Editing SSC/Stability/BiPolytropes/PlannedApproach (section)

=Reconciliation=

<table border="0" cellpadding="8" align="right">
<tr>
  <th align="center">Figure 7:  Conflicting Instability Regions</th>
</tr>
<tr>
  <td align="center" colspan="10">[[File:EigenvectorStability.png|400px|Marginally unstable models]]</td>
</tr>
</table>
Figure 7, shown here on the right, is identical to the right-hand panel of [[#Figure4|Figure 4, as displayed above]].  From a standard global, free-energy analysis &#8212; such as the one [[#Virial_Stability_Evaluation|summarized above]] &#8212; we have determined that the red-dashed curve shown in the right panel of Figure 6 divides the <math>~q-\nu</math> plane into dynamically stable (below and to the right) and unstable (above and to the left) regions.


==Variational Principle==

===Setup===

Let's follow the guidelines of the [[SSC/SynopsisStyleSheet|variational principle]].  Instead of starting with the form of the LAWE given above, namely,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

we will start with a form that is more amenable to the variational principle, namely,
<div align="center">
<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_0}\biggl[ r_0^4 \gamma P_0 ~\frac{dx}{dr_0} \biggr] 
+\biggl[ \omega^2 \rho_0 r_0^4 + (3\gamma - 4) r_0^3 \frac{dP_0}{dr_0} \biggr] x \, .
</math>
  </td>
</tr>
</table>
</div>

<table border="1" cellpadding="8" align="center" width="85%"><tr><td align="left">
<font color="red">'''ASIDE:'''</font>  Let's show that these two expressions are equivalent.  Remembering that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dP_0}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- g_0 \rho_0 = - \biggl(\frac{GM_r}{r_0^2}\biggr)\rho_0 \, ,</math>
  </td>
</tr>
</table>
the second expression becomes,
<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>~
r_0^4 \gamma P_0 ~\frac{d^2x}{dr_0^2} 
+ ~\gamma \frac{dx}{dr_0}\biggl[ 4r_0^3 P_0 - r_0^4 g_0 \rho_0\biggr] 
+\biggl[ \omega^2 \rho_0 r_0^4 - (3\gamma - 4) r_0^3 g_0 \rho_0\biggr] x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r_0^4 \gamma P_0 \biggl\{\frac{d^2x}{dr_0^2} 
+ \biggl[ \frac{4}{r_0} - \frac{g_0 \rho_0}{P_0} \biggr] \frac{dx}{dr_0}
+\frac{\rho_0}{\gamma P_0 }\biggl[ \omega^2 - (3\gamma - 4) \frac{g_0}{r_0}  \biggr] x 
\biggr\}\, .
</math>
  </td>
</tr>
</table>
Hence, we must multiply the first expression through by <math>~r_0^4 \gamma P_0</math> in order to obtain the second expression.
</td></tr></table>


From [[#Foundation|above]], we realize that multiplying the second expression through by <math>~(K_c/G)\rho_c^{-4 / 5}</math> gives,

<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>~
r_0^4\gamma P_0 \biggl\{
\frac{d^2x}{dr*^2} + \biggl[ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr]  x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(r^*)^4 [K_c^{1 / 2}/(G^{1 / 2} \rho_c^{2/5}) ]^4 \gamma P^* [K_c\rho_c^{6/5}] \biggl\{
\frac{d^2x}{dr*^2} + \biggl[ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr]  x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{K_c^{3}}{ G^{2} \rho_c^{2/5}} \biggr]  (r^*)^4 \gamma P^* \biggl\{
\frac{d^2x}{dr*^2} + \biggl[ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl[ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr]  x 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

That is, multiplying the second expression through by,  <math>~(K_c/G)\rho_c^{-4 / 5} \cdot G^2 \rho_c^{2 / 5}/K_c^3 = G/(K_c^2 \rho_c^{2 / 5})</math> , should give a desirable, totally dimensionless version of the LAWE. Remembering that,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\rho_0}{\rho_c}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{r_0}{[K_c^{1/2}/(G^{1/2}\rho_c^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P_0}{K_c\rho_c^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~M_r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_c^{1/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{dP^*}{dr^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{M^*_r \rho^*}{(r^*)^2} </math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K_c^5}{G^3}\biggr]^{1 / 2}</math>
  </td>
</tr>

</table>
</div>
let's try it.

