Editing SSC/Structure/PolytropesEmbedded/Other


==Additional, Numerically Constructed Polytropic Configurations==
As has been detailed in an [[SSC/Structure/Polytropes#Polytropic_Spheres|accompanying chapter]], using numerical techniques we have solved the Lane-Emden equation, and thereby discerned the internal structural profiles, for polytropes having a wide variety of polytropic indexes.  The righthand panel of Figure 3 presents a diagram in which the mass-radius "sequences" corresponding to eight different polytropic indexes have been drawn.

<div align="center" id="DFBsequences">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="2"><br />[[File:DataFileButton02.png|right|75px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 3: &nbsp; Mass-Radius Behavior of Various Polytropic Sequences</th>
</tr>
<tr>
  <td align="center">
[[File:Stahler_MRdiagram1.png|350px|center|Stahler (1983) Figure 17 (edited)]]
  </td>
  <td align="center">
[[File:DFBsequenceB.png|350px|Combined DFB Sequences]]
  </td>
</tr>
</table>
</div>

==Turning Points==

===Limiting Pressure Along M<sub>1</sub> Sequence===
As is illustrated in the figures presented above, when an equilibrium sequence is constructed for any bounded (pressure-truncated) configuration having <math>~n > 3</math>, the sequence exhibits multiple "turning points."  For example, when moving along the R-P sequence [[SSC/Structure/PolytropesEmbedded#WhitworthFig1b|displayed in Figure 1]] for <math>~n=5</math> configurations, the external pressure monotonically climbs to a maximum value, <math>~P_\mathrm{max}</math>, then "turns around" and steadily decreases thereafter.  [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981b)] separately derived an expression that pinpoints the location of the  <math>~P_\mathrm{max}</math> turning point along an R-P sequence &#8212; Kimura refers to this as an "M<sub>1</sub> sequence" because the configuration's mass is held fixed while the external pressure and the system's corresponding equilibrium radius is varied.  The turning point is located along the sequence at the point where,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d P_e}{d R_\mathrm{eq}} \biggr|_M
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
0\, ,
</math>
  </td>
</tr>
</table>
</div>
or, just as well, where,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{d \ln P_e}{d\ln R_\mathrm{eq}} \biggr|_M
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
0\, .
</math>
  </td>
</tr>
</table>
</div>
In what follows, we examine the expressions derived by both authors in order to show that they are identical to one another as well as to re-express the result in a form that conforms to our own adopted notation.

====Horedt's Derivation====

Appreciating that Horedt's notation for the surface pressure of an equilibrium configuration &#8212; which equals the applied external pressure <math>~P_e</math> &#8212; is <math>~\tilde{p}</math>, and his notation for <math>~R_\mathrm{eq}</math> is <math>~\tilde{r}</math>, the requisite expression from Horedt's paper [see also equation (13) in [http://adsabs.harvard.edu/abs/1974A%26A....33..195V Viala &amp; Horedt (1974)]] is the one displayed in the following boxed image:   
<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center">
Excerpt from [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] 
  </td>
</tr>
<tr><td align="left">

<table border="0" align="center">
<tr><td align="center">
[[File:HoredtEq00.png|450px|center|Viala &amp; Horedt (1974) Expressions]]
<!-- [[Image:AAAwaiting01.png|450px|center|Viala &amp; Horedt (1974) Expressions]]-->
</td></tr>
<tr><td align="left">
where,
</td></tr>
<tr><td align="center">
[[File:HoredtEq01.png|300px|center|Viala &amp; Horedt (1974) Expressions]] 
<!-- [[Image:AAAwaiting01.png|300px|center|Viala &amp; Horedt (1974) Expressions]] -->
</td></tr>
</table>

