Editing SSC/VirialStability (section)

=Strategy=
==Scaling Parameters==
We want to vary the total radius, <math>R \,</math>, of the configuration and look for extrema in the free energy, while holding the following parameters fixed:  <math>M_\mathrm{tot} \,</math>, <math>c_s^2 \,</math> (or <math>K_c \,</math>), <math>\nu \,</math>, and <math>q \,</math>.  So we need to rewrite the expressions for <math>W \,</math> and <math>U \,</math> such that everything is constant except for <math>R \,</math>, or <math>\chi \equiv R/R_0 \,</math>.  And, in order to put everything explicitly in terms of the fixed parameters just specified, let's go ahead and define the length, time, and density against which all dimensional quantities will ultimately be scaled.

<table border="1" cellpadding="10" align="center">
<tr>
  <td colspan="3" align="center">
<font size="+1"><b>Chosen Scaling Parameters</b></font>
  </td>
</tr>

<tr>
  <th>
&nbsp;
  </th>
  <th align="center">
Polytropic Core
  </th>
  <th align="center">
Isothermal Core
  </th>
</tr>

<tr>
   <td align="center">
<math>R_0 \,</math>
  </td>
   <td align="center">
<math>\biggl[ \biggl( \frac{K_c}{G} \biggr)^{n_c} M_\mathrm{tot}^{1-n_c} \biggr]^{1/(3-n_c)} \,</math>
  </td>
   <td align="center">
<math>\frac{GM_\mathrm{tot}}{c_s^2} \,</math>
  </td>
</tr>

<tr>
   <td align="center">
<math>t_0 \,</math>
  </td>
   <td align="center">
<math>\biggl[ \biggl( \frac{K_c^3}{M_\mathrm{tot}^2} \biggr)^{n_c} G^{-(2n_c+3)} \biggr]^{1/(6-2n_c)} \,</math>
 </td>
   <td align="center">
<math>\frac{GM_\mathrm{tot}}{c_s^3} \,</math>
  </td>
</tr>

<tr>
   <td align="center">
<math>\rho_0 \equiv \frac{3M_\mathrm{tot}}{4\pi R_0^3} \,</math>
  </td>
   <td align="center">
<math>\frac{3}{4\pi} \biggl[ \frac{G^3 M_\mathrm{tot}^2}{K_c^3} \biggr]^{n_c/(3-n_c)} \,</math>
  </td>
   <td align="center">
<math>\frac{3}{4\pi} \biggl[ \frac{c_s^6}{G^3 M_\mathrm{tot}^2} \biggr] \,</math>
  </td>
</tr>

</table>


==Rescaled Energy Expressions==
===Isothermal Core===
In the case of an isothermal core, the expression for <math>W</math> shown above needs only a slight modification to put it in the appropriate form, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>W \,</math> 
  </td>
  <td align="left">
<math>
= - \frac{3}{5}  M_\mathrm{tot} c_s^2 \chi^{-1} \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) \, ,
</math>
  </td>
</tr>

</table>

In the case of an isothermal core, the expression for the total internal energy may be rewritten as,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\frac{U}{M_\mathrm{tot}} \,
</math>
  </td>
  <td align="center">
<math>= \,</math>
  </td>
  <td align="left">
<math>
\nu c_s^2 \ln(\rho_c/\rho_0)  + (1-\nu) n_e K_e \rho_e^{1/n_e}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\nu c_s^2 [\ln(\rho_c/\bar{\rho}) + \ln(\bar\rho/\rho_0)]   + (1-\nu) n_e K_e \rho_0^{1/n_e} \biggl[\biggl( \frac{\rho_e}{\bar\rho} \biggr) \biggl( \frac{\bar\rho}{\rho_0} \biggr) \biggr]^{1/n_e} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\nu c_s^2 [\ln(\nu/q^3) -3 \ln\chi]   + (1-\nu) n_e K_e \rho_0^{1/n_e} \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} \chi^{-3/n_e} \, .
</math>
  </td>
</tr>
</table>
Hence (in the case of an isothermal core),

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{M_\mathrm{tot} c_s^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
 - \frac{3}{5}  \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) \chi^{-1} 
+ \nu [\ln(\nu/q^3) -3 \ln\chi]   + (1-\nu) n_e \biggl[ \frac{K_e \rho_0^{1/n_e}}{c_s^2} \biggr] \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} \chi^{-3/n_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mathfrak{G}_0^* - A \chi^{-1} - C \ln\chi  + D \chi^{-3/n_e} \, ,
</math>
  </td>
</tr>