<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>~G K_c^{-2} \rho_c^{-2 / 5} \biggl\{
\frac{d}{dr_0}\biggl[ r_0^4 \gamma P_0 ~\frac{dx}{dr_0} \biggr] 
+\biggl[ \omega^2 \rho_0 r_0^4 + (3\gamma - 4) r_0^3 \frac{dP_0}{dr_0} \biggr] x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~G K_c^{-2} \rho_c^{-2 / 5} \biggl\{ \biggl( \frac{K_c^2 \rho_c^{2/5}}{G} \biggr)
\frac{d}{dr^*}\biggl[ (r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr] 
+\biggl[ \omega^2 \biggl(\frac{K_c^2}{G^2 \rho_c^{8/5}}\biggr) \rho_c \rho^* (r^*)^4 + (3\gamma - 4)\biggl(\frac{K_c }{G \rho_c^{4/5}}\biggr)K_c\rho_c^{6/5} (r^*)^3 \frac{dP^*}{dr^*} \biggr] x 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{dr^*}\biggl[ (r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr] 
+\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] x \, .
</math>
  </td>
</tr>
</table>

Now, guided by the [[SSC/SynopsisStyleSheet#Stability|accompanying summary]], if we multiply through by <math>~4\pi x dr^*</math> and integrate over the entire volume, we obtain the ''governing variational relation'', namely,
<!-- OLD VERSION; IGNORE

<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  x\cdot
d\biggl[ (r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr] 
+ \int_0^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 4\pi  x(r^*)^4 \gamma P^* ~\frac{dx}{dr^*} \biggr]_0^{R^*} 
- \int_0^{R^*}4\pi  \biggl[ (r^*)^4 \gamma P^* ~\biggl( \frac{dx}{dr^*} \biggr)^2\biggr] dr^*
+ \int_0^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4  \biggr] 4\pi x^2 dr^*
- \int_0^{R^*}\biggl[ (3\gamma - 4)M^*_r \rho^*  \biggr] 4\pi x^2 r^* 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^*)^2 P^*  dr^*
+ \int_0^{R^*} (3\gamma - 4)  x^2\biggl( -\frac{M^*_r}{r^*}\biggr)  4\pi \rho^*  (r^*)^2 dr^*
-\biggl[ 4\pi  x^2 (r^*)^3 \gamma P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_0^{R^*} 
+ \int_0^{R^*} 4\pi \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4   x^2 dr^* \, .
</math>
  </td>
</tr>
</table>
OLD VERSION; IGNORE-->

<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^*_\mathrm{core}}4\pi  x\cdot
d\biggl[ (r^*)^4 \gamma_c P^* ~\frac{dx}{dr^*} \biggr] 
+ \int_0^{r^*_\mathrm{core}}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma_c - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\int_{r^*_\mathrm{core}}^{R^*}4\pi  x\cdot
d\biggl[ (r^*)^4 \gamma_e P^* ~\frac{dx}{dr^*} \biggr] 
+ \int_{r^*_\mathrm{core}}^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4 + (3\gamma_e - 4)(r^*)^3 \frac{dP^*}{dr^*} \biggr] 4\pi x^2 dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 4\pi  x(r^*)^4 \gamma_c P^* ~\frac{dx}{dr^*} \biggr]_0^{r^*_\mathrm{core}} 
- \int_0^{r^*_\mathrm{core}}4\pi  \biggl[ (r^*)^4 \gamma_c P^* ~\biggl( \frac{dx}{dr^*} \biggr)^2\biggr] dr^*
- \int_0^{r^*_\mathrm{core}}\biggl[ (3\gamma_c - 4)M^*_r \rho^*  \biggr] 4\pi x^2 r^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[ 4\pi  x(r^*)^4 \gamma_e P^* ~\frac{dx}{dr^*} \biggr]_{r^*_\mathrm{core}}^{R^*} 
- \int_{r^*_\mathrm{core}}^{R^*}4\pi  \biggl[ (r^*)^4 \gamma_e P^* ~\biggl( \frac{dx}{dr^*} \biggr)^2\biggr] dr^*
- \int_{r^*_\mathrm{core}}^{R^*}\biggl[ (3\gamma_e - 4)M^*_r \rho^*  \biggr] 4\pi x^2 r^* dr^*
+ \int_{0}^{R^*}\biggl[ \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4  \biggr] 4\pi 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^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 \gamma_c 4\pi (r^*)^2 P^*  dr^*
+ \int_0^{r^*_\mathrm{core}} (3\gamma_c - 4)  x^2\biggl( -\frac{M^*_r}{r^*}\biggr)  4\pi \rho^*  (r^*)^2 dr^*
-\biggl[ 4\pi  x^2 (r^*)^3 \gamma_c P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_0^{r^*_\mathrm{core}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \int_{r^*_\mathrm{core}}^{R^*}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 \gamma_e 4\pi (r^*)^2 P^*  dr^*
+ \int_{r^*_\mathrm{core}}^{R^*} (3\gamma_e - 4)  x^2\biggl( -\frac{M^*_r}{r^*}\biggr)  4\pi \rho^*  (r^*)^2 dr^*
-\biggl[ 4\pi  x^2 (r^*)^3 \gamma_e P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_{r^*_\mathrm{core}}^{R^*} 
+ \int_{0}^{R^*} 4\pi \biggl( \frac{\omega^2}{G\rho_c}\biggr) \rho^* (r^*)^4   x^2 dr^* \, .
</math>
  </td>
</tr>
</table>