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

That is, from Horedt's work we have,
<div align="center">
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~\frac{dP_e}{dR_\mathrm{eq}}\biggr|_M ~~\rightarrow ~~ \frac{d\tilde{p}}{d \tilde{r}}</math>
  </td>
  <td align="center">
<math>~\sim</math>
  </td>
  <td align="left">
<math>~\frac{(3-n)(n+1)(\tilde\theta^')^2 + (2n+2)\tilde\theta^{n+1}}{(1-n)\tilde\xi f^' + (3-n)(n+1)(\tilde\theta^')^2} \, .</math>
  </td>
</tr>
</table>
</div>
Let's independently derive this relation, starting from Horedt's equilibrium expressions for <math>~\tilde{r}</math> and <math>~\tilde{p}</math>, as [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|summarized above]].  (For purposes of simplification, we will for the most part drop the tilde notation.)
<div align="center">
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~ \frac{1}{R_\mathrm{Horedt}} \cdot \frac{d\tilde{r}}{d \tilde\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\xi}\biggl[ \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(-\xi^2\theta^')^{(1-n)/(n-3)}\biggl[ 1 +\frac{(1-n)}{(n-3)} \cdot \xi (-\xi^2\theta^')^{-1} (-\xi^2\theta^')^' \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(n-3)(n+1)}\cdot (-\xi^2\theta^')^{(1-n)/(n-3)} \biggl[ (n-3)(n+1)
 +(n-1)\cdot (\theta^')^{-2} \xi f^'
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(-\xi^2\theta^')^{(1-n)/(n-3)} }{(3-n)(n+1)(\theta^')^{2}} 
\biggl[ (3-n)(n+1)(\theta^')^{2} +(1-n) \xi f^' \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \frac{1}{P_\mathrm{Horedt}} \cdot \frac{d\tilde{p}}{d \tilde\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\xi}\biggl[ \tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
\biggl[ f^' + f \cdot \frac{2(n+1)}{(n-3)}( -\tilde\xi^2 \tilde\theta' )^{-1} ( -\tilde\xi^2 \tilde\theta' )^' \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}
\biggl[ f^' - \frac{2(n+1)}{(n-3)(n+1)} \cdot \frac{f\cdot f^'}{(\theta^')^2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{f^' ( -\tilde\xi^2 \tilde\theta' )^{(3n+1)/(n-3)} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} }{(3-n)(n+1)(\theta^')^2} 
\biggl[ (3-n)(n+1)(\theta^')^2 + 2(n+1) \theta^{n+1} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

The ratio of these two expressions gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_\mathrm{Horedt}}{P_\mathrm{Horedt}} \cdot \frac{dP_e}{dR_\mathrm{eq}}\biggr|_M </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f^' ( -\tilde\xi^2 \tilde\theta' )^{(3n+1)/(n-3)}  
\biggl\{ \frac{(3-n)(n+1)(\theta^')^2 + 2(n+1) \theta^{n+1}}{(3-n)(n+1)(\theta^')^{2} +(1-n) \xi f^' } \biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
completing our task, as the term inside the curly braces exactly matches the equation excerpt from Horedt's work, as displayed above.

====Kimura's Derivation====

Appreciating that Kimura uses the subscript "1," rather than a tilde, to identify equilibrium parameter values, the requisite expression is equation (22) from [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura's "Paper II,"] as displayed in the following boxed image:   
<div align="center">
<table border="1" align="center" cellpadding="4">
<tr>
  <td align="center">
Excerpts (edited) from [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981b)] 
  </td>
</tr>
<tr><td align="left">

<table border="0" align="center">
<tr><td align="center">
[[File:KimuraEq00.png|500px|center|Kimura (1981b) Expressions]]
<!-- [[Image:AAAwaiting01.png|500px|center|Kimura (1981b) Expressions]] -->
</td></tr>
<tr><td align="left">
where,
</td></tr>
<tr><td align="center">
[[File:KimuraEq01.png|500px|center|Kimura (1981b) Expressions]] 
<!-- [[Image:AAAwaiting01.png|500px|center|Kimura (1981b) Expressions]] -->
</td></tr>
</table>

