Editing SSC/Virial/Polytropes (section)

==Isolated Nonrotating Adiabatic Configuration==
For a nonrotating configuration <math>~(C=J=0)</math> that is not influenced by the effects of a bounding external medium <math>~(D=P_e = 0)</math>, the statement of virial equilibrium is,
<div align="center">
<math>
3 B\chi_\mathrm{eq}^{3 -3\gamma_g} -~3A\chi_\mathrm{eq}^{-1}  = 0 \, .
</math>
</div>

Hence, one equilibrium state exists for each value of <math>~\gamma_g</math> and it occurs where,

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

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{4-3\gamma_g} = \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{4-3\gamma_g} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{A}{B} \, .
</math>
  </td>
</tr>
</table>
</div>

<div align="center" id="TwoPointsOfView">
<table border="1" width="100%" cellpadding="8">
<tr>
  <th align="center" colspan="2">
Two Points of View
  </th>
</tr>

<tr>
  <td align="center" colspan="1">
In terms of <math>~K</math> and <math>~M_\mathrm{limit} ~(= M_\mathrm{tot})</math>
  </td>
  <td align="center" colspan="1">
In terms of <math>~P_c</math> and <math>~M_\mathrm{limit} ~(= M_\mathrm{tot})</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="1">

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

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{4-3\gamma_g}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\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
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times 
\biggl\{ 
\frac{3}{4\pi} 
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{-{\gamma_g}}
\cdot \frac{1}{\mathfrak{f}_A}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~R_\mathrm{eq}^{4-3\gamma_g}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3\cdot 5} 
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \frac{1}{\mathfrak{f}_M} \biggr]_\mathrm{eq}^{2-{\gamma_g}}
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggl[
\frac{GM_\mathrm{tot}^{2-\gamma_g}}{K} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~\frac{KR_\mathrm{eq}^{4-3\gamma_g}}{GM_\mathrm{limit}^{2-\gamma_g}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{5} \biggl( \frac{4\pi}{3} \biggr)^{\gamma_g-1} \frac{\mathfrak{f}_W}{\mathfrak{f}_A \mathfrak{f}_M^{2-\gamma_g}}  
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
&#8212; &#8212; &#8212; &#8212; &#8212; &#8212; &nbsp; &nbsp; or, 
inverted and setting <math>~\gamma_g = 1 + 1/n</math> &nbsp; &nbsp; &#8212; &#8212; &#8212; &#8212; &#8212; &#8212;
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
<math>~
4\pi \biggl( \frac{G}{K} \biggr)^n M_\mathrm{limit}^{n-1} R_\mathrm{eq}^{3-n}
=
\biggl( \frac{5 \mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggr)^n \frac{3}{\mathfrak{f}_M}
</math>
  </td>
</tr>

</table>

  </td>
  <td align="center" colspan="1">

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

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq}^{4-3\gamma_g}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\biggl\{ 
\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
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times 
\biggl\{ 
\frac{3} {4\pi}
\biggl[ \biggl( \frac{P_\mathrm{norm}}{P_c} \biggr)\chi^{-3\gamma} \biggr]_\mathrm{eq} 
\cdot \frac{1}{\mathfrak{f}_A}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\chi_\mathrm{eq}^{4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\biggl( \frac{P_\mathrm{norm}}{P_c} \biggr)
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{P_c R_\mathrm{eq}^{4}}{P_\mathrm{norm} R_\mathrm{norm}^4} \biggl( \frac{M_\mathrm{tot}}{M_\mathrm{limit}} \biggr)^{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\frac{3}{20\pi} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{P_c R_\mathrm{eq}^{4}}{G M_\mathrm{limit}^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\frac{3}{20\pi} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}
</math>
  </td>
</tr>
</table>

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

According to the solution shown in the left-hand column, for fluid with a given specify entropy content, the equilibrium mass-radius relationship for adiabatic configurations is,
<div align="center">
<math>
M_\mathrm{tot}^{(\gamma_g - 2)} \propto R_\mathrm{eq}^{(3\gamma_g -4)} \, .
</math>
</div>
We see that, for <math>~\gamma_g=2</math>, the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass.  Conversely, for <math>~\gamma_g = 4/3</math>, the mass of the configuration is independent of the radius.  For <math>~\gamma_g > 2</math> or <math>~\gamma_g <  4/3</math>, configurations with larger mass (but the same specific entropy) have larger equilibrium radii.  However, for <math>~\gamma_g</math> in the range, <math>~2 > \gamma_g > 4/3</math>, configurations with larger mass have smaller equilibrium radii.  (Note that the related result for [[SSC/Virial/Isothermal#Virial_Equilibrium_of_Isothermal_Spheres|isothermal configurations]] can be obtained by setting <math>~\gamma_g = 1</math> in this adiabatic solution, because <math>~K = c_s^2</math> when  <math>~\gamma_g = 1</math>.)