Editing SSC/Virial/Polytropes/Pt2 (section)

====P-V Diagram for Unity Form Factors====

Writing the coefficient, <math>B</math>, in terms of the average sound speed and setting the radial scale factor equal to the equilibrium radius of an isolated adiabatic sphere, that is, setting,
<div align="center">
<math>
R_0 = \frac{GM}{5\bar{c_s}^2} \, ,
</math>
</div>
the equation governing the radii of ''adiabatic'' equilibrium states becomes, 
<div align="center">
<math>
\chi^4 - \frac{1}{\Pi_a} \chi^{(4-3\gamma_g)} + \frac{1}{\Pi_a} = 0 \, ,
</math>
</div>
where,
<div align="center">
<math>
\Pi_a \equiv \frac{4\pi P_e G^3 M^2}{3\cdot 5^3 \bar{c_s}^8} \, .
</math>
</div>
As in the isothermal case, for a given choice of configuration mass and sound speed, this parameter, <math>\Pi_a</math>, can be viewed as a dimensionless external pressure.  Alternatively, for a given choice of <math>P_e</math> and <math>\bar{c_s}</math>, <math>\Pi_a^{1/2}</math> can represent a dimensionless mass; or, for a given choice of <math>M</math> and <math>P_e</math>, <math>\Pi_a^{-1/8}</math> can represent a dimensionless sound speed.  Here we will view it as a dimensionless external pressure.  

Unlike the isothermal case, for an arbitrary value of the adiabatic exponent, <math>\gamma_g</math>, it isn't possible to invert this equation to obtain an analytic expression for <math>\chi</math> as a function of <math>\Pi_a</math>.  But we can straightforwardly solve for <math>\Pi_a</math> as a function of <math>\chi</math>.  The solution is,
<div align="center">
<math>
\Pi_a = \frac{\chi^{(4- 3\gamma_g)} - 1}{\chi^4} \, .
</math>
</div>
For physically relevant solutions, both <math>\chi</math> and <math>\Pi_a</math> must be nonnegative.  Hence, as is illustrated by the curves in Figure 4, the physically allowable range of equilibrium radii is,
<div align="center">
<math>
1 \le \chi \le \infty \, ~~~~~\mathrm{for}~ \gamma_g < 4/3 \, ;
</math>

<math>
0 < \chi \le 1 \, ~~~~~~\mathrm{for}~ \gamma_g > 4/3 \, .
</math>
</div>

<div align="center">
<table border="2" cellpadding="8">
<tr>
  <td align="center" colspan="2">
'''Figure 4:''' <font color="darkblue">Equilibrium Adiabatic P-V Diagram </font> 
  </td>
</tr>
<tr>
  <td valign="top" width=450 rowspan="1">
The curves trace out the function,
<div align="center">
<math>
\Pi_a = (\chi^{4-3\gamma_g} - 1)/\chi^4 \, ,
</math>
</div>
for six different values of <math>\gamma_g</math> (<math>2, ~5/3, ~7/5, ~6/5, ~1, ~2/3</math>, as labeled)  and show the dimensionless external pressure, <math>\Pi_a</math>, that is required to construct a nonrotating, self-gravitating, uniform density, adiabatic sphere with an equilibrium radius <math>\chi</math>.  The mathematical solution becomes unphysical wherever the pressure becomes negative.

The solid red curve, drawn for the case <math>\gamma_g = 1</math>, is identical to the solid black (isothermal) curve displayed above in Figure 1.
 </td>
  <td align="center" bgcolor="white">
[[File:AdabaticBoundedSpheres_Virial.jpg|450px|center|Equilibrium Adiabatic P-R Diagram]]
  </td>
</tr>
</table>
</div>

Each of the <math>\Pi_a(\chi)</math> curves drawn in Figure 4 exhibits an extremum.  In each case this extremum occurs at a configuration radius, <math>\chi_\mathrm{extreme}</math>, given by,
<div align="center">
<math>
\frac{\partial\Pi_a}{\partial\chi} = 0 \, ,
</math>
</div>
that is, where,
<div align="center">
<math>
4 - 3\gamma_g \chi^{4-3\gamma_g} = 0 ~~~~\Rightarrow ~~~~~ \chi_\mathrm{extreme} = \biggl[ \frac{4}{3\gamma_g} \biggr]^{1/(4-3\gamma_g)} \, .
</math>
</div>
For each value of <math>\gamma_g</math>, the corresponding dimensionless pressure is,
<div align="center">
<math>
\Pi_a \biggr|_\mathrm{extreme} = \biggl(\frac{4}{3\gamma} - 1 \biggr) \biggl[ \frac{3\gamma_g}{4} \biggr]^{4/(4-3\gamma_g)} \, .
</math>
</div>

Note, first, that for <math>\gamma_g > 4/3</math>, an equilibrium configuration with a positive radius can be constructed for all physically realistic &#8212; that is, for all positive &#8212; values of <math>\Pi_a</math>.  Also, consistent with the behavior of the curves shown in Figure 4, the extremum arises in the regime of physically relevant &#8212; ''i.e.,'' positive &#8212; pressures only for values of <math>\gamma_g < 4/3</math>; and in each case it represents a ''maximum'' limiting pressure.