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

Drawing on the additional parameter and variable definitions provided in our [[SSC/Structure/PolytropesEmbedded#Kimura.27s_Presentation|discussion of Kimura's presentation, above]], we can rewrite this key expression as,
<div align="center">
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~\frac{R_\mathrm{eq}}{P_e} \cdot \frac{dP_e}{dR_\mathrm{eq}}\biggr|_M ~~\rightarrow ~~ \frac{d\ln{p_1}}{d \ln{r_1}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{v_G \cdot h_G}{k_G} \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v_G</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{[1-2(n+1)^{-1}]} =\frac{ 2(n+1)}{n-1} \, ,
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~h_G</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{u_G} \biggl[ \frac{\zeta \theta^n}{\phi^'} \biggr]_1 - \frac{1}{v_G} \biggl[ \frac{\zeta \phi^'}{\theta} \biggr]_1
=
\frac{(n-1)}{(n-3)} \biggl[ \frac{\tilde\xi \tilde\theta^n}{-\tilde\theta^'} \biggr] - 
\frac{(n-1)}{2(n+1)} \biggl[ \frac{(n+1)\tilde\xi (-\tilde\theta^')}{\tilde\theta} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{(n-1)\tilde\xi}{2(n+1)(n-3)\tilde\theta (-\tilde\theta^')}
\biggl\{ 2(n+1) \tilde\theta^{n+1} + (3-n) (n+1) (-\tilde\theta^')^2 \biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~k_G</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1-
\frac{1}{u_G} \biggl[ \frac{\zeta \theta^n}{\phi^'} \biggr]_1 
=1-
\frac{(n-1)}{(n-3)} \biggl[ \frac{\tilde\xi \tilde\theta^n}{-\tilde\theta^'} \biggr] 
=
\frac{1}{ (n-3) (-\tilde\theta^') } \biggl\{ (n-3)(- \tilde\theta^') - (n-1) \tilde\xi \tilde\theta^n \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{1}{ (n+1)(n-3) (-\tilde\theta^')^2 } \biggl\{ (n-3)(n+1) (-\tilde\theta^')^2 
- (n-1)\tilde\xi [(n+1) \tilde\theta^n (-\tilde\theta^')] \biggr\} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{-1}{ (n+1)(n-3) (-\tilde\theta^')^2 } \biggl\{ (3-n)(n+1) (-\tilde\theta^')^2 
+ (1-n)\tilde\xi [(n+1) \tilde\theta^n (\tilde\theta^')] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, from Kimura's work we find,
<div align="center">
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~\frac{R_\mathrm{eq}}{P_e} \cdot \frac{dP_e}{dR_\mathrm{eq}}\biggr|_M </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n+1)\tilde\xi \tilde\theta^'}{\tilde\theta}
\biggl\{ \frac{2(n+1) \tilde\theta^{n+1} + (3-n) (n+1) (\tilde\theta^')^2}{(3-n)(n+1) (\tilde\theta^')^2 
+ (1-n)\tilde\xi [(n+1) \tilde\theta^n \tilde\theta^'] } \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Appreciating that <math>~f^' = [(n+1)\tilde\theta^n \tilde\theta^']</math>, we see that the expression inside the curly braces here matches exactly the expression inside the curly braces that was obtained through Horedt's derivation, as it should!  The prefactor is different in the two expressions only because Kimura's result is for a logarithmic derivative whereas Horedt's derivation is not; the ratio of the two prefactors is, simply, the ratio,
<div align="center">
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~\frac{R_\mathrm{eq}/R_\mathrm{Horedt}}{P_e/P_\mathrm{Horedt}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\tilde\xi}{\tilde\theta_n^{n+1}}\cdot ( -\tilde\xi^2 \tilde\theta' )^{[(1-n)-2(n+1)]/(n-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\tilde\xi}{\tilde\theta_n^{n+1}}\cdot ( -\tilde\xi^2 \tilde\theta' )^{-(3n+1)/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>

In a [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#KimuraApplication|separate discussion]], specifically focused on the <math>~n=5</math> mass-radius relationship, we show how Kimura's analysis of turning points can be usefully applied.

====Location of Pressure Limit====

Now we can identify the location along the M<sub>1</sub> sequence where the turning point set by <math>~P_\mathrm{max}</math> occurs by setting the numerator of this expression equal to zero, specifically,
<div align="center">
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>~2(n+1) \tilde\theta^{n+1} + (3-n) (n+1) (\tilde\theta^')^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, .
</math>
  </td>
</tr>
</table>
</div>
This means that the equilibrium model that sits at the <math>~P_\mathrm{max}</math> turning point will have,
<div align="center">
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>~\frac{\tilde\theta^{n+1}}{(\tilde\theta^')^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(n-3)}{2} \, .
</math>
  </td>
</tr>
</table>
</div>

