Editing SSCpt1/Virial/FormFactors/Pt3

=Structural Form Factors (Pt 3)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" bgcolor="lightblue" width="33%"><br />[[SSCpt1/Virial/FormFactors|Part I:&nbsp; Synopsis]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="33%"><br />[[SSCpt1/Virial/FormFactors/Pt2|Part II:&nbsp; n = 5 Polytrope]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"><br />[[SSCpt1/Virial/FormFactors/Pt3|Part III:&nbsp; n = 1 Polytrope]]
&nbsp;
  </td>
</tr>
</table>

==Second Detailed Example (n = 1)==


===Foundation (n = 1)===
We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations|our more general introductory discussion]]:
<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr><th align="center" colspan="2">
Adopted Normalizations <math>(n=1; ~\gamma=2)</math>
</th></tr>
<tr><td align="center" colspan="2">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl(\frac{K}{G}\biggr)^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl( \frac{G^3 M_\mathrm{tot}^2}{K^2}\biggr)  </math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
----
  </td>
</tr>

<tr>
  <td align="right">
<math>E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 = 
\biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot}^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi}\biggl( \frac{G}{K} \biggr)^{3/2} M_\mathrm{tot} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c^2_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl( \frac{G^3}{K} \biggr)^{1/2} M_\mathrm{tot}  </math>
  </td>
</tr>
</table>

</td>
</tr>

<tr><th align="left" colspan="2">
Note that the following relations also hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}} 
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</th></tr>
</table>
</div>


As is detailed in our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of the properties of ''isolated'' polytropes]], in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{1}</math>, where,
<div align="center">
<math>
a_{1} = \biggl( \frac{K}{2\pi G} \biggr)^{1/2}  \, ,
</math>
</div>
the radial profile of various physical variables is as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{r}{(K/G)^{1/2}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\rho}{\rho_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\sin\xi}{\xi} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{P}{K\rho_0^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\sin\xi}{\xi} \biggr)^{2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{M_r}{(K/G)^{3/2}\rho_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{2}{\pi} \biggr)^{1/2} (\sin\xi - \xi \cos\xi) \, .</math>
  </td>
</tr>
</table>
</div>
Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that &#8212; see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope|accompanying discussion]] &#8212; in ''isolated'' <math>n=1</math> polytropes, the total mass is given by the expression,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2}  \rho_0  
~~~~\Rightarrow ~~~~
\rho_0 = \biggr[ \frac{G^3}{2\pi K^3} \biggr]^{1/2}   M_\mathrm{tot} \, .</math>
</div>
Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
<div align="center" id="NormalizedProfiles1">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r^\dagger \equiv \frac{r}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho^\dagger \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{2^3 \pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^\dagger \equiv \frac{P}{P_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2\pi} \biggl( \frac{\sin\xi}{\xi}\biggr)^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{M_r}{M_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\pi}  (\sin\xi - \xi \cos\xi) 
\, .</math>
  </td>
</tr>
</table>
</div>

===Mass1 (n = 1)===
While we already know the expression for the <math>M_r</math> profile, having copied it from our [[SSC/Structure/Polytropes#.3D_1_Polytrope|discussion of detailed force-balanced models of ''isolated'' polytropes]], let's show how that profile can be derived by integrating over the density profile.  After employing the ''norm''-subscripted quantities, as defined above, to normalize the radial coordinate and the mass density in our [[SSCpt1/Virial#Normalize|introductory discussion of the virial theorem]], we obtained the following integral defining the, 

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_r(r^\dagger)  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{tot} \int_0^{r^\dagger}  3(r^\dagger)^2 \rho^\dagger dr^\dagger  \, .
</math>
  </td>
</tr>
</table>
</div>
Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math> gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot} } </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3 \int_0^{\xi}  \frac{\xi^2}{2\pi} \biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2} \frac{\sin\xi}{\xi} \cdot \frac{d\xi }{(2\pi)^{1/2}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3( 2\pi)^{-3/2}\biggl( \frac{2^3\pi}{3^2} \biggr)^{1/2}   \int_0^{\xi}  \xi \sin\xi d\xi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\pi} (\sin\xi - \xi\cos\xi) \, .