Energy Normalization:

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\gamma - 1)dU_\mathrm{int}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi r^2 P dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi \biggl[ \frac{K_c^{1 / 2}}{G^{1 / 2} \rho_c^{2 / 5}} \biggr]^3 K_c \rho_c^{6 / 5} (r^*)^2 P^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi \biggl[ \frac{K_c^{5}}{G^{3}} \biggr]^{1 / 2} (r^*)^2 P^* dr^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{K_c^{5}}{G^{3}} \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~dW_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{GM_r}{r} \biggr) 4\pi r^2 \rho dr
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{GM^*_r}{r^*} \biggr) 4\pi (r^*)^2 \rho^* dr^* \biggl[ \frac{K_c}{G\rho_c^{4 / 5}} \biggr]\rho_c \biggl[ \frac{ K_c^{3 / 2} }{G^{3 / 2}\rho_c^{1 / 5}  } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\biggl( \frac{M^*_r}{r^*} \biggr) 4\pi (r^*)^2 \rho^* dr^*  E_\mathrm{norm}
</math>
  </td>
</tr>
</table>

<span id="VariationPrincipleRelation">Hence, the ''dimensionless'' governing variational relation becomes,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_0^{R^*} \biggl( \frac{2\pi}{3}\biggr)\sigma_c^2(r^*)^2   x^2 dM_r^* </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 \int_0^{r^*_\mathrm{core}}  \gamma_c(\gamma_c - 1) x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- \int_0^{r^*_\mathrm{core}} (3\gamma_c - 4)  x^2 dW^*_\mathrm{grav}
+ \biggl[ 4\pi  x^2 (r^*)^3 \gamma_c P^* \biggl( - \frac{d\ln x}{d\ln r^*} \biggr) \biggr]_0^{r^*_\mathrm{core}} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\gamma_c}{3} \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
+ 4\pi  x_i^2 (r_\mathrm{core}^*)^3 \gamma_c P_i^* \biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{core}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2 \gamma_e}{3}\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 \gamma_e P_i^* \biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{env}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\gamma_c}{3} \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (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>~
+ \frac{2 \gamma_e}{3}\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 P_i^*\biggl[ \gamma_c \biggl\{ \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{core}  - \gamma_e \biggl\{ \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{env} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\gamma_c}{3} \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (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>~
+ \frac{2 \gamma_e}{3}\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 P_i^*\biggl[ 3(\gamma_e - \gamma_c) \biggr] \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~dM^*_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi (r^*)^2 \rho^* dr^*  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~dU^*_\mathrm{int}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(\gamma-1)} \biggl[4\pi (r^*)^2 P^*  dr^* \biggr] = \biggl[ \frac{2}{3(\gamma - 1)} \biggr]dS^*_\mathrm{therm}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~dW^*_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\biggl( \frac{M^*_r}{r^*} \biggr) 4\pi (r^*)^2 \rho^* dr^*  \, .</math>
  </td>
</tr>
</table>

