Editing SSC/VirialStability (section)

=Old and Probably Irrelevant Discussion=

==Summary Expressions (New)==
In the above derivations, we have adopted the notation, 
<div align="center">
<math>
\rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R_0^3} \, .
</math>
</div>
Now, guided by the [[SSCpt1/Virial#Nonrotating_Configuration_Embedded_in_an_External_Medium|earlier discussion of pressure-bounded isothermal spheres]], we choose the following normalization energy and radius:
<div align="center">
<math>
E_0 = 3M_\mathrm{tot} c_s^2 
</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>
R_0 = \frac{GM_\mathrm{tot}}{5c_s^2} \, .
</math>
</div>
Also, by analogy, it is useful to define the dimensionless parameter,
<div align="center">
<math>
\Pi_I \equiv \frac{K_e \rho_\mathrm{norm}^{1/n_e}}{c_s^2}
= \frac{K_e}{c_s^2} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr]^{1/n_e}
= \biggl( \frac{3\cdot 5^3}{4\pi} \biggr)^{1/n_e} \frac{K_e}{c_s^2} \biggl[ \frac{c_s^6}{G^3 M_\mathrm{tot}^2} \biggr]^{1/n_e} \, .
</math>
</div>
(It is worth noting that if we set <math>n_e = -1</math>, the dimensionless parameter <math>\Pi_I</math> becomes identical to the parameter <math>\Pi</math> as defined [[SSCpt1/Virial#P-V_Diagram|in the context of our discussion of the Bonnor-Ebert sphere]].  But in order to complete the analogy with the Bonnor-Ebert sphere discussion, we would also need to change the sign  on the last term in the above expression for the free energy because in the earlier discussion the external pressure was an external, confining condition whereas here it is included as an internal energy of the system.)


<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="3">
Relevant Expressions for Isothermal Core
  </th>
</tr>

<tr>
  <td align="center">
<math>
\frac{\rho_e}{\rho_c}= \frac{\mu_e}{\mu_c}
</math>
  </td>
  <td align="center">
<math>
\frac{q^3}{\nu}\biggl( \frac{1 - \nu}{1- q^3} \biggr) 
</math>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>
\chi \equiv \frac{R}{R_0} 
</math>
  </td>
  <td align="center">
<math>
\Pi_I^{n_e/3}  [\nu^{-n_e}\ (1-\nu)^{n_e+1} ]^{1/3} q^{n_e} (1-q^3)^{-(n_e+1)/3} 
</math>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>
\chi^3 = \biggl(\frac{R}{R_0} \biggr)^{3}
</math>
  </td>
  <td align="center">
<math>
\Pi_I^{n_e}  \biggl( \frac{\rho_e}{\rho_c}\biggr)^{n_e + 1} \biggl[ q^3 + \biggl( \frac{\rho_e}{\rho_c} \biggr)(1-q^3)  \biggr]^{-1} 
</math>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{A}{E_0} 
</math>
  </td>
  <td align="center">
<math>
\nu^2 \xi_s \biggl\{ 1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\}
</math>
  </td>
  <td align="left">
<math>= \biggl( \frac{\nu^2}{q} \biggr) \biggl\{ f_A(\nu,q) \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{B_e}{E_0} 
</math>
  </td>
  <td align="center">
<math>
\Pi_I \biggl( \frac{n_e}{3} \biggr) (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e}  (\xi_s^3 - 1)^{-1/n_e}
</math>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{B_I}{E_0} 
</math>
  </td>
  <td align="center">
<math>\nu </math>
  </td>
  <td align="left">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{\mathfrak{G}}{E_0} 
</math>
  </td>
  <td align="center">
<math>
- \frac{A}{E_0} \chi^{-1}  - \frac{B_I}{E_0} \ln\chi + \frac{B_e}{E_0} \chi^{-3/n_e} 
</math>
  </td>
  <td align="left">
<math>
= \nu \biggl[ - \biggl( \frac{\nu}{q} \biggr) \frac{f_A(\nu,q)}{\chi} - \ln\chi + \frac{n_e}{3} \biggl(\frac{1}{q^3}-1\biggr)\biggr]
</math>
  </td>
</tr>
</table>