===Other Limits===

In a similar fashion, [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981b)] derived mathematical expressions that identify the location of other turning points along equilibrium sequences of bounded polytropic configurations.  An M<sub>1</sub> sequence &#8212; as displayed, for example, in the set of P-R diagrams shown in [[SSC/Structure/PolytropesEmbedded#WhitworthFig1b|Figure 1, above]] &#8212; exhibits not only an "extremal of p<sub>1</sub>" but also an "extremal of r<sub>1</sub>."  As we have [[SSC/Structure/PolytropesEmbedded#Location_of_Pressure_Limit|just reviewed]], the first of these is identified by setting <math>~(d\ln p_1/d\ln r_1)_{M} = 0</math> or, using Kimura's more compact terminology, the first occurs at a location that satisfies the condition, 
<div align="center">
<math>h_G = 0 \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
that is, where &hellip; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~\tilde\theta^{n+1} (\tilde\theta^')^{-2} = (n-3)/2 \, .</math>
</div>
Similarly, Kimura points out that an "extremal in r<sub>1</sub>" along an M<sub>1</sub> sequence occurs at a location that satisfies the condition,
<div align="center">
<math>k_G = 0 \, ,</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
that is, where &hellip; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~\tilde\xi \tilde\theta^{n} (-\tilde\theta^')^{-1} = (n-3)/(n-1) \, .</math>
</div>

As is illustrated by the plots presented in [[SSC/Structure/PolytropesEmbedded#Stahler1983Fig17|Figure 2, above]], turning points also arise in the mass-radius relationship of bounded polytropic configurations having <math>~n > 3</math>.  These are identified by Kimura as "p<sub>1</sub> sequences" because the external pressure is held fixed while the system's mass and corresponding equilibrium radius is varied.  In &sect;5 of his [http://adsabs.harvard.edu/abs/1981PASJ...33..299K "Paper II,"] Kimura points out that the same two conditions &#8212; namely, <math>~h_G = 0</math> and <math>~k_G = 0</math> &#8212; also identify the location of extrema in M<sub>1</sub> along, respectively, p<sub>1</sub> sequences and r<sub>1</sub> sequences.


We can also identify extrema in r<sub>1</sub> along p<sub>1</sub> sequences by setting <math>~(\dot{p}_1/p_1) = 0</math> in [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura's] equation (17), then substituting the resulting expression for the function <math>~Z</math>, namely,
<div align="center">
<math>~Z = v_1 \, ,</math>
</div>
into his equations (15) and (16).  The ratio of these two resulting expressions gives,
<div align="center">
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>~\frac{d\ln M_1}{d \ln r_1}\biggr|_{p_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{u_1 -(u_G/v_G)v_1}{1 - v_1/v_G}
=
[u_1 v_G - u_G v_1][v_G - v_1]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{2(n+1)}{(n-1)} \cdot \frac{\xi \theta^n}{(-\theta^')}  - \frac{(n-3)}{(n-1)} \cdot \frac{(n+1)\xi (-\theta^')}{\theta} \biggr] 
\biggl[\frac{2(n+1)}{(n-1)} - \frac{(n+1)\xi (-\theta^')}{\theta}  \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\xi }{(-\theta^')} \biggl[ \frac{2 \theta^{n+1}  - (n-3) (-\theta^')^2 }{2\theta - (n-1)\xi (-\theta^')  } \biggr]
</math>
  </td>
</tr>
</table>
</div>

<span id="TurningPointXmax">As has just been reviewed, the condition <math>~h_G=0</math> results from setting the numerator of this expression equal to zero and identifies extrema in M<sub>1</sub> along p<sub>1</sub> sequences.  In addition, now, we can identify the condition for extrema in r<sub>1</sub> along p<sub>1</sub> sequences by setting the denominator to zero.</span>  The condition is,
<div align="center">
<math>~\frac{\xi (-\theta^')}{\theta} = \frac{2}{(n-1)} \, .</math>
</div>