Or, for inclusion in our accompanying ''[[SSC/SynopsisStyleSheet#Bipolytropes|Tabular Overview]]'',
<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>

===Implementation===

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="15">
Table 3  
  </th>
</tr>
<tr>
  <td rowspan="3" align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td rowspan="3" align="center"><math>~\xi_i</math></td>
  <td colspan="6" align="center" bgcolor="lightblue">Core</td>
  <td colspan="6" align="center" bgcolor="lightgreen">Envelope</td>
  <td align="center">Virial</td>
</tr>
<tr>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dS^*_\mathrm{therm}</math></td>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dW^*_\mathrm{grav}</math></td>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dS^*_\mathrm{therm}</math></td>
  <td colspan="3" align="center"><font size="-1">Integrals over</font><p></p><math>~dW^*_\mathrm{grav}</math></td>
  <td rowspan="2" align="center">Numerical<p></p><math>~\biggl[\frac{2\mathfrak{s}_\mathrm{tot}}{|\mathfrak{w}_\mathrm{tot}|}-1\biggr] </math></td>
</tr>
<tr>
  <td align="center">Analytic<br /><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center">'''TERM1'''</td>
  <td align="center">Analytic<br /><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center">'''TERM2'''</td>
  <td align="center">Analytic<br /><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center">'''TERM3'''</td>
  <td align="center">Analytic<br /><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center">'''TERM4'''</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">1.6686460157</td>
  <td align="right">3.021916335</td>
  <td align="center">3.021921</td>
  <td align="center">0.116389175</td>
  <td align="center">-3.356583022</td>
  <td align="center">-3.35666</td>
  <td align="center">-2.649752079</td>
  <td align="center">1.47780476</td>
  <td align="center">1.47791</td>
  <td align="center">1.0720821</td>
  <td align="center">-5.642859167</td>
  <td align="center">-5.642820</td>
  <td align="center">-1.91142893</td>
  <td align="center">0.000020</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.27925811317</td>
  <td align="right">4.241287117</td>
  <td align="center">4.241410819</td>
  <td align="center">0.440878529</td>
  <td align="center">-6.074241035</td>
  <td align="center">-6.074317546</td>
  <td align="center">-4.150731169</td>
  <td align="center">4.284931508</td>
  <td align="center">4.28547195</td>
  <td align="center">1.44651932</td>
  <td align="center">-10.97819621</td>
  <td align="center">-10.97847622</td>
  <td align="center">-0.92598634</td>
  <td align="center">0.000057</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.560146865247</td>
  <td align="right">4.639705843</td>
  <td align="center">4.6399114</td>
  <td align="center">0.6794857</td>
  <td align="center">-7.125754184</td>
  <td align="center">-7.125854025</td>
  <td align="center">-4.5487829</td>
  <td align="center">11.72861751</td>
  <td align="center">11.730381</td>
  <td align="center">1.51410084</td>
  <td align="center">-25.61089252</td>
  <td align="center">-25.6115597</td>
  <td align="center">-0.4496513</td>
  <td align="center">0.000097</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.582007485476</td>
  <td align="right">4.667042505</td>
  <td align="center">4.667254935</td>
  <td align="center">0.700414598</td>
  <td align="center">-7.200966267</td>
  <td align="center">-7.201068684</td>
  <td align="center">-4.57274936</td>
  <td align="center">13.15887139</td>
  <td align="center">13.1608467</td>
  <td align="center">1.51408246</td>
  <td align="center">-28.45086152</td>
  <td align="center">-28.45153761</td>
  <td align="center">-0.4170461</td>
  <td align="center">0.000101</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.6274239687695</td>
  <td align="right">4.722277318</td>
  <td align="center">4.722504339</td>
  <td align="center">0.744964507</td>
  <td align="center">-7.354156963</td>
  <td align="center">-7.3542507</td>
  <td align="center">-4.61961058</td>
  <td align="center">17.1374434</td>
  <td align="center">17.1399773</td>
  <td align="center">1.51055838</td>
