Editing SSC/Virial/PolytropesEmbedded/FirstEffortAgain/Pt1 (section)

===Mass-Radius Relationship===
Third, [[Appendix/References|Chandrasekhar [C67]]] shows &#8212; see his Equation (72), p. 98 &#8212; that the general mass-radius relationship for isolated spherical polytropes is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~GM^{(n-1)/n} R^{(3-n)/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{(n+1)K}{(4\pi)^{1/n}} \biggl[ - \xi^{(n+1)/(n-1)} \frac{d\Theta_H}{d\xi} \biggr]^{(n-1)/n}_{\xi=\xi_1} \, ,
</math>
  </td>
</tr>
</table>
</div>
which we choose to rewrite as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4\pi \biggl( \frac{G}{K}\biggr)^n M^{(n-1)} R^{(3-n)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~(n+1)^n\biggl[ \xi^{(n+1)} (-\Theta^')^{(n-1)}\biggr]_{\xi=\xi_1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \biggl( \frac{\xi}{\Theta^'} \biggr)_{\xi=\xi_1} \biggl[ (n+1) \xi ~(-\Theta^') \biggr]^n_{\xi=\xi_1} \, .
</math>
  </td>
</tr>
</table>
</div>

By comparison, the expression for the equilibrium radius that has been derived, above, from an analysis of extrema in the free energy function &#8212; specifically, see the last expression in the left-hand column of the [[SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]] &#8212; we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~4\pi \biggl( \frac{G}{K}\biggr)^n M^{(n-1)} R_\mathrm{eq}^{(3-n)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{3}{\mathfrak{f}_M} \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, it appears as though, quite generally,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~ \frac{1}{\mathfrak{f}_M} \biggl( \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \biggl( \frac{\xi}{3\Theta^'} \biggr)_{\xi=\xi_1} \biggl[ (n+1) \xi ~(-\Theta^') \biggr]^n_{\xi=\xi_1}  \, .
</math>
  </td>
</tr>
</table>
</div>
Or, taking into account the expressions for <math>~\mathfrak{f}_M</math> and <math>~\mathfrak{f}_W</math> that have just been uncovered, we conclude that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ (n+1) \xi ~(-\Theta^') \biggr]_{\xi=\xi_1}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{\mathfrak{f}_A}{\mathfrak{f}_W}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{(n+1) }{3\cdot 5} ~\xi_1^2 \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{(n+1) }{3\cdot 5} ~\xi_1^2 \biggl\{ \frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\xi_1}  \biggr\}\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\xi_1}  \, .
</math>
  </td>
</tr>
</table>
</div>