[On <font color="red">8 November 2013</font>, J. E. Tohline wrote: I just confirmed that the simpler expression for the normalized total free energy, <math>\mathfrak{G}/E_0</math>, matches the more complicated version.  I don't like the result because the third term in the free energy -- the one contributed by the internal energy of the envelope -- is independent of the radius of the configuration, <math>\chi</math>; it works out this way because my expression for <math>\Pi_I</math> has a radial dependence that exactly cancels out the explicit radial dependence that appears in the more complicated expression.  But maybe it's okay after all because this expression is intended to show how the free energy varies across the <math>(q,\nu)</math> plane, and the effect of <math>\Pi_I</math> appears implicitly through the specification of <math>\chi</math>, or visa versa.]


==Summary Expressions (Old)==
In the above derivations, we have adopted the notation, 
<div align="center">
<math>
\rho_\mathrm{norm} \equiv \frac{3M_\mathrm{tot}}{4\pi R_0^3} \, .
</math>
</div>
Now, guided by the dimensional aspects of the various coefficients in the free energy expression, we choose the following normalization energy and radius:
<div align="center">
<math>
E_0 = M_\mathrm{tot} K_e \rho_\mathrm{norm}^{1/n_e} 
</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>
R_0 = \frac{GM_\mathrm{tot}}{K_e \rho_\mathrm{norm}^{1/n_e}} \, .
</math>
</div>
When combined with the expression for <math>\rho_\mathrm{norm}</math>, these become,
<div align="center">
<math>
E_0 = \biggl[ \biggl( \frac{4\pi}{3} \biggr) G^3 M_\mathrm{tot}^{5-n_e} K_e^{-n_e}\biggr]^{1/(3-n_e)} 
</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>
R_0 = \biggl[ \frac{3M_\mathrm{tot}}{4\pi} \biggl( \frac{K_e}{GM_\mathrm{tot}} \biggr)^{n_e} \biggr]^{1/(3-n_e)}  \, .
</math>
</div>
So, the primary scales are determined after specifying two parameters:  <math>M_\mathrm{tot}</math> and <math>K_e</math>.
We also obtain,
<div align="center">
<math>
\kappa_I \equiv \frac{c_s^2}{K_e \rho_\mathrm{norm}^{1/n_e}}  = \frac{M_\mathrm{tot} c_s^2}{E_0} = 
c_s^2 \biggl[ \biggl( \frac{3}{4\pi} \biggr) \frac{K_e^{n_e}}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(3-n_e)}  \, .
</math>
</div>


<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="2">
Relevant Expressions for Isothermal Core
  </th>
</tr>

<tr>
  <td align="center">
<math>
\frac{\rho_e}{\rho_c}
</math>
  </td>
  <td align="center">
<math>
\frac{q^3}{\nu}\biggl( \frac{1 - \nu}{1- q^3} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>
\chi \equiv \frac{R}{R_0} 
</math>
  </td>
  <td align="center">
<math>
q^{n_e} (1-q^3)^{-(n_e+1)/3} \biggl[\nu^{-n_e}\ (1-\nu)^{n_e+1} \kappa_I^{-n_e} \biggr]^{1/3} 
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{A}{E_0} 
</math>
  </td>
  <td align="center">
<math>
\biggl( \frac{3}{5} \biggr)\nu^2 \xi_s \biggl\{ 1 
+ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{B_e}{E_0} 
</math>
  </td>
  <td align="center">
<math>
n_e(1 - \nu)^{1+1/n_e} \xi_s^{3/n_e}  (\xi_s^3 - 1)^{-1/n_e}
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{B_I}{E_0} 
</math>
  </td>
  <td align="center">
<math>
3 \kappa_I \nu 
</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>
\frac{\mathfrak{G}}{E_0} 
</math>
  </td>
  <td align="center">
<math>
- \frac{A}{E_0} \chi^{-1}  - \frac{B_I}{E_0} \ln\chi + \frac{B_e}{E_0} \chi^{-3/n_e} 
</math>
  </td>
</tr>
</table>

Subsequently, we will also find it useful to have expressions for the following coefficient ratios:

<table border="0" align="center">

<tr>
  <td align="right">
<math>\frac{n_e A}{3 B_e}</math>  
</td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{GM^2_\mathrm{tot}}{5R_0} \biggr) \nu^2 \xi_s \biggl\{ f(\rho_e/\rho_c , \xi_s) \biggr\}
\biggl[K_e M_\mathrm{tot}  \rho_\mathrm{norm}^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e}  (\xi_s^3 - 1)^{-1/n_e}\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{GM_\mathrm{tot}}{5K_e \rho_\mathrm{norm}^{1/n_e} R_0} \biggr) \nu^2 \xi_s \biggl\{ f(\rho_e/\rho_c , \xi_s) \biggr\}
(1 - \nu)^{-(1+1/n_e)} q^{3/n_e}  \biggl(\frac{1}{q^3} - 1 \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{GM_\mathrm{tot}}{5K_e \rho_\mathrm{norm}^{1/n_e} R_0} \biggr) 
\nu^2 q^{-1} [ (1 - \nu)^{-(n_e+1)} (1 - q^3 ) ]^{1/n_e} \biggl\{ f(\rho_e/\rho_c , \xi_s) \biggr\} \, ;
</math>
  </td>
</tr>
</table>


<table border="0" align="center">
<tr>
  <td align="right">
<math>\frac{n_e B_I}{3 B_e}</math>  
</td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{tot} c_s^2 \nu \biggl[K_e M_\mathrm{tot}  \rho_\mathrm{norm}^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e}  (\xi_s^3 - 1)^{-1/n_e}\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
 </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\kappa_I  \nu (1 - \nu)^{-(1+1/n_e)} q^{3/n_e}  \biggl(\frac{1}{q^3} - 1 \biggr)^{1/n_e} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
 </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\kappa_I  \nu \biggl[ (1 - \nu)^{-(n_e+1)} (1-q^{3})  \biggr]^{1/n_e} \, ;
</math>
  </td>
</tr>
</table>


<table border="0" align="center">

<tr>
  <td align="right">
<math>\frac{n_e B_c}{n_c B_e}</math>  
</td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[K_c M_\mathrm{tot}  \rho_\mathrm{norm}^{1/n_c} \nu^{1+1/n_c}\xi_s^{3/n_c}  \biggr]
\biggl[K_e M_\mathrm{tot}  \rho_\mathrm{norm}^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e}  (\xi_s^3 - 1)^{-1/n_e}\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\kappa_A \nu^{1+1/n_c}\xi_s^{3/n_c}  \biggl[ (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e}  (\xi_s^3 - 1)^{-1/n_e}\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\kappa_A \biggl[\nu^{1+1/n_c} (1 - \nu)^{-(1+1/n_e)}  \biggr]  \biggl[ \xi_s^{3(1/n_c-1/n_e)}   (\xi_s^3 - 1)^{1/n_e}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\kappa_A \biggl[\nu^{1+1/n_c} (1 - \nu)^{-(1+1/n_e)}  \biggr]  \biggl[ q^{3(1/n_e-1/n_c)}   \biggl( \frac{1}{q^3} - 1\biggr)^{1/n_e}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\kappa_A \biggl[\nu^{1+1/n_c} (1 - \nu)^{-(1+1/n_e)}  \biggr]  \biggl[ q^{-3n_e/n_c}   (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>
\kappa_A \biggl[\nu^{1+1/n_c} q^{-3/n_c}   \biggr]  \biggl[  (1 - \nu)^{-(n_e+1)} (1-q^3) \biggr]^{1/n_e} \, .
</math>
  </td>
</tr>
</table>

==Derivatives of Free Energy==
<div align="center">
<math>
\frac{\partial\mathfrak{G}}{\partial \chi} = A \chi^{-2} -(1-\delta_{\infty n_c}) \frac{3}{n_c} B_c \chi^{-(1+3/n_c)} 
- \delta_{\infty n_c} B_I \chi^{-1} -\frac{3}{n_e} B_e \chi^{-(1+3/n_e)} \, ;
</math>

<math>
\frac{\partial^2\mathfrak{G}}{\partial \chi^2} = -2 A \chi^{-3} + (1-\delta_{\infty n_c}) \frac{3}{n_c} \biggl(1+\frac{3}{n_c}\biggr) B_c \chi^{-(2+3/n_c)} 
+ \delta_{\infty n_c} B_I \chi^{-2} + \frac{3}{n_e} \biggl(1+\frac{3}{n_e}\biggr) B_e \chi^{-(2+3/n_e)} \, .
</math>
</div>