</math>
  </td>
</tr>
</table>
</div>
As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above.  Notice that if we decide to truncate an <math>n=1</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> &#8212; as in the discussion that follows &#8212; the mass of this truncated configuration will be, simply,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot} }  = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
  </td>
</tr>
</table>
</div>

===Mass2 (n = 1)===

Alternatively, as has been laid out in our [[SSCpt1/Virial#Summary_of_Normalized_Expressions|accompanying summary of normalized expressions that are relevant to free-energy calculations]],
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_0} \biggr]  dx \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>M_\mathrm{limit}</math> is the "total" mass of the polytropic configuration that is truncated at <math>R_\mathrm{limit}</math>; keep in mind that, here,
<div align="center">
<math>M_\mathrm{tot} = \biggr[ \frac{2\pi K^3}{G^3} \biggr]^{1/2}  \rho_0  \, ,</math>
</div>
is the total mass of the ''isolated'' <math>n=1</math> polytrope, that is, a polytrope whose ''Lane-Emden'' radius extends all the way to <math>\xi_1 = \pi</math>.  In our discussions of truncated polytropes, we often will use <math>\tilde\xi \le \xi_1</math> to specify the truncated radius in terms of the familiar, dimensionless Lane-Emden radial coordinate, so here we will set,
<div align="center">
<math>R_\mathrm{limit} = a_1 \tilde\xi ~~~~\Rightarrow ~~~~ x = \frac{r}{R_\mathrm{limit}} = \frac{a_1 \xi}{a_1 \tilde\xi} = \frac{\xi}{\tilde\xi} \, .</math>
</div>
Hence, in terms of the desired integration coordinate, <math>x</math>, the density profile provided above becomes,

<div align="center" id="rhoofx1">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>\frac{\rho(x)}{\rho_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\sin(\tilde\xi x)}{\tilde\xi x} \, ,</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>
and the integral defining <math>M_r(x)</math> becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi} \int_0^{x}  x \sin(\tilde\xi x)  dx </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{3}{\tilde\xi^3} 
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, . </math>
  </td>
</tr>
</table>
</div>
In this case, integrating "all the way out to the surface" means setting <math>r = R_\mathrm{limit}</math> and, hence, <math>x = 1</math>; by definition, it also means <math>M_r(x) = M_\mathrm{limit}</math>.  Therefore we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\frac{3}{\tilde\xi^3} [\sin \tilde\xi  - \tilde\xi \cos \tilde\xi  ] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{\tilde\xi^3} [\sin \tilde\xi  - \tilde\xi \cos \tilde\xi  ]   \, .</math>
  </td>
</tr>
</table>
</div>

Using this expression for the mean-to-central density ratio along with the expression for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>, derived in the preceding subsection, we also can state that for truncated <math>n=1</math> polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) 
\biggl\{ \frac{[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ]}{( \sin \tilde\xi  - \tilde\xi \cos \tilde\xi  )} \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] </math>
  </td>
</tr>
</table>
</div>
By making the substitution, <math>x \rightarrow \xi/\tilde\xi</math>, this expression becomes identical to the <math>M_r/M_\mathrm{tot}</math> [[#NormalizedProfiles1|profile presented just before the "Mass1" subsection]], above.  In summary, then, we have the following two equally valid expressions for the <math>M_r</math> profile &#8212; one expressed as a function of <math>\xi</math> and the other expressed as a function of <math>x</math>:

<div align="center" id="2MassProfiles">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">

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

<tr>
  <td align="right">
<math>\frac{M_r(\xi)}{M_\mathrm{tot}}</math> 
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\pi} (\sin\xi - \xi \cos\xi  ) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}</math> 
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\pi} [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
  </td>
</tr>
</table>

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

===Mean-to-Central Density (n = 1)===

[[SSCpt1/Virial/FormFactors#Mean-to-Central_Density|Following the line of reasoning provided above]], we can use the just-derived central-to-mean density ratio to specify one of the structural form factors.  Specifically,
<div align="center">
<math>~\mathfrak{f}_M\biggr|_{n=1} = \frac{\bar\rho}{\rho_c} = \frac{3}{\tilde\xi^3} [\sin \tilde\xi  - \tilde\xi \cos \tilde\xi  ] \, .</math>
</div>

===Gravitational Potential Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the gravitational potential energy is,

