Editing SSC/SynopsisStyleSheet (section)

=Spherically Symmetric Configurations Synopsis (Using Style Sheet)=


==Structure==
===Tabular Overview===

{| class="Synopsis1A" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
|+ style="height:30px;" | <font size="+1">'''Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion'''</font> &#8212; adiabatic index, <math>\gamma</math>
|-
! colspan="2" |

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>dV = 4\pi r^2 dr</math> 
  </td>
  <td align="center">
and   
  </td>
  <td align="left">
&nbsp;&nbsp;&nbsp;<math>dM_r = \rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r ~~ \propto ~~ R^{-1}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>U_\mathrm{int}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr ~~ \propto ~~ R^{3-3\gamma}</math>
  </td>
</tr>
</table>

|- 
! style="background-color:lightgreen;" colspan="2"|<b><font size="+1">Equilibrium Structure</font></b>
|- 
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2460;</font></b>&nbsp; &nbsp;<b>Detailed Force Balance</b>  
! style="text-align:center; background-color:lightblue" |<b><font color="maroon" size="+1">&#x2462;</font></b>&nbsp; &nbsp;<b>Free-Energy Identification of Equilibria</b>
|- 
! style="vertical-align:top; text-align:left;" |Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of
<div align="center">
<font color="maroon"><b>Hydrostatic Balance</b></font><br />
{{ Math/EQ_SShydrostaticBalance01 }}
</div>
for the radial density distribution, <math>\rho(r)</math>.
! style="vertical-align:top; text-align:left;" rowspan="3"|The Free-Energy is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{G}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>W_\mathrm{grav} + U_\mathrm{int} + P_eV</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-a \biggl(\frac{R}{R_0}\biggr)^{-1} + b\biggl(\frac{R}{R_0}\biggr)^{3-3\gamma}+ c\biggl(\frac{R}{R_0}\biggr)^3 \, .</math>
  </td>
</tr>
</table>
Therefore, also,  
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R_0 ~\frac{\partial\mathfrak{G}}{\partial R}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a\biggl(\frac{R}{R_0}\biggr)^{-2} +(3-3\gamma)b\biggl(\frac{R}{R_0}\biggr)^{2-3\gamma} + 3c\biggl(\frac{R}{R_0}\biggr)^2</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{R_0}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math>
  </td>
</tr>
</table>
Equilibrium configurations exist at extrema of the free-energy function, that is, they are identified by setting <math>d\mathfrak{G}/dR = 0</math>.  Hence, equilibria are defined by the condition,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>W_\mathrm{grav} + 3(\gamma-1)U_\mathrm{int} - 3P_eV\, .</math>
  </td>
</tr>
</table>
|- 
! style="text-align:center; background-color:#ffff99;" |<b><font color="maroon" size="+1">&#x2461;</font></b>&nbsp; &nbsp;<b>Virial Equilibrium</b>
|-
! style="vertical-align:top; text-align:left;" |
Multiply the hydrostatic-balance equation through by <math>rdV</math> and integrate over the volume:
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\int_0^R r\biggl(\frac{dP}{dr}\biggr)dV - \int_0^R r\biggl(\frac{GM_r \rho}{r^2}\biggr)dV</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\int_0^R 4\pi r^3 \biggl(\frac{dP}{dr}\biggr) dr - \int_0^R \biggl(\frac{GM_r}{r}\biggr)dM_r</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\int_0^R\biggl[ \frac{d}{dr}\biggl( 4\pi r^3P \biggr) - 12\pi r^2 P\biggr] dr + W_\mathrm{grav}</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^R 3\biggl[ 4\pi r^2 P \biggr]dr - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3(\gamma-1)U_\mathrm{int}  + W_\mathrm{grav} - \biggl[ 3PV \biggr]_0^R \, .</math>
  </td>
</tr>
</table>
|}

===Pointers to Relevant Chapters===

<!-- BACKGROUND MATERIAL -->

<font size="+1" color="maroon"><b>&#x24EA; </b></font> Background Material:
{| class="Synopsis1B" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[PGE#Principal_Governing_Equations|Principal Governing Equations]] (PGEs) in most general form being considered throughout this H_Book
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|PGEs in a form that is relevant to a study of the ''Structure, Stability, &amp; Dynamics'' of [[SSCpt1/PGE|spherically symmetric systems]]
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[SR#Supplemental_Relations|Supplemental relations]] &#8212; see, especially, [[SR#Barotropic_Structure|barotropic equations of state]]
|}

<!-- DETAILED FORCE BALANCE -->

<font size="+1" color="maroon"><b>&#x2460; </b></font> Detailed Force Balance:
{| class="Synopsis1C" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Derivation of the equation of Hydrostatic Balance]], and a description of several standard strategies that are used to determine its solution &#8212; see, especially, what we refer to as [[SSCpt2/SolutionStrategies#Technique_1|Technique 1]]
|}

