Editing SSC/VirialEquilibrium/UniformDensity (section)

==Adiabatic Evolution of Pressure-truncated Sphere==
Here we seek to determine the equilibrium radius of a non-rotating configuration (<math>~J = 0</math>) that undergoes adiabatic compression/expansion (<math>\delta_{1\gamma_g} =~0</math>) and that is embedded in a hot, tenuous external medium whose confining pressure, <math>~P_e</math>, truncates the configuration. 
===Solution===
In this case, virial equilibrium implies that <math>~\chi_\mathrm{eq}</math> is given by the root(s) of the equation,
<div align="center">
<math>
3(\gamma_g-1) B\chi^{3 -3\gamma_g} ~ -~A\chi^{-1}  -~ 3D\chi^3 = 0 \, .
</math>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~D</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
(\gamma_g-1) B\chi_\mathrm{eq}^{-3\gamma_g} -  \frac{A}{3}\chi_\mathrm{eq}^{-4}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~\frac{4\pi}{3} R_0^3 P_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ K M_\mathrm{tot} \biggl( \frac{3M_\mathrm{tot} }{4\pi R_0^3} \biggr)^{\gamma_g - 1}  \cdot \mathfrak{f}_A \biggr] \chi_\mathrm{eq}^{-3\gamma_g} 
- \biggl[ \frac{1}{5} \frac{GM_\mathrm{tot} ^2}{R_0} \cdot \mathfrak{f}_W  \biggr] \chi_\mathrm{eq}^{-4} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ P_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ K \biggl( \frac{3M_\mathrm{tot} }{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g}  \cdot \mathfrak{f}_A \biggr] 
- \biggl[ \biggl(\frac{3}{20\pi} \biggr) \frac{GM_\mathrm{tot} ^2}{R_\mathrm{eq}^4} \cdot \mathfrak{f}_W  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
P_c \cdot \mathfrak{f}_A - \biggl(\frac{3}{20\pi} \biggr) \frac{GM_\mathrm{tot} ^2}{R_\mathrm{eq}^4} \cdot \mathfrak{f}_W  \, ,
</math>
  </td>
</tr>

</table>
</div>
where, in the last step as was [[SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model|recognized above]], we have set,
<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>P_c \biggl( \frac{4\pi R_\mathrm{eq}^3}{3M_\mathrm{tot}} \biggr)^{\gamma_g} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, for any external pressure, <math>~P_e < P_c</math>, the pressure-confined equilibrium radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
 ~R_\mathrm{eq}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl(\frac{3}{20\pi} \biggr) \frac{GM_\mathrm{tot} ^2}{P_c} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} 
\biggl( 1  -  \frac{P_e}{P_c} \cdot \frac{1}{\mathfrak{f}_A} \biggr)^{-1} \biggr]^{1/4} \, .
</math>
  </td>
</tr>
</table>
</div>

===Comparison with Detailed Force-Balance Model===
It is reasonable to ask how close this virial expression for the equilibrium radius is to the [[SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium|exact result]].  As before, from a free-energy analysis alone, the best we can do is assume that both structural form factors, <math>~\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, are of order unity.  But we can do better than this.  To begin with, because <math>~\rho</math> is uniform throughout the configuration, <math>~\mathfrak{f}_W = 1</math>, even though the configuration is truncated by the imposed external pressure.  We need to reassess how <math>~\mathfrak{f}_A</math> is evaluated, however, because the pressure does not drop to zero at the surface of the configuration.

Going back to our [[SSCpt1/Virial#Energy_Content_for_a_System_of_a_Given_Size_and_Internal_Structure|original definition of the thermodynamic energy reservoir for spherically symmetric adiabatic systems]],
<div align="center">
<math>\mathfrak{W}_A = \frac{1}{({\gamma_g}-1)}  \int_0^R  4\pi r^2 P dr 
 \, ,</math>
</div>
we begin by normalizing the radial coordinate to <math>~R_0</math>, the radius of the isolated (''i.e.,'' not truncated) sphere, because we know [[SSC/Structure/UniformDensity#Summary|from the detailed force-balanced solution]] that, structurally, the pressure varies with <math>~r</math> inside the configuration as,
<div align="center">
<math>\frac{P(x)}{P_c} = 1 - x^2 \, ,</math>
</div>
where, <math>~x \equiv r/R_0</math>.  Integrating only out to the edge of the ''truncated'' sphere, which we will identify as <math>~R_e</math> and, correspondingly, 
<div align="center">
<math>~x_e \equiv \frac{R_e}{R_0} = \biggl( 1 - \frac{P_e}{P_c} \biggr)^{1/2} \, ,</math>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{W}_A</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\frac{4\pi P_c R_0^3}{({\gamma_g}-1)}  \int_0^{x_e}  ( 1-x^2 ) x^2 dx</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi P_c R_0^3}{({\gamma_g}-1)}  \biggl[ \frac{x^3}{3}-\frac{x^5}{5} \biggr]_0^{x_e}
= \frac{P_c }{({\gamma_g}-1)} \biggl( \frac{4\pi R_e^3}{3} \biggr)  \biggl[ 1-\frac{3}{5}x_e^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{M_\mathrm{tot} }{({\gamma_g}-1)} \biggl( \frac{P_c}{\rho_c} \biggr)  \biggl[ 1-\frac{3}{5}\biggl(1 - \frac{P_e}{P_c}\biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, in the case of a pressure-truncated, uniform-density sphere, we surmise that the relevant structural form factor is,
<div align="center">
<math>
\mathfrak{f}_A = 1-\frac{3}{5}\biggl(1 - \frac{P_e}{P_c}\biggr) = \frac{2}{5} + \frac{3}{5}\frac{P_e}{P_c} \, .
</math>
</div>
Plugging this expression for <math>~\mathfrak{f}_A</math> along with <math>~\mathfrak{f}_W = 1</math> into the [[SSC/VirialEquilibrium/UniformDensity#Comparison_with_Detailed_Force-Balance_Model_2|just-derived virial equilibrium solution]] gives,

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

<tr>
  <td align="right">
<math>~\biggl(\frac{3}{20\pi} \biggr) \frac{GM_\mathrm{tot} ^2}{R_\mathrm{eq}^4} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~P_c \cdot \biggl[ \frac{2}{5} + \frac{3}{5}\frac{P_e}{P_c} \biggr] - P_e 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\frac{2}{5} P_c \biggl( 1 - \frac{P_e}{P_c} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ R_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{3}{2^3\pi} \biggr) \frac{G M^2}{P_c} \biggl( 1 - \frac{P_e}{P_c} \biggr)^{-1} \biggr]^{1/4} \, . 
</math>
  </td>
</tr>

</table>
</div>
This result exactly matches the solution for the equilibrium radius of a pressure-truncated, uniform-density sphere that has been [[SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium|derived elsewhere]].