==Equilibrium Condition==
We obtain the equilibrium radius, <math>\chi_E</math>, when <math>\partial\mathfrak{G}/\partial\chi = 0</math>.  Hence,
the relation governing the equilibrium radius is,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
A \chi_E^{-2} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(1-\delta_{\infty n_c}) \frac{3}{n_c} B_c \chi_E^{-1- 3/n_c)} 
+\delta_{\infty n_c} B_I \chi_E^{-1} +\frac{3}{n_e} B_e \chi_E^{-1-3/n_e)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ \frac{n_e A}{3B_e}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(1-\delta_{\infty n_c}) \frac{n_e B_c}{n_c B_e} \chi_E^{1- 3/n_c} 
+\delta_{\infty n_c} \frac{n_e B_I}{3B_e} \chi_E + \chi_E^{1-3/n_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ \chi_E^{1-3/n_e}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\alpha - (1-\delta_{\infty n_c}) \beta \chi_E^{1- 3/n_c} - \delta_{\infty n_c} \beta_I \chi_E \, ,
</math>
  </td>
</tr>

</table>
</div>
where,
<div align="center">
<math>\alpha \equiv \frac{n_e A}{3B_e} \, ; ~~~ \beta \equiv \frac{n_e B_c}{n_c B_e} \, ; ~~~ \beta_I \equiv \frac{n_e B_I}{3B_e} \, .</math> 
</div>

==Stability==
At this equilibrium radius, the second derivative of the free energy has the value,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\chi_E^3 \biggl( \frac{n_e}{3B_e} \biggr) \frac{\partial^2\mathfrak{G}}{\partial \chi^2} \biggr|_E
</math>
  </td>
  <td align="right">
<math>
=
</math>
  </td>
  <td align="left">
<math>
-2 A\biggl( \frac{n_e}{3B_e} \biggr)  + (1-\delta_{\infty n_c}) \biggl( \frac{n_e}{3B_e} \biggr) \frac{3}{n_c} \biggl(1+\frac{3}{n_c}\biggr) B_c \chi_E^{1-3/n_c} 
+ \delta_{\infty n_c} \biggl( \frac{n_e}{3B_e} \biggr) B_I \chi_E + \frac{3}{n_e}\biggl( \frac{n_e}{3B_e} \biggr)  \biggl(1+\frac{3}{n_e}\biggr) B_e \chi_E^{1-3/n_e}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
<math>
=
</math>
  </td>
  <td align="left">
<math>
-2 \alpha + (1-\delta_{\infty n_c}) \beta \biggl(1+\frac{3}{n_c}\biggr) \chi_E^{1-3/n_c} 
+ \delta_{\infty n_c} \beta_I \chi_E + \biggl(1+\frac{3}{n_e}\biggr) \chi_E^{1-3/n_e} \, ,
</math>
  </td>
</tr>

</table>
</div>
which, when combined with the condition for equilibrium gives,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\chi_E^3 \biggl( \frac{n_e}{3B_e} \biggr) \frac{\partial^2\mathfrak{G}}{\partial \chi^2} \biggr|_E
</math>
  </td>
  <td align="right">
<math>
=
</math>
  </td>
  <td align="left">
<math>
-2 \alpha + (1-\delta_{\infty n_c}) \beta \biggl(1+\frac{3}{n_c}\biggr) \chi_E^{1-3/n_c} 
+ \delta_{\infty n_c} \beta_I \chi_E + \biggl(1+\frac{3}{n_e}\biggr) \biggl[  \alpha - (1-\delta_{\infty n_c}) \beta \chi_E^{1- 3/n_c} - \delta_{\infty n_c} \beta_I \chi_E \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="right">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\alpha \biggl(\frac{3}{n_e}-1\biggr) + (1-\delta_{\infty n_c}) \beta \biggl(\frac{3}{n_c}-\frac{3}{n_e}\biggr) \chi_E^{1-3/n_c} 
-  \delta_{\infty n_c} \beta_I \biggl(\frac{3}{n_e}\biggr)\chi_E 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~ \chi_E^3 \biggl( \frac{n_e^2}{3B_e} \biggr) \frac{\partial^2\mathfrak{G}}{\partial \chi^2} \biggr|_E
</math>
  </td>
  <td align="right">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\alpha (3-n_e) - (1-\delta_{\infty n_c}) 3\beta \biggl(1 - \frac{n_e}{n_c} \biggr) \chi_E^{1-3/n_c} -  \delta_{\infty n_c} 3\beta_I \chi_E \, .
</math>
  </td>
</tr>