<!-- VIRIAL EQUILIBRIUM -->

<font size="+1" color="maroon"><b>&#x2461; </b></font> Virial Equilibrium:
{| class="Synopsis1D" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|Formal derivation of the multi-dimensional, [[VE#Second-Order_Tensor_Virial_Equations|2<sup>nd</sup>-order tensor virial equations]] 
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[VE#Scalar_Virial_Theorem|Scalar Virial Theorem]], as appropriate for spherically symmetric configurations
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[VE#Generalization|Generalization]] of scalar virial theorem to include the bounding effects of a hot, tenuous external medium
|}

==Stability==
===Isolated &amp; Pressure-Truncated Configurations===

{| class="Synopsis1E" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
|- 
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis: &nbsp; Applicable to Isolated & Pressure-Truncated Configurations</b></font>
|- 
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2463;</font></b>&nbsp; &nbsp;<b>Perturbation Theory</b>
! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">&#x2466;</font></b>&nbsp; &nbsp;<b>Free-Energy Analysis of Stability</b>

|- 
! style="vertical-align:top; text-align:left;" |
Given the radial profile of the density and pressure in the equilibrium configuration, solve the [[SSC/VariationalPrinciple#Ledoux_and_Pekeris_.281941.29|eigenvalue problem defined]] by the,
<div align="center">
<font color="#770000">'''LAWE: &nbsp; Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d}{dr}\biggl[ r^4 \gamma P ~\frac{dx}{dr} \biggr] 
+\biggl[ \omega^2 \rho r^4 + (3\gamma - 4) r^3 \frac{dP}{dr} \biggr] x 
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. II, &sect;3.7.1, p. 174, Eq. (3.145)
</td></tr>
</table>
</div>
to find one or more radially dependent, radial-displacement eigenvectors, <math>x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>\omega^2</math>.
! style="vertical-align:top; text-align:left;" rowspan="5"|
The second derivative of the free-energy function is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R_0^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2a\biggl(\frac{R}{R_0}\biggr)^{-3} + (3-3\gamma)(2-3\gamma)b \biggl(\frac{R}{R_0}\biggr)^{1-3\gamma} + 6c\biggl(\frac{R}{R_0}\biggr)
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{R_0}{R} \biggr)^2\biggl[
2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V 
\biggr] \, .
</math>
  </td>
</tr>
</table>
Evaluating this second derivative for an equilibrium configuration &#8212; that is by calling upon the (virial) equilibrium condition to set the value of the internal energy &#8212; we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>3(\gamma-1)U_\mathrm{int}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3P_e V - W_\mathrm{grav} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2W_\mathrm{grav} - (2-3\gamma)\biggl[3P_e V - W_\mathrm{grav}   \biggr] + 6P_e V 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(4-3\gamma)W_\mathrm{grav} + 3^2\gamma P_e V \, .
</math>
  </td>
</tr>
</table>
Note the similarity with <b><font color="maroon" size="+1">&#x2465;</font></b>.
----


Alternatively, recalling that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~3(\gamma - 1)U_\mathrm{int}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2S_\mathrm{therm} \, ,
</math>
  </td>
</tr>
</table>
the conditions for virial equilibrium and stability, may be written respectively as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>3P_e V</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2S_\mathrm{therm}+ W_\mathrm{grav} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4W_\mathrm{grav} + 6\gamma S_\mathrm{therm}  \, .
</math>
  </td>
</tr>
</table>


|- 
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2464;</font></b>&nbsp; &nbsp;<b>Variational Principle</b>

|- 
! style="vertical-align:top; text-align:left;" |
Multiply the LAWE through by <math>4\pi x dr</math>, and integrate over the volume of the configuration gives the,
<div align="center">
<font color="#770000">'''Governing Variational Relation</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_0^R 4\pi r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr
- \int_0^R 4\pi (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- 4\pi \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_0^R x^2 \biggl(\frac{d\ln x}{d\ln r}\biggr)^2 \gamma 4\pi r^2P  dr
- \int_0^R (3\gamma - 4)x^2 \biggl( - \frac{GM_r}{r}  \biggr) 4\pi \rho r^2 dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ \biggr[\gamma 4\pi r^3 Px^2 \biggl(-\frac{d\ln x}{d\ln r}\biggr) \biggr]_0^R
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .
</math>
  </td>
</tr>
</table>
</div>
Now, by setting <math>(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>P = P_e</math> at the surface, in which case this relation becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\omega^2</math> 
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\gamma (\gamma -1) \int_0^R x^2 \bigl(\frac{d\ln x}{d\ln r}\bigr)^2 dU_\mathrm{int}
- \int_0^R (3\gamma - 4)x^2 dW_\mathrm{grav}
+ 3^2 \gamma  x^2  P_eV}{ \int_0^R  x^2 r^2  dM_r} 
</math>
  </td>
</tr>
</table>
</div>