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

<tr>
  <td align="right">
<math>~\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~  3\cdot 5 \int_0^1 \biggl\{ \int_0^x  \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\}  \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, .</math>
  </td>
</tr>
</table>
</div>
From the derivations already presented, above, for <math>~n=1</math> polytropic configurations, we know all of the functions under this integral.  We know, for example, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl\{ \int_0^x  \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\tilde\xi^3} 
[\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \, .</math>
  </td>
</tr>
</table>
</div>
Hence, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], we have,

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

<tr>
  <td align="right">
<math>~\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
\frac{3\cdot 5}{\tilde\xi^4} \int_0^1 \biggl\{ [\sin(\tilde\xi x) - (\tilde\xi x)\cos(\tilde\xi x) ] \biggr\}  
\sin(\tilde\xi x) dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
\frac{3\cdot 5}{\tilde\xi^4} \biggl\{ \frac{x}{2} - \frac{\sin(2\tilde\xi x)}{4\tilde\xi} - \tilde\xi\biggl[
\frac{\sin(2\tilde\xi x) - 2\tilde\xi x\cos(2\tilde\xi x)}{8\tilde\xi^2}
\biggr] \biggr\}_0^1  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~  
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Now that we have expressions for, both, <math>~\mathfrak{f}_M</math> and <math>~\mathfrak{f}_W</math>, we can determine an analytic expression for the normalized gravitational potential energy for truncated, <math>~n=1</math> polytropes.  As is shown in [[SSCpt1/Virial#Structural_Form_Factors|a companion discussion]],
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- 3\mathcal{A} \chi^{-1}  \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\chi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
 <td align="left">
<math>
\frac{R_\mathrm{limit}}{R_\mathrm{norm}} = \biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, .
</math>
  </td>
</tr>
</table>
</div>
A summary of derived expressions, from above, gives,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~\frac{3}{\tilde\xi^3} [\sin \tilde\xi  - \tilde\xi \cos \tilde\xi  ]  \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~  
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\pi} (\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, .
</math>
  </td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~\frac{1}{5} \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^{2} \mathfrak{f}_W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

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

===Thermal Energy (n = 1)===
As presented at the [[#Structural_Form_Factors|top of this page]], the structural form factor associated with determination of the configuration's thermal energy is,

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

<tr>
  <td align="right">
<math>\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr]  x^2 dx \, ,</math>
  </td>
</tr>
</table>
</div>
Given that an expression for the normalized pressure profile, <math>P/P_0</math>, has already [[#Foundation_2|been provided, above]], we can carry out the integral immediately.  Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{P(\xi)}{P_0} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\sin\xi}{\xi}\biggr)^{2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{P(x)}{P_0} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{\sin(\tilde\xi x)}{(\tilde\xi x)}\biggr]^{2} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, with the aid of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], the relevant integral gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{\tilde\xi^2} \int_0^1 \sin^2(\tilde\xi x) dx </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\frac{3}{\tilde\xi^2} 
\biggl[\frac{x}{2}- \frac{\sin(2\tilde\xi x)}{4\tilde\xi} \biggr]_0^1 
= \frac{3}{2^2\tilde\xi^3}  \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>


<table border="1" width="90%" align="center" cellpadding="10">
<tr><td align="left">

<font color="maroon">'''ASIDE:'''</font> Having this expression for <math>\mathfrak{f}_A</math> allows us to determine an analytic expression for the coefficient, <math>\mathcal{B}</math>, that appears in our general expression for the free energy, and that can be straightforwardly used to obtain an expression for the thermal energy content of <math>n=1 (\gamma=2)</math> polytropic configurations.  From our [[SSCpt1/Virial#Gathering_it_all_Together|accompanying introductory discussion]], we have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\mathcal{B}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr)
\biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{2}
\cdot \mathfrak{f}_A \, .