</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathfrak{G}_0^* \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
\nu \ln(\nu/q^3)   \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>A \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
\frac{3}{5} \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>C \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
3 \nu  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>D \, </math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
(1-\nu) n_e \biggl[ \frac{K_e \rho_0^{1/n_e}}{c_s^2} \biggr] \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} 
= (1-\nu) n_e \biggl( \frac{K_e}{c_s^2} \biggr) \biggl[  \frac{3}{4\pi} \biggl(\frac{c_s^6}{G^3M^2_\mathrm{tot}}\biggr)   \biggr]^{1/n_e} \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} \, .
</math>
  </td>
</tr>
</table>

===Polytropic Core===
In the case of a polytropic core,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{M_\mathrm{tot} K_c \rho_0^{1/n_c}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- A \chi^{-1}  + B_c \chi^{-3/n_c} + B_e \chi^{-3/n_e} \, ,
</math>
  </td>
</tr>

</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
\frac{3}{5} \biggl[ \frac{GM_\mathrm{tot}}{R_0 K_c \rho_0^{1/n_c}} \biggr] \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) =
\frac{3}{5} \biggl( \frac{4\pi}{3} \biggr)^{1/n_c} \biggl( \frac{\nu^2}{q} \biggr) f(\nu,q) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_c \,</math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
n_c \nu \biggl( \frac{\nu}{q^3}\biggr)^{1/n_c} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_e \, </math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
n_e (1-\nu) \biggl[ \biggl( \frac{K_e }{K_c}\biggr)\rho_0^{1/n_e - 1/n_c} \biggr] \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} 
\, .
</math>
  </td>
</tr>
</table>

==Temperature Across the Interface==
In order to ensure that the temperature of the envelope is the same as the temperature of the core when there is a drop in the mean molecular weight at the interface, we need to have,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{P_e}{(\rho_e/\mu_e)}\biggr]_E</math>
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math>\biggl[ \frac{P_c}{(\rho_c/\mu_c)} \biggr]_E \, .</math>
  </td>
</tr>
</table>
Note that this reflects the same physical condition as the constraint that is placed on the enthalpy at the interface when we analyze the detailed structure of <math>n_c = 5</math>, <math>n_e=1</math> bipolytropes (see the [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|middle of Table 1 in the accompanying discussion]]), namely,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{\mu_e H_e}{(n_e+1)}\biggr]_i</math>
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math>\biggl[ \frac{\mu_c H_c}{(n_c+1)} \biggr]_i \, .</math>
  </td>
</tr>
</table>


In the case of an isothermal core, this implies,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{K_e}{ c_s^2}</math>
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math>\biggl[\rho_e^{-1/n_e}\biggr]_E</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~~ \biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{K_e \rho_0^{1/n_e}}{ c_s^2}</math>
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math> \biggl[\biggl( \frac{\rho_e}{\bar\rho} \biggr) \biggl( \frac{\bar\rho}{\rho_0} \biggr)_E \biggr]^{-1/n_e}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1-\nu}{1-q^3} \biggr)^{-1/n_e} \biggl( \frac{\bar\rho}{\rho_0} \biggr)_E^{-1/n_e}
= \biggl( \frac{1-\nu}{1-q^3} \biggr)^{-1/n_e} \chi_E^{3/n_e}
</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="5" align="center">
<tr>
  <td colspan="3" align="center">
Not Necessarily Useful
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~~ \chi_E^{3/n_e}</math>
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{K_e}{ c_s^2} \biggl[  \frac{3}{4\pi} \biggl(\frac{c_s^6}{G^3M^2_\mathrm{tot}}\biggr)   \biggr]^{1/n_e} 
\biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c}\biggr) \biggl[  \frac{D}{(1-\nu)n_e M_\mathrm{tot} c_s^2}  \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c}\biggr) \biggl[  \frac{3\nu}{(1-\nu)n_e}  \biggl(\frac{D}{C}\biggr)\biggr]</math>
  </td>
</tr>
</table>
Therefore, the coefficient <math>D\,</math> becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>D \, </math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
n_e (1-\nu) \biggl[ \frac{K_e \rho_0^{1/n_e}}{c_s^2} \biggr] \biggl( \frac{1-\nu}{1-q^3} \biggr)^{1/n_e} 
= n_e (1-\nu) \biggl( \frac{\mu_c }{\mu_e}\biggr) \chi_E^{3/n_e} \, .
</math>
  </td>
</tr>

</table>


In the case of a polytropic core, this implies,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{K_e}{ K_c}</math>
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math>\rho_0^{1/n_c} \biggl[\rho_e^{-1/n_e}\biggr]_E</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~~ \biggl( \frac{\mu_e}{\mu_c}\biggr) \biggl( \frac{K_e }{ K_c} \biggr) \rho_0^{(1/n_e-1/n_c)}</math>
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math> \biggl[\biggl( \frac{\rho_e}{\bar\rho} \biggr) \biggl( \frac{\bar\rho}{\rho_0} \biggr)_E \biggr]^{-1/n_e}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
   <td align="center">
<math>= \, </math>
  </td>
  <td align="left">
<math>
\biggl( \frac{1-\nu}{1-q^3} \biggr)^{-1/n_e} \biggl( \frac{\bar\rho}{\rho_0} \biggr)_E^{-1/n_e}
= \biggl( \frac{1-\nu}{1-q^3} \biggr)^{-1/n_e} \chi_E^{3/n_e}
</math>
  </td>
</tr>

