Editing SSC/VirialEquilibrium/UniformDensity (section)

==Adiabatic Evolution of an Isolated 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 not confined by an external medium (<math>P_e = 0~</math>). 
===Solution===
In this case, the statement of virial equilibrium is simplified considerably.  Specifically, <math>~\chi_\mathrm{eq}</math> is given by the root(s) of the equation,
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<table border="0" cellpadding="5" align="center">

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<math>~A\chi_\mathrm{eq}^{-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3(\gamma_g-1) B\chi_\mathrm{eq}^{3 -3\gamma_g} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~~\chi_\mathrm{eq}^{3\gamma_g-4} </math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3(\gamma_g-1) B}{A} 
</math>
  </td>
</tr>

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

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>
\biggl( \frac{1}{R_0} \biggr)^{3\gamma_g-4} \biggl[ 5\biggl( \frac{3}{4\pi} \biggr)^{\gamma_g-1} \biggr(\frac{K}{G}\biggr) 
M^{(\gamma_g-2)} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_W} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

In other words,
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<math>
R_\mathrm{eq} = \biggl[ 5\biggl( \frac{3}{4\pi} \biggr)^{\gamma_g-1} \biggr(\frac{K}{G}\biggr) 
M^{(\gamma_g-2)} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_W} \biggr]^{1/(3\gamma_g-4)} \, .
</math>
</div>

===Comparison with Detailed Force-Balance Model===
This derived solution will look more familiar if, instead of <math>~K</math>, we express the solution in terms of the central pressure,
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<math>P_c = K\rho_0^{\gamma_g} \, ,</math>
</div>
where, for this uniform-density sphere, <math>~\rho_0 = 3M_\mathrm{tot}/(4\pi R_\mathrm{eq}^3)</math>.  Hence,
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<table border="0" cellpadding="5" align="center">
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  <td align="right">
<math>~K</math>
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<math>~=</math>
  </td>
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<math>P_c \biggl( \frac{4\pi R_\mathrm{eq}^3}{3M_\mathrm{tot}} \biggr)^{\gamma_g} \, ,</math>
  </td>
</tr>
</table>
</div>
and the solution takes the form,
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<math>R_\mathrm{eq}^{3\gamma_g - 4}</math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>5\biggl( \frac{4\pi}{3} \biggr) \biggr(\frac{P_c R_\mathrm{eq}^{3\gamma_g}}{GM^2_\mathrm{tot}}\biggr) 
\cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_W} </math>
  </td>
</tr>

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  <td align="right">
<math>\Rightarrow ~~~~ R_\mathrm{eq}^{4} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>\biggl( \frac{3}{20\pi} \biggr) \biggr(\frac{GM^2_\mathrm{tot}}{P_c}\biggr) 
\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \, .</math>
  </td>
</tr>
</table>
</div>
Or, solving for the central pressure,
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<math>~P_c</math>
  </td>
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<math>~=</math>
  </td>
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<math>\biggl( \frac{3}{20\pi} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr) \frac{GM^2_\mathrm{tot}}{R_\mathrm{eq}^{4} }
\, .</math>
  </td>
</tr>
</table>
</div>
This should be compared with our [[SSC/Structure/UniformDensity#Summary|detailed force-balance solution]] of the interior structure of an isolated, nonrotating, uniform-density sphere, which gives the precise expression,
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<math>~P_c</math>
  </td>
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<math>~=</math>
  </td>
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<math>\biggl( \frac{3}{8\pi}  \biggr) \frac{GM^2_\mathrm{tot}}{R_\mathrm{eq}^{4} }
\, .</math>
  </td>
</tr>
</table>
</div>
The expression for <math>~P_c</math> derived from our identification of an extremum in the free energy is identical to the expression derived from the more precise, detailed force-balance analysis, except that the leading numerical coefficients differ by a factor of <math>~(5\mathfrak{f}_A/2\mathfrak{f}_W)</math>.  

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 knowing the detailed force-balance solution allows us to evaluate both form factors.  From our [[SSCpt1/Virial#FormFactors|introductory discussion of the free energy function]], their respective definitions are,
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<math>~\mathfrak{f}_W</math>
  </td>
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<math>~\equiv</math>
  </td>
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<math>~  3\cdot 5 \int_0^1 \biggl\{ \int_0^x  \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\}  \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ 3\int_0^1 \biggl[ \frac{P(x)}{P_c}\biggr]  x^2 dx \, .</math>
  </td>
</tr>
</table>
</div>
Now, because the configuration under discussion has a uniform density, we should set <math>~\rho(x)/\rho_c = 1</math> in the definition of <math>~\mathfrak{f}_W</math> which, after evaluation of the nested integrals, gives <math>~\mathfrak{f}_W = 1</math>.  But, instead of being uniform throughout the configuration, in the [[SSC/Structure/UniformDensity#Summary|detailed force-balance model]], the pressure drops from the center to the surface according to the relation,
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<math>\frac{P(x)}{P_c} = 1 - x^2 \, .</math>
</div>
Integrating over this function, in accordance with the definition of <math>~\mathfrak{f}_A</math>, gives,
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<math>~\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
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<math>~ 3\int_0^1 (1-x^2) x^2 dx = 3 \biggl[ \frac{x^3}{3} - \frac{x^5}{5} \biggr]_0^1 = \frac{2}{5} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, the ratio,
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<math>
\frac{5\mathfrak{f}_A}{2\mathfrak{f}_W} = 1 \, ,
</math>
</div>
which brings into perfect agreement the two separate determinations of the equilibrium expressions for <math>~R_\mathrm{eq}</math> and <math>~P_c</math> in terms of one another and the total mass.  

This demonstrates that the free-energy approach to determining the equilibrium radius of a spherical configuration is only handicapped by its inability to precisely nail down values of the structural form factors.  But this is not a severe limitation as the (dimensionless) form factors are generally of order unity.  In contrast, the free-energy analysis brings with it a capability to readily evaluate the global stability of equilibrium configurations.