  <td align="center">-36.36528446</td>
  <td align="center">-36.36591543</td>
  <td align="center">-0.3524855</td>
  <td align="center">0.000110</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="left">2.7357711469398</td>
  <td align="right">4.84592201</td>
  <td align="center">4.846185027</td>
  <td align="center">0.857001395</td>
  <td align="center">-7.70305421</td>
  <td align="center">-7.703178009</td>
  <td align="center">-4.7163542</td>
  <td align="center">37.84289623</td>
  <td align="center">37.8479208</td>
  <td align="center">1.47966673</td>
  <td align="center">-77.67458196</td>
  <td align="center">-77.67408155</td>
  <td align="center">-0.2194152</td>
  <td align="center">0.000128</td>
</tr>
<tr>
  <td colspan="15" align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{s}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{r^*_\mathrm{core}}  dS^*_\mathrm{therm}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM1'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{w}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_0^{r^*_\mathrm{core}}  dW^*_\mathrm{grav}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM2'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{s}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_{r^*_\mathrm{core}}^{R^*} dS^*_\mathrm{therm}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM3'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dS^*_\mathrm{therm}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{w}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_{r^*_\mathrm{core}}^{R^*} dW^*_\mathrm{grav}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM4'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}</math>
  </td>
</tr>
<tr>
  <td colspan="7" align="center">NOTE:  In all integrals, the fractional radial-displacement function, <math>~x</math>, has been normalized<br />to unity at the center of the spherical model.</td>
</tr>
</table>
  </td>
</tr>
</table>
</div>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{2\pi \cdot \mathbf{TERM5} } \biggl\{
\frac{2\gamma_c}{3} [\mathbf{TERM1}]
- (3\gamma_c - 4) [\mathbf{TERM2}]
+ \frac{2 \gamma_e}{3} [\mathbf{TERM3}]
- (3\gamma_e - 4)   [\mathbf{TERM4}]
- 4\pi  x_i^2 (r_\mathrm{core}^*)^3 P_i^* [3(\gamma_e - \gamma_c)] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.036312577\times \biggl[0.093111340 - 1.059900832 + 1.4294428 - (-3.82285786) - 1.782200484\times (2.4) \biggr] = 0.036312577\times [0.00819] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0002973 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.020704186\times \biggl[0.352702823 - 1.660292468 + 1.928692426 - (-1.85197268) - 1.029029184\times (2.4) \biggr] = 0.020704186\times [0.003405] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0000705 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 0.345}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.019377703\times \biggl[0.54358856 - 1.81951316 + 2.01880112 - (-0.8993026) - 0.68747441\times (2.4) \biggr] = 0.019377703\times [0.003585] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0000695 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1 / 3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.019467503\times \biggl[0.560331678 - 1.81099744 + 2.018776613 - (-0.8340922) - 0.658354814\times (2.4) \biggr] = 0.019467503\times [1.602203052 - 1.580051555] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.00043123 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 0.309}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.019781992\times \biggl[0.595971606 - 1.847844232 + 2.01407784 - (-0.704971) - 0.60905486\times (2.4) \biggr] = 0.019781992\times [1.467176214 - 1.461731665] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.0001077 \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[  \sigma_c^2 \biggr]^\mathrm{VP}_{\mu_e/\mu_c = 1 / 4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.021344909\times \biggl[0.685601116 - 1.88654168 + 1.972888973 - (-0.4388304) - 0.499262049\times (2.4) \biggr] = 0.021344909\times [1.210778809 - 1.198228918] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0.000267876 \, .</math>
  </td>
</tr>
</table>