</table>
Therefore, the coefficient <math>B_e\,</math> becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>B_e \, </math>
  </td>
  <td align="center">
<math>\equiv \,</math>
  </td>
  <td align="left">
<math>
n_e (1-\nu) \biggl( \frac{\mu_c }{\mu_e}\biggr) \chi_E^{3/n_e} \, .
</math>
  </td>
</tr>

</table>


==Pressure Across the Interface==

We will relate <math>K_e</math> to <math>K_c</math> by demanding that initially the pressure is identical in both layers.  
The relevant algebraic relation will depend on whether the core is isothermal (<math>n_c = \infty</math>), or whether it
has a finite polytropic index and therefore adjusts adiabatically to compressions or expansions.

===Isothermal Core===
Guided by the interface conditions presented in [[SSC/Structure/BiPolytropes#Interface_Conditions|Table 2 of our accompanying discussion of the structure of bipolytropes]], the condition for pressure balance in the case of an isothermal core should be,
<div align="center">
<math>\frac{c_s^2}{K_e \rho_0^{1/n_e}} = \biggl(\frac{\rho_c}{\rho_0}\biggr)^{1/n_e} \biggl( \frac{\rho_e}{\rho_c} \biggr)^{1+1/n_e} 
= \biggl(\frac{\rho_c}{\rho_0}\biggr)^{1/n_e} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{1+1/n_e} \, .</math>
</div>

Actually, in the full structural solution,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>\frac{c_s^2}{K_e \rho_0^{1/n_e}}</math>
  </td>
  <td align="center">
<math>= </math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\rho_c}{\rho_0} \biggr)^{1/n_e} e^{\psi_i}
\biggl( \frac{\rho_e}{\rho_c} \biggr)^{1+1/n_e} \phi_i^{1+n_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>= </math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\rho_c}{\rho_0} \biggr)^{1/n_e} e^{\psi_i}
\biggl[ \frac{\mu_e}{\mu_c} e^{-\psi_i} \phi_i^{-{n_e}}\biggr]^{1+1/n_e} \phi_i^{1+n_e} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>= </math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\rho_c}{\rho_0} e^{-\psi_i} \biggr)^{1/n_e} 
\biggl[ \frac{\mu_e}{\mu_c} \biggr]^{1+1/n_e} \, .
</math>
  </td>
</tr>
</table>
</div>

===Adiabatic Core===
Guided by the interface conditions presented in [[SSC/Structure/BiPolytropes#Interface_Conditions|Table 2 of our accompanying discussion of the structure of bipolytropes]], the condition for pressure balance in the case of polytropic core should be,
<div align="center">
<math>\biggl(\frac{K_c}{K_e}\biggr) \rho_0^{1/n_c -1/n_e} = \biggl(\frac{\rho_c}{\rho_0}\biggr)^{1/n_e - 1/n_c} \biggl( \frac{\rho_e}{\rho_c} \biggr)^{1+1/n_e} 
= \biggl(\frac{\rho_c}{\rho_0}\biggr)^{1/n_e - 1/n_c} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{1+1/n_e} \, .</math>
</div>

Actually, in the full structural solution,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>\biggl(\frac{K_c}{K_e}\biggr) \rho_0^{1/n_c -1/n_e} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\rho_0^{1/n_c -1/n_e} \rho_c^{-(1+1/n_c)} \theta_i^{-(1+n_c)} \rho_e^{1+1/n_e} \phi_i^{1+n_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl( \frac{\rho_c}{\rho_0} \biggr)^{1/n_e - 1/n_c} \theta_i^{-(1+n_c)} \biggl( \frac{\rho_e}{\rho_c}\biggr)^{1+1/n_e} \phi_i^{1+n_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl( \frac{\rho_c}{\rho_0} \biggr)^{1/n_e - 1/n_c} \biggl[ \frac{\mu_e}{\mu_c} \theta_i^{n_c} 
 \biggr]^{1+1/n_e} \theta_i^{-(1+n_c)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl( \frac{\rho_c}{\rho_0} \theta_i^{n_c}\biggr)^{1/n_e - 1/n_c} \biggl[ \frac{\mu_e}{\mu_c}  
 \biggr]^{1+1/n_e} \, .
</math>
  </td>
</tr>

</table>
</div>