
</tr>

</table>
</div>
</span>

=Key Parallel References (printed texts spanning 5+ decades)=

* [<b><font color="red">C67</font></b>] [http://adsabs.harvard.edu/abs/1967aits.book.....C '''Chandrasekhar, S.''' 1967] (originally, 1939), An Introduction to the Study of Stellar Structure (New York: Dover)
** <span id="C67note_EOSideal0A">EQ_EOSideal0A</span> — In C67, the ideal gas equation of state is initially written in terms of the specific volume {{Template:Math/VAR_SpecificVolume01}}, instead of the mass density {{Template:Math/VAR_Density01}}; also, it is initially assumed that {{Template:Math/MP_MeanMolecularWeight}} = 1. Both {{Template:Math/VAR_Density01}} and {{Template:Math/MP_MeanMolecularWeight}} are introduced in §III.1, Eq.(5).
** <span id="C67note_SSLaneEmden01">EQ_SSLaneEmden01</span> — At the end of his Chapter IV, C67 writes an extensive history of the earliest work on stellar structure pointing especially the origins of the so-called Lane-Emden equation. He points out, for example, that [http://gallica.bnf.fr/ark:/12148/bpt6k152556/f340.image.r=Annalen%20der%20Physic.langEN Ritter (1880)] actually published this governing differential equation prior to Emden.

* [<b><font color="red">LL75</font></b>] [http://adsabs.harvard.edu/abs/1959flme.book.....L '''Laundau, L. D. & Lifshitz, E. M.''' 1975 (originally, 1959)], Fluid Mechanics (New York: Pergamon Press)
** <span id="LL75note_Continuity01">EQ_Continuity01</span> — LL75 present the Eulerian, rather than the Lagrangian form of the Continuity equation.
** <span id="LL75note_Euler01">EQ_Euler01</span> — In the Euler equation, LL75 do not initially include a source term to account for a gradient in the Newtonian gravitational potential, {{Template:Math/VAR_NewtonianPotential01}}; a term representing acceleration due to gravity, <math>\vec{g} = -\nabla\Phi</math>, is introduced in Eq.(2.4), but in LL75 this is intended primarily to describe gravity at the surface of the Earth.
** <span id="LL75note_FirstLaw01">EQ_FirstLaw01</span> — LL75's Eq.(2.5) must be combined with their discussion of what they refer to as ''the familiar thermodynamic relation'' (between LL75 Eqs. 2.8 and 2.9) in order to appreciate the similarity with our expression.
** <span id="LL75note_Poisson01">EQ_Poisson01</span> — In LL75, the symbol <math>\Delta</math>, rather than <math>\nabla^2</math>, is used to represent the Laplacian spatial operator.
** <span id="LL75note_EOSideal0A">EQ_EOSideal0A</span> — In LL75, the ideal gas equation of state is written in terms of the specific volume {{Template:Math/VAR_SpecificVolume01}}, as well as in terms of the mass density {{Template:Math/VAR_Density01}}.

* [<b><font color="red">ST83</font></b>] [http://adsabs.harvard.edu/abs/1983bhwd.book.....S '''Shapiro, S. L. & Teukolsky, S. A.''' 1983], Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA
** <span id="ST83note_Euler01">EQ_Continuity01</span> — ST83 present the Eulerian, rather than the Lagrangian form of the Continuity equation.

* [<b><font color="red">H87</font></b>] [http://adsabs.harvard.edu/abs/1987stme.book.....H '''Huang, K.''' 1987] (originally 1963), Statistical Mechanics (New York: John Wiley & Sons)
** <span id="H87note_Euler01">EQ_Continuity01</span> — H87 presents the Eulerian, rather than the Lagrangian form of the Continuity equation, and the variable <math>\vec{u}</math> is used instead of {{Template:Math/VAR_VelocityVector01}} to represent the velocity.
** <span id="H87note_Continuity01">EQ_Euler01</span> — H87 presents the Eulerian, rather than the Lagrangian form of the Euler equation, and the variable <math>\vec{u}</math> is used instead of {{Template:Math/VAR_VelocityVector01}} to represent the velocity. Furthermore, to match the source term in our version of the Euler equation, we must set H87's applied acceleration, <math>\vec{F}/m = -\nabla</math>{{Template:Math/VAR_NewtonianPotential01}}.
** <span id="H87note_FirstLaw01">EQ_FirstLaw01</span> — H87 begins a discussion of the 1<sup>st</sup> Law of Thermodynamics in the first section of the first chapter, but it does not appear in the form we present (relevant for a "dilute gas") until Eq.(4.31).

* [<b><font color="red">BT87</font></b>] '''Binney, J. & Tremaine, S.''' 1987, Galactic Dynamics (Princeton, NJ: Princeton University Press)

* [<b><font color="red">KW94</font></b>] '''Kippenhahn, R. & Weigert, A.''' 1994, Stellar Structure and Evolution (New York: Springer-Verlag)
** <span id="KW94note_Continuity01">EQ_Continuity01</span> — KW94 present the Eulerian, rather than the Lagrangian form of the Continuity equation.
** <span id="KW94note_FirstLaw01">EQ_FirstLaw01</span> — In KW94, the symbol <math>u</math> instead of {{Template:Math/VAR_SpecificInternalEnergy01}} is used to represent the specific internal energy.
** <span id="KW94note_EOSideal0A">EQ_EOSideal0A</span> — In KW94, the ideal gas equation of state is actually first introduced in §2.2, Eq.(27), but it is seriously discussed in Chapter 13. KW94 provide a particularly nice explanation of how to calculate the model parameter, {{Template:Math/MP_MeanMolecularWeight}}.
** <span id="KW94note_SShydrostaticBalance">EQ_SShydrostaticBalance01</span> — In KW94, the hydrostatic balance equation is expressed in terms of <math>dP/dM_r</math> instead of <math>dP/dr</math>; and the second term on the right-hand-side allows for a net radial acceleration.