|- 
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2465;</font></b>&nbsp; &nbsp;<b>Approximation: &nbsp; Homologous Expansion/Contraction</b>

|- 
! style="vertical-align:top; text-align:left;" |
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>x</math> = constant, and the governing variational relation gives,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\omega^2 \int_0^R  r^2  dM_r</math> 
  </td>
  <td align="center">
<math>\leq</math>
  </td>
  <td align="left">
<math>
(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma   P_eV \, .
</math>
  </td>
</tr>
</table>
</div>

|}

===Bipolytropes===

{| class="Synopsis1F" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
|- 
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis: &nbsp; Applicable to Bipolytropic Configurations</b></font>
|- 
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2467;</font></b>&nbsp; &nbsp;<b>Variational Principle</b>
! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">&#x2469;</font></b>&nbsp; &nbsp;<b>Free-Energy Analysis of Stability</b>

|- 
! style="vertical-align:top; text-align:left;" |
<div align="center">
<font color="#770000">'''Governing Variational Relation'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^*} (x r^*)^2  dM_r^* </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\gamma_c (\gamma_c-1) \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ ~\gamma_e (\gamma_e-1) \int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^*  \, .
</math>
  </td>
</tr>
</table>
</div>
! style="vertical-align:top; text-align:left;" rowspan="3"|
As we have detailed in an [[SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives|accompanying discussion]], the first derivative of the relevant free-energy expression is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R ~\frac{\partial \mathfrak{G}}{\partial R}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2S_\mathrm{tot} + W_\mathrm{tot} 
\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>S_\mathrm{tot} \equiv S_\mathrm{core} + S_\mathrm{env}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp;
  </td>
  <td align="left">
<math>W_\mathrm{tot} \equiv W_\mathrm{core} + W_\mathrm{env} \, ;</math>
  </td>
</tr>
</table>
and the second derivative of that free-energy function is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\biggl[
W_\mathrm{tot} + (3\gamma_c - 2) S_\mathrm{core} + (3\gamma_e-2)S_\mathrm{env} 
\biggr] \, .
</math>
  </td>
</tr>
</table>

----


This stability criterion may be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
2[(3\gamma_c -4) S_\mathrm{core}
+ (3\gamma_e -4) S_\mathrm{env} ] \, .
</math>
  </td>
</tr>
</table>
Hence, in bipolytropes, the marginally unstable equilibrium configuration (second derivative of free-energy set to zero) will be identified by the model that exhibits the ratio,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{S_\mathrm{core}}{S_\mathrm{env}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(3\gamma_e - 4)}{(4 - 3\gamma_c)} \, .
</math>
  </td>
</tr>
</table>

See the [[SSC/Stability/BiPolytropes#What_to_Expect_for_Equilibrium_Configurations|accompanying discussion]].

----

If &#8212; based for example on <b><font color="maroon" size="+1">&#x2466;</font></b> &#8212; we make the reasonable assumption that, in equilibrium, the statements,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp;
  </td>
  <td align="left">
<math>2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,</math>
  </td>
</tr>
</table>
hold separately, then we satisfy the virial equilibrium condition, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2S_\mathrm{tot} + W_\mathrm{tot} \, ,</math>
  </td>
</tr>
</table>
and the second derivative of the relevant free-energy function can be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(W_\mathrm{core} + W_\mathrm{env})
+ (3\gamma_c - 2) (3P_i V_\mathrm{core} - W_\mathrm{core}) 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ (3\gamma_e-2)(-3P_i V_\mathrm{core} - W_\mathrm{env})
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^2 P_i V_\mathrm{core}(\gamma_c - \gamma_e) 
+ (4-3\gamma_c ) W_\mathrm{core}
+ (4-3\gamma_e)W_\mathrm{env} \, .
</math>
  </td>
</tr>
</table>

Note the similarity with <b><font color="maroon" size="+1">&#x2468;</font></b> &#8212; temporarily, see [[SSC/Stability/BiPolytropes#Revised_Free-Energy_Analysis|this discussion]].

|- 
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2468;</font></b>&nbsp; &nbsp;<b>Approximation: &nbsp; Homologous Expansion/Contraction</b>

|- 
! style="vertical-align:top; text-align:left;" |
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^R  r^2  dM_r</math> 
  </td>
  <td align="center">
<math>\leq</math>
  </td>
  <td align="left">
<math>
(4- 3\gamma_c) W_\mathrm{core}+ (4- 3\gamma_e) W_\mathrm{env}+ 3^2 (\gamma_c - \gamma_e)   P_i V_\mathrm{core} \, .
</math>
  </td>
</tr>
</table>
</div>

|}