<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="11">
Table 4
  </th>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center">Analytic<br /><math>~\mathfrak{m}_\mathrm{tot}</math></td>
  <td align="center">Numerical<br /><math>~\mathfrak{m}_\mathrm{tot}</math></td>
  <td align="center">'''TERM5'''</td>
  <td align="center"><math>~x_i</math></td>
  <td align="center"><math>~r^*_\mathrm{core}</math></td>
  <td align="center"><math>~P^*_i</math></td>
  <td align="center"><math>~\biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{core}</math></td>
  <td align="center"><math>~\biggl\{ - \frac{d\ln x}{d\ln r^*}\biggr|_i \biggr\}_\mathrm{env}</math></td>
  <td align="center"><math>~\biggl[\sigma_c^2\biggr]^\mathrm{VP}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">1.6686460157</td>
  <td align="right">4.818155928</td>
  <td align="center">4.818145</td>
  <td align="center">13.14874521</td>
  <td align="center">0.814374698</td>
  <td align="center">1.153014872</td>
  <td align="center">0.139506172</td>
  <td align="center">+0.455871977</td>
  <td align="center">+1.47352</td>
  <td align="center">0.0002973</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.27925811317</td>
  <td align="right">9.020985415</td>
  <td align="center">9.021084268</td>
  <td align="center">23.0612702</td>
  <td align="center">0.653665497</td>
  <td align="center">1.574940686</td>
  <td align="center">0.049058481</td>
  <td align="center">+1.059668912</td>
  <td align="center">+1.835801347</td>
  <td align="center">0.0000705</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.560146865247</td>
  <td align="right">17.41399388</td>
  <td align="center">17.4141672</td>
  <td align="center">24.63990825</td>
  <td align="center">0.563043202</td>
  <td align="center">1.769031527</td>
  <td align="center">0.030957085</td>
  <td align="center">+1.552125296</td>
  <td align="center">+2.131275177</td>
  <td align="center">0.0000695</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.582007485476</td>
  <td align="right">18.8449906</td>
  <td align="center">18.8451614</td>
  <td align="center">24.52624951</td>
  <td align="center">0.555549156</td>
  <td align="center">1.78413696</td>
  <td align="center">0.029889634</td>
  <td align="center">+1.600041467</td>
  <td align="center">+2.16002488</td>
  <td align="center">0.0004312</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.6274239687695</td>
  <td align="right">22.61541791</td>
  <td align="center">22.6155686</td>
  <td align="center">24.13633675</td>
  <td align="center">0.539776219</td>
  <td align="center">1.815519219</td>
  <td align="center">0.027798189</td>
  <td align="center">+1.705239188</td>
  <td align="center">+2.223143513</td>
  <td align="center">0.0001077</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="left">2.7357711469398</td>
  <td align="right">39.12088278</td>
  <td align="center">39.1208075</td>
  <td align="center">22.36902623</td>
  <td align="center">0.501037082</td>
  <td align="center">1.890385851</td>
  <td align="center">0.023427588</td>
  <td align="center">+1.991720516</td>
  <td align="center">+2.39503231</td>
  <td align="center">0.0002679</td>
</tr>
<tr>
  <td colspan="11" align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{m}_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}dM_r^*</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~I_\mathrm{sphere}^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}(r^*)^2   dM_r^*</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
'''TERM5'''
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^{R^*}(r^*)^2   x^2 dM_r^*</math>
  </td>
</tr>
<tr>
  <td colspan="11" align="center">NOTE:  In the '''TERM5''' integral (as elsewhere), the fractional radial-displacement function, <math>~x</math>, <br />has been normalized to unity at the center of the spherical model.</td>
</tr>
</table>
  </td>

</tr>
</table>

==Revised Free-Energy Analysis==
If we set <math>~x </math> = constant in the [[#VariationPrincipleRelation|variational principle relation]], we have,
<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^*)^2dM_r^* </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  dW^*_\mathrm{grav}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} dW^*_\mathrm{grav}
- 4\pi  (r_\mathrm{core}^*)^3 P_i^*\biggl[ 3(\gamma_e - \gamma_c) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~~~ \biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 I_\mathrm{sphere}^* </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(4 - 3\gamma_c )\mathfrak{w}_\mathrm{core}
+ (4 - 3\gamma_e )  \mathfrak{w}_\mathrm{env}
+ P_i^*V^*_\mathrm{core}\biggl[ 3^2(\gamma_c - \gamma_e) \biggr] \, ,
</math>
  </td>
</tr>
</table>
Notice the similarity between this last expression and the pair of expressions &#8212; numbered <b><font color="maroon" size="+1">&#x2465;</font></b> and <b><font color="maroon" size="+1">&#x2466;</font></b> &#8212; that arise in the context of [[SSC/SynopsisStyleSheet#Stability|pressure-truncated polytropes]].