</math>
  </td>
</tr>
</table>
</div>
The various factors in the definition of <math>\mathcal{B}</math> and <math>S_\mathrm{therm}</math> are (see above),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\chi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
\biggl(\frac{1}{2\pi}\biggr)^{1/2} \tilde\xi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot} }\biggr)\frac{1}{\mathfrak{f}_M} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\tilde\xi^3}{3\pi} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>~  
\frac{3}{2^2\tilde\xi^3}  \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\mathcal{B}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3}{2^2 \pi} \biggr) \biggl[ \frac{\tilde\xi^3}{3\pi} \biggr]^2
\frac{3}{2^2\tilde\xi^3}  \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>
</table>
</div>
and (see [[VE#Adiabatic_Systems|here]] and [[SSCpt1/Virial#Structural_Form_Factors|here]]),
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3}{2}(\gamma - 1)\biggl[ \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}}\biggr] 
= \frac{3}{2} \cdot \chi^{3(1-\gamma)} \mathcal{B}
= \frac{3}{2} \cdot \chi^{-3} \mathcal{B}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3 \tilde\xi^3}{2^5\pi^3} \biggr) \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr]
\biggl[ (2\pi)^{3/2} \tilde\xi^{-3} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

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

===Summary (n = 1)===
In summary, for <math>~n=1</math> structures we have,

<div align="center">
<table border="1" align="center" cellpadding="10">
<tr><th align="center">
Structural Form Factors (n = 1)
</th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\frac{3}{\tilde\xi^3} [\sin \tilde\xi  - \tilde\xi \cos \tilde\xi  ] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3}{2^2\tilde\xi^3}  \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>
</table>

</td></tr>
<tr><th align="center">
Free-Energy Coefficients (n = 1)
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\mathcal{A}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
\frac{1}{2^3\cdot 3\pi^2} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathcal{B}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\tilde\xi^3}{2^4\pi^3} \biggr) \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>
</table>
<tr><th align="center">
Normalized Energies (n = 1)
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3^2}{2^{7}\pi^3} \biggr)^{1/2} \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
- \biggl(\frac{1}{2^5\pi^3} \biggr)^{1/2} \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]  
</math>
  </td>
</tr>
</table>

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

===Reality Checks (n = 1)===
====Expectation from Stahler's Equilibrium Models====
If we add twice the thermal energy to the gravitational potential energy, we obtain,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>2\biggl(\frac{S_\mathrm{therm}}{E_\mathrm{norm}}\biggr)+ \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} \biggl\{ \biggl[6\tilde\xi  - 3\sin(2\tilde\xi ) \biggr]
- \biggl[ 4\tilde\xi - 3 \sin(2\tilde\xi) + 2\tilde\xi \cos(2\tilde\xi ) \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
\biggl( \frac{1}{2^{5}\pi^3} \biggr)^{1/2} 2\tilde\xi\biggl\{ 1-\cos(2\tilde\xi ) \biggr\}
= \biggl( \frac{1}{2\pi^3} \biggr)^{1/2} \tilde\xi \sin^2(\tilde\xi ) \, .