===Some Tabulated Values===

<div align="center" id="Table3">
<table border="1" cellpadding="8" align="center">
<tr><th align="center" colspan="14">Table 3: &nbsp; Turning-Point Locations along M-R Sequences of Pressure-Truncated Polytropes</th></tr>
<tr>
  <td align="center" rowspan="2">n</td>
  <td align="center" colspan="6"><font color="yellow" size="+2">&#x25CF;</font> Maximum Radius <font color="yellow" size="+2">&#x25CF;</font></td>
  <td align="center" colspan="6"><font color="darkgreen" size="+2">&#x25CF;</font> Maximum Mass <font color="darkgreen" size="+2">&#x25CF;</font></td>
</tr>
<tr>
  <td align="center"><math>~\tilde\xi</math></td>
  <td align="center"><math>~\tilde\theta</math></td>
  <td align="center"><math>~\biggl|\frac{d\theta}{d\xi}\biggr|_\tilde\xi</math></td>
  <td align="center"><math>~\frac{(n-1)}{2}\biggl[ \frac{\xi}{\theta} \biggl|\frac{d\theta}{d\xi}\biggr|~\biggr]_\tilde\xi</math></td>
  <td align="center"><math>~\frac{R}{R_\mathrm{SWS}}</math>
  <td align="center"><math>~\frac{M}{M_\mathrm{SWS}}</math>
  <td align="center"><math>~\tilde\xi</math></td>
  <td align="center"><math>~\tilde\theta</math></td>
  <td align="center"><math>~\biggl|\frac{d\theta}{d\xi}\biggr|_\tilde\xi</math></td>
  <td align="center"><math>~\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2 \biggr]_\tilde\xi</math></td>
  <td align="center"><math>~\frac{R}{R_\mathrm{SWS}}</math>
  <td align="center"><math>~\frac{M}{M_\mathrm{SWS}}</math>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">2.172</td>
  <td align="center">0.5387</td>
  <td align="center">0.2496</td>
  <td align="center">1.006</td>
  <td align="center">0.5717</td>
  <td align="center">1.726</td>
  <td align="center">6.89684862</td>
  <td align="center">0.0</td>
  <td align="center">-0.04242976</td>
  <td align="center">--</td>
  <td align="center">0.0</td>
  <td align="center">2.9583456</td>
</tr>
<tr>
  <td align="center">3.05</td>
  <td align="center">2.162</td>
  <td align="center">0.5437</td>
  <td align="center">0.2479</td>
  <td align="center">1.010</td>
  <td align="center">0.5704</td>
  <td align="center">1.715</td>
  <td align="center">5.034</td>
  <td align="center">0.1152</td>
  <td align="center">0.07842</td>
  <td align="center">0.973</td>
  <td align="center">0.2707</td>
  <td align="center">2.829</td>
</tr>
<tr>
  <td align="center">3.5</td>
  <td align="center">2.050</td>
  <td align="center">0.5930</td>
  <td align="center">0.2340</td>
  <td align="center">1.011</td>
  <td align="center">0.5630</td>
  <td align="center">1.594</td>
  <td align="center">3.910</td>
  <td align="center">0.2788</td>
  <td align="center">0.1126</td>
  <td align="center">0.994</td>
  <td align="center">0.4180</td>
  <td align="center">2.311</td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center"><math>~\sqrt{3}</math></td>
  <td align="center"><math>~\frac{1}{\sqrt{2}}</math></td>
  <td align="center"><math>~\frac{1}{\sqrt{24}}</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~\biggl( \frac{3\cdot 5}{2^4\pi}\biggr)^{1 / 2}</math></td>
  <td align="center"><math>~\biggl( \frac{3\cdot 5^3}{2^6\pi}\biggr)^{1 / 2}</math></td>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~\frac{1}{2}</math></td>
  <td align="center"><math>~\frac{1}{8}</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~\biggl( \frac{3^2\cdot 5}{2^6\pi}\biggr)^{1 / 2}</math></td>
  <td align="center"><math>~\biggl( \frac{3^4\cdot 5^3}{2^{10}\pi}\biggr)^{1 / 2}</math></td>
</tr>
<tr>
  <td align="center">6</td>
  <td align="center">1.6</td>
  <td align="center">0.7510</td>
  <td align="center">0.1884</td>
  <td align="center">1.003</td>
  <td align="center">0.5404</td>
  <td align="center">1.301</td>
  <td align="center">2.7</td>
  <td align="center">0.5811</td>
  <td align="center">0.1221</td>
  <td align="center">0.999</td>
  <td align="center">0.4802</td>
  <td align="center">1.635</td>
</tr>
</table>
</div>