</table>
</div>

Finally, the equilibrium configuration is stable as long as this second derivative is positive.  Hence, for a bipolytrope with an 
isothermal core (<math>\delta_{\infty n_c} = 1</math>), the configuration is stable as long as,
<div align="center">
<math>
\chi_E <  \frac{\alpha (3-n_e)}{3\beta_I} \, .
</math>
</div>
In the adiabatic case (<math>\delta_{\infty n_c} = 0</math>), the configuration is stable as long as,
<div align="center">
<math>
\chi_E^{1-3/n_c} <  \frac{\alpha n_c (3-n_e)}{3\beta (n_c-n_e)} \, .
</math>
</div>


==Examples==
===Isothermal Core with <math>n=3/2</math> Envelope===
When the core is isothermal and <math>n_e = 3/2</math>, the equilibrium condition is:
<div align="center">
<math>
\chi_E^{-1} =  \alpha - \beta_I \chi_E \, ,
</math>

<math>
\Rightarrow ~~~~ \beta_I \chi_E^2 -  \alpha \chi_E + 1 = 0 \, ,
</math>

<math>
\Rightarrow ~~~~ \chi_E = \frac{1}{2\beta_I} \biggl[ \alpha \pm \sqrt{\alpha^2 - 4\beta_I} \biggr] = \frac{\alpha}{2\beta_I} \biggl[ 1 \pm \sqrt{1 - \frac{4\beta_I}{\alpha^2}} \biggr] \, .
</math>
</div>

At the same time, the condition for stability is,
<div align="center">
<math>
\chi_E <  \frac{\alpha}{2\beta_I} \, .
</math>
</div>

===Isothermal Core with <math>n=1</math> Envelope===
When the core is isothermal and <math>n_e = 1</math>, the equilibrium condition is:
<div align="center">
<math>
\chi_E^{-2} =  \alpha - \beta_I \chi_E \, ,
</math>

<math>
\Rightarrow ~~~~ \chi_E^3 -  \frac{\alpha}{\beta_I} \chi_E^2 + \frac{1}{\beta_I} = 0 \, .
</math>
</div>
(We need to solve this cubic equation.)

At the same time, the condition for stability is,
<div align="center">
<math>
\chi_E <  \frac{2\alpha}{3\beta_I} \, .
</math>
</div>

===Old (and probably incorrect) cases===
====Envelope with <math>n=3/2</math>====
If we choose an <math>n_e = 3/2</math> envelope, we obtain stability for,
<div align="center">
<math>
\chi_E <  \frac{2\beta}{\alpha}\, .
</math>
</div>
In this case, the equilibrium radius condition is,
<div align="center">
<math>
\chi_E^2 - \alpha \chi_E + \beta =0 
</math>

<math>
\Rightarrow ~~~~ \chi_E = \frac{1}{2}\biggl[\alpha \pm \biggl(  \alpha^2 -4\beta \biggr)^{1/2}   \biggr] = 
\frac{\alpha}{2}\biggl[1 \pm \biggl(  1 -\frac{4\beta}{\alpha^2} \biggr)^{1/2}   \biggr]
</math>

</div>


====Envelope with <math>n=1</math>====
If, instead, we choose an <math>n_e = 1</math> envelope, we obtain stability for,
<div align="center">
<math>
\chi_E <  \sqrt{\frac{3\beta}{\alpha} }\, .
</math>
</div>
In this case, the equilibrium radius condition is,
<div align="center">
<math>
\alpha = \chi_E + \beta \chi_E^{-2} \, ,
</math>

<math>
\Rightarrow ~~~~ \chi_E^3 - \alpha \chi_E^2 + \beta = 0 \, .
</math>

</div>