</math>
  </td>
</tr>
</table>
</div>
For embedded polytropes, this should be compared against the expectation (prediction) [[#Generic_Reality_Check|provided by Stahler's equilibrium models, as detailed above]].  Given that, for <math>n=1</math> polytropes &#8212; see the [[#Mass1_.28n_.3D_1.29|"Mass1" discussion above]] and our accompanying [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D1.29|tabular summary of relevant properties]],
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>
\frac{M_\mathrm{limit}}{M_\mathrm{tot}} = \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)
</math>
  </td>

  <td align="center">
&nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp;
  </td>

  <td align="right">
<math>
\theta_1 = \frac{\sin\tilde\xi}{\tilde\xi}
</math>
  </td>

  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
  </td>

  <td align="right">
<math>
-\frac{d\theta_1}{d\xi} \biggr|_{\tilde\xi} = \frac{1}{\tilde\xi^2}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \, ,
</math>
  </td>
</tr>
</table>

</div>

the expectation is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)} 
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{2}{\pi} \biggr]^{-1/2} \biggl[ \frac{1}{\pi}(\sin\tilde\xi - \tilde\xi \cos\tilde\xi) \tilde\xi^2(\sin\tilde\xi - \tilde\xi \cos\tilde\xi)^{-1}\biggr]^{2}
\biggl( \frac{\sin\tilde\xi}{\tilde\xi}\biggr)^2 \tilde\xi^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1}{2\pi^3}\biggr)^{1/2} \tilde\xi \sin^2\tilde\xi \, .
</math>
  </td>
</tr>
</table>
</div>
This precisely matches our sum of the thermal and gravitational potential energies, as just determined using our expressions for the structural form factors, giving us additional confidence that our form-factor expressions are correct.

====Compare With General Expressions Based on VH74 Work====
Based on the general expressions [[#PTtable|derived above]] in the context of {{ VH74hereafter }}, for the specific case of <math>n=1</math> polytropic configurations, the three structural form factor should be,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{3\cdot 5}{4\tilde\xi^2} 
\biggl[\tilde\theta^{2} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \biggl[ 3\tilde\theta^{2} +  
3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \, .
</math>
  </td>
</tr>
</table>
Also, remember that,

<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>\frac{\sin\xi}{\xi}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~~\theta^' \equiv \frac{d\theta}{d\xi}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\cos\xi}{\xi} - \frac{\sin\xi}{\xi^2} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~~(\theta^' )^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{1}{\xi^4} \biggl[ \xi\cos\xi - \sin\xi \biggr]^2
=\frac{1}{\xi^4} \biggl[ \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr] \, .
</math>
  </td>
</tr>
</table>
Now, let's look at the structural form factors, one at a time.  First, we have,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{\xi^3} \biggl[\sin\xi  - \xi\cos\xi \biggr]</math>
  </td>
</tr>
</table>

which matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]].  Next,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~ \frac{3\cdot 5}{4\xi^2} \biggl[ \frac{\sin^2\xi}{\xi^2} 
+  \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi  - \xi\cos\xi \biggr) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 \sin^2\xi
+  3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi  - \xi\cos\xi \biggr) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{3\cdot 5}{4\xi^6} \biggl[ \xi^2 +  2\xi^2\cos^2\xi - 3 \xi\sin\xi \cos\xi  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{3\cdot 5}{4\xi^6} \biggl\{ \xi^2 +  \xi^2[1+\cos(2\xi)] - \frac{3}{2} \xi\sin(2\xi)  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{3\cdot 5}{8\xi^6} \biggl[ 4\xi^2 +  2\xi^2 \cos(2\xi) - 3 \xi\sin(2\xi)  \biggr] \, ,
</math>
  </td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]].  Finally,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \biggl[ \frac{3\sin^2\xi}{\xi^2} 
+  \frac{3}{\xi^4} \biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- \frac{3\sin\xi}{\xi^4} \biggl(\sin\xi  - \xi\cos\xi \biggr) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+  3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi + \sin^2\xi \biggr)
- 3\sin\xi \biggl(\sin\xi  - \xi\cos\xi \biggr) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2\xi^4} \biggl[ 3\xi^2 \sin^2\xi
+  3\biggl( \xi^2\cos^2\xi - 2\xi\sin\xi \cos\xi \biggr)
+ 3\xi \sin\xi \cos\xi  \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{2\xi^4} \biggl[ \xi^2 - \xi \sin\xi \cos\xi  \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{2^2\xi^3} \biggl[ 2\xi - \sin(2\xi) \biggr] \, ,</math>
  </td>
</tr>
</table>
which also matches the expression presented in the [[#Summary_.28n.3D1.29|summary table, above]].  So this adds support to the deduction, above, that {{ VH74hereafter }} have provided us with the information necessary to develop general expressions for the three structural form factors.

