Editing SSC/Virial/Isothermal (section)

====P-V Diagram====
Returning to the dimensionless form of this expression and multiplying through by <math>~[-\chi/(3D)]</math>, we obtain,
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<math>
\chi^4 - \frac{B_I}{D} \chi + \frac{A}{D} = 0 \, .
</math>
</div>
Now, taking a cue from the solution presented above for an isolated isothermal configuration, we choose to set the previously unspecified scale factor, <math>~R_0</math>, to,  
<div align="center">
<math>
R_0 = \frac{GM}{5c_s^2} \, ,
</math>
</div>
in which case <math>~B_I = A</math>, and the quartic equation governing the radii of equilibrium states becomes, simply,
<div align="center">
<math>
\chi^4 - \frac{\chi}{\Pi} + \frac{1}{\Pi} = 0 \, ,
</math>
</div>
where,
<div align="center">
<math>
\Pi \equiv \frac{D}{B_I} = \frac{4\pi R_0^3 P_e}{3Mc_s^2} = \frac{4\pi P_e G^3 M^2}{3\cdot 5^3 c_s^8} \, .
</math>
</div>
For a given choice of <math>~P_e</math> and <math>~c_s</math>, <math>~\Pi^{1/2}</math> can represent a dimensionless mass, in which case,
<div align="center">
<math>
M = \Pi^{1/2} \biggl( \frac{3\cdot 5^3}{2^2\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .
</math>
</div>
Alternatively, for a given choice of configuration mass and sound speed, this parameter, <math>~\Pi</math>, can be viewed as a dimensionless external pressure; or, for a given choice of <math>~M</math> and <math>~P_e</math>, <math>~\Pi^{-1/8}</math> can represent a dimensionless sound speed.  In most of what follows we will view <math>~\Pi</math> as a dimensionless external pressure.  

The above quartic equation can be rearranged immediately to give the external pressure that is required to obtain a particular configuration radius, namely,
<div align="center">
<math>
\Pi = \frac{(\chi - 1)}{\chi^4} \, .
</math>
</div>
The resulting behavior is shown by the black curve in Figure 2.

<div align="center">
<table border="2" cellpadding="8">
<tr>
  <td align="center" colspan="2">
'''Figure 2:''' <font color="darkblue">Equilibrium Isothermal P-V Diagram </font> 
  </td>
</tr>
<tr>
  <td valign="top" width=450 rowspan="1">
The black curve traces out the function,
<div align="center">
<math>~
\Pi = (\chi - 1)/\chi^4 \, ,
</math>
</div>
and shows the dimensionless external pressure, <math>~\Pi</math>, that is required to construct a nonrotating, self-gravitating, isothermal sphere with an equilibrium radius <math>~\chi</math>.  The pressure becomes negative at radii <math>~\chi < 1</math>, hence the solution in this regime is unphysical.

[[SSCpt1/Virial#Visual_Representation|Figure 1]] displays the free energy surface that "lies above" the two-dimensional parameter space <math>~(1.2 < \chi < 1.51</math>; <math>~0.103 < \Pi < 0.104)</math> that is identified here by the thin, red rectangle.
 </td>
  <td align="center" bgcolor="white">
[[File:Bonnor1956Fig1.jpg|450px|center|Equilibrium P-R Diagram]]
  </td>
</tr>

</table>
</div>
In the absence of self-gravity (''i.e.,'' <math>~A=0</math>), the product of the external pressure and the volume should be constant.  The corresponding relation, <math>~\Pi = \chi^{-3}</math>, is shown by the blue dashed curve in the figure.  As the figure illustrates, when gravity is included the P-V relationship pulls away from the PV = constant curve at sufficiently small volumes.  Indeed, the curve turns over at a finite pressure, <math>~\Pi_\mathrm{max}</math>, and for every value of <math>~\Pi < \Pi_\mathrm{max}</math> a second, more compact equilibrium configuration appears.  The location of <math>~\Pi_\mathrm{max}</math> along the curve is identified by setting <math>~\partial\Pi/\partial\chi = 0</math>, that is, it occurs where,
<div align="center">
<math>
\frac{\partial\Pi}{\partial\chi} = -4 \chi^{-5}(\chi - 1) + \chi^{-4} = 0 \, , 
</math>

<math>
\Rightarrow ~~~~~ \chi = \frac{2^2}{3} \approx 1.333333 \, .
</math>
</div>
<span id="BonnorEbertMass">Hence,</span>
<div align="center">
<math>~\Pi_\mathrm{max} = \biggl( \frac{2^2}{3} \biggr)^{-4} \biggl( \frac{2^2}{3}-1 \biggr) = \frac{3^3}{2^8} \approx 0.105469\, ;</math>
</div>
therefore, from above,
<div align="center">
<math>~
M_\mathrm{max} = \biggl( \frac{3^4\cdot 5^3}{2^{10}\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2}
\approx 1.77408 \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .
</math>
</div>