* [<b><font color="red">HK94</font></b>] '''Hansen, C. J. & Kawaler, S. D.''' 1994, Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer)

* [<b><font color="red">P00</font></b>] '''Padmanabhan, T.''' 2000, Theoretical Astrophysics. Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics. Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)
** <span id="P00note_Poisson01">EQ_Poisson01</span> — See also Vol.I: §10.4, Eq.(10.58).

* [<b><font color="red">BLRY07</font></b>] <span id="BLRY07">'''Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W.''' 2007,</span> Numerical Methods in Astrophysics <font size="-1">An Introduction</font> (New York: Taylor & Francis)

=Other Equations with Assigned Templates=

<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
<td colspan=8>
To insert a given equation into any Wiki document, type ...<br /><center>
{{ Template:Math/<i><font color="red">Template_Name</font></i> }}</center>
</td>
</tr>

<tr>
<th width="15%">
<font color="red">Template_Name</font>
</th>
<th width="40%">
<font color="red">Resulting Equation</font>
</th>
<td colspan=6 align="center">
<font color="red">Description</font>
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Continuity02|EQ_Continuity02]]
</td>
<td align="center">
{{Template:Math/EQ Continuity02}}
</td>
<td colspan=6>
Eulerian (and Conservative) form of the continuity equation.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Euler02|EQ_Euler02]]
</td>
<td align="center">
{{Template:Math/EQ Euler02}}
</td>
<td colspan=6>
Eulerian form of the Euler equation.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Euler03|EQ_Euler03]]
</td>
<td align="center">
{{Template:Math/EQ Euler03}}
</td>
<td colspan=6>
Conservative form of the Euler equation.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Euler04|EQ_Euler04]]
</td>
<td align="center">
{{Template:Math/EQ Euler04}}
</td>
<td colspan=6>
Euler equation in terms of vorticity.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ FirstLaw02|EQ_FirstLaw02]]
</td>
<td align="center">
{{Template:Math/EQ FirstLaw02}}
</td>
<td colspan=6>
Adiabatic form of the 1<sup>st</sup> Law of Thermodynamics.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Polytrope01|EQ_Polytrope01]]
</td>
<td align="center">
{{Template:Math/EQ Polytrope01}}
</td>
<td colspan=6>
Polytropic equation of state.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ Polytrope02|EQ_Polytrope02]]
</td>
<td align="center">
{{Template:Math/EQ Polytrope02}}
</td>
<td colspan=6>
Enthalpy in a polytrope.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_Polytrope03|EQ_Polytrope03]]
</td>
<td align="center">
{{Math/EQ_Polytrope03}}
</td>
<td colspan=6>
Density in terms of enthalpy for polytrope.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ_EOSideal00|EQ_EOSideal00]]
</td>
<td align="center">
{{Template:Math/EQ_EOSideal00}}
</td>
<td colspan=6>
Alternate form of the ideal gas equation of state.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ EOSideal02|EQ_EOSideal02]]
</td>
<td align="center">
{{Template:Math/EQ EOSideal02}}
</td>
<td colspan=6>
Alternate form of the ideal gas equation of state.
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ TRApproximation|EQ_TRApproximation]]
</td>
<td align="center">
{{Template:Math/EQ TRApproximation}}
</td>
<td colspan=6>
Gravitational potential exterior to an axisymmetric torus,<br />in the [[User:Tohline/Apps/DysonWongTori#TRApproximation|Thin Ring (TR) Approximation]].
</td>
</tr>

<tr>
<td>
[[Template:Math/EQ CT99Axisymmetric|EQ_CT99Axisymmetric]]
</td>
<td align="center">
{{Template:Math/EQ CT99Axisymmetric}}
</td>
<td colspan=6>
Gravitational potential of any axisymmetric mass distribution.
</td>
</tr>

</table>
</div>

 <br />
{{SGFfooter}}

Template:Math/MP PolytropicIndex

2023-12-12T19:35:46Z

Xxsrm: Created page with "<span title="Polytropic index"><math>n</math></span>"

Template:Math/MP PolytropicConstant

2023-12-12T19:35:06Z

Xxsrm: Created page with "<span title="Polytropic constant"><math>K_\mathrm{n}</math></span>"

<span title="Polytropic constant"><math>K_\mathrm{n}</math></span>

Template:Math/MP MeanMolecularWeight

2023-12-12T19:34:18Z

Xxsrm: Created page with "<span title="Mean molecular weight"><math>\bar{\mu}</math></span>"

Template:Math/MP ElectronMolecularWeight

2023-12-12T19:33:33Z

Xxsrm: Created page with "<span title="Molecular weight of electrons"><math>\mu_e</math></span>"