==Fiddling Around==
NOTE (from Tohline on 17 March 2015):  Chronologically, this "Fiddling Around" subsection was developed before our discovery of the {{ VH74hereafter }} derivations.  It put us on track toward the correct development of general expressions for the structural form factors that are applicable to pressure-truncated polytropic spheres.  But this subsection's conclusions are superseded by the {{ VH74hereafter }} work.

In this subsection, for simplicity, we will omit the "tilde" over the variable <math>\xi</math>.  In the case of <math>n=1</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~\mathfrak{f}_A - \theta^{n+1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2^2\xi^3}  \biggl[6\xi  - 3\sin(2\xi ) \biggr] 
- \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2^2\xi^3}  \biggl[4\xi  - 3\sin(2\xi ) + 2\xi\cos(2\xi) \biggr] \, .
</math>
  </td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{2^3 \xi^5}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\biggl[ 4\xi - 3\sin(2\xi) + 2\xi \cos(2\xi ) \biggr] \, .
</math>
  </td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{2 \xi^2}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
  </td>
</tr>

</table>
</div>

Similarly, in the case of <math>n = 5</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta^{n+1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
(1 + \ell^2)^{-3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl( \frac{2^3}{3} \ell^{3}  \biggr) \biggl[ \mathfrak{f}_A - \theta^{n+1} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[ \tan^{-1}(\ell ) + \ell (\ell^4-1) (1+\ell^2)^{-3} ]  - \biggl( \frac{2^3}{3} \ell^{3}  \biggr) (1 + \ell^2)^{-3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\tan^{-1}(\ell ) + \ell \biggl(\ell^4-\frac{8}{3}\ell^2 - 1 \biggr) (1+\ell^2)^{-3} \, .
</math>
  </td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{2^4}{5} \cdot \ell^{5} \biggr) \mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell )  \, .
</math>
  </td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{2\cdot 3}{5} \cdot \ell^{2} \biggr) \mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\mathfrak{f}_A - \theta^{n+1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~\biggl( \frac{2\xi^2}{5} \biggr) \mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
  </td>
</tr>

</table>
</div>
This is pretty amazing!  Both examples produce almost exactly the same relationship between the two structural form factors, <math>\mathfrak{f}_A</math> and <math>\mathfrak{f}_W</math>.  I think that we are well on our way toward nailing down the generic, analytic relationship and, in turn, a generally applicable mass-radius relationship for pressure-truncated polytropic configurations.

Okay &hellip; here is the final piece of information.  In the case of isolated polytropes, we know that the correct expressions for the structural form factors are as summarized in the following table:
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} 
</math>
  </td>
</tr>
</table>

</td>
</tr>
</table>
</div>
We notice, from this, that the ratio,

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

<tr>
  <td align="right">
<math>\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} \cdot \frac{5-n}{3^2\cdot 5} \biggl[ \frac{\xi}{\Theta^'} \biggr]^{2}_{\tilde\xi} 
</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 }{3\cdot 5} \, .
</math>
  </td>
</tr>
</table>
Even in the case of the two pressure-truncated polytropes, analyzed above, this ratio proves to give the correct prefactor on <math>\mathfrak{f}_W</math>.  So we ''suspect'' that the universal relationship between the two form factors is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[ \frac{(n+1) \xi^2 }{3\cdot 5} \biggr] \mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
  </td>
</tr>

</table>
</div>

=See Also=


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