Editing SSCpt1/Virial (section)

====Implementation====
=====Normalize=====
We will now judiciously introduce our adopted normalizations into the [[#Expressions_for_Various_Energy_Terms|above-defined free-energy term expressions]], using asterisks to denote dimensionless variables that have been accordingly normalized; for example, 
<div align="center">
<math>
r^* \equiv \frac{r}{R_\mathrm{norm}} \, , ~~~~~~ P^* \equiv \frac{P}{P_\mathrm{norm}} \, , ~~~~~~ </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}} \, .
</math>
</div>

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_r(r^*)  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
R_\mathrm{norm}^3 \rho_\mathrm{norm} \int_0^{r^*}  4\pi (r^*)^2 \rho^* dr^* 
= M_\mathrm{tot} \int_0^{r^*}  3(r^*)^2 \rho^* dr^*  \, .
</math>
  </td>
</tr>
</table>
</div>

<font color="red">Confinement by External Pressure (Normalized Volume):</font>  
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>P_e V</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>E_\mathrm{norm} \biggl[ \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) 
\biggl(\frac{R_\mathrm{limit}}{R_\mathrm{norm}}\biggr)^3 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

<font color="red">Normalized Gravitational Potential Energy:</font>  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 4\pi GM_\mathrm{tot} R_\mathrm{norm}^2 \rho_\mathrm{norm} \int_0^{\chi=R_\mathrm{limit}^*} \biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* 
</math>
  </td>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- E_\mathrm{norm} \int_0^{\chi = R_\mathrm{limit}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* \, .
</math>
  </td>
</tr>
</table>
</div>

<font color="red">Normalized Reservoir of Thermodynamic Energy:</font>  
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{S}_I</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>E_\mathrm{norm}  \int_0^{\chi=R_\mathrm{limit}^*} 3 \ln (\rho^*) (r^*)^2 \rho^* dr^* \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{S}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{E_\mathrm{norm}}{({\gamma_g}-1)}  \int_0^{\chi=R_\mathrm{limit}^*}  4\pi (r^*)^2 P^* dr^* \, .</math>
  </td>
</tr>
</table>
</div>

<font color="red">Normalized Rotational Kinetic Energy:</font>  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>T_\mathrm{rot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\pi \dot\varphi_c^2 R_\mathrm{norm}^5 \rho_\mathrm{norm} 
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3  d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}}  (\rho^*)  dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5^2\pi}{2^2} \biggr) \biggl[ \frac{J^2 R_\mathrm{norm} \rho_\mathrm{norm}}{M_\mathrm{tot}^2} \biggr] \chi_\mathrm{eq}^{-4}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3  d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}}  (\rho^*)  dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3\cdot 5^2}{2^4} \biggr) \biggl[  \frac{J^2}{M_\mathrm{tot}} \biggl(\frac{E_\mathrm{norm} }{G M_\mathrm{tot}^2 }\biggr)^2 \biggr] \chi_\mathrm{eq}^{-4}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3  d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}}  (\rho^*)  dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>E_\mathrm{norm}
\biggl( \frac{3^2\cdot 5^2}{2^6 \pi} \biggr) \biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4}  \biggr] \chi_\mathrm{eq}^{-4}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3  d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}}  (\rho^*)  dz^* \, ,
</math>
  </td>
</tr>
</table>
</div>
where, 
<div align="center">
<math>\dot\varphi_c \equiv \frac{5J}{2M_\mathrm{tot} R_\mathrm{eq}^2} = 
\frac{5}{2} \biggl[ \frac{J}{M_\mathrm{tot} R_\mathrm{norm}^2} \biggr] \chi_\mathrm{eq}^{-2} \, ,</math>
</div>
is a characteristic rotation frequency in the equilibrium configuration whose value is set once the system's total angular momentum, <math>J</math>, is specified.

=====Separate Time &amp; Space=====
Our intent is to vary the size of the configuration <math>(R_\mathrm{limit})</math> while holding the (properly normalized) internal structural profile fixed, so let's separate the spatial integral over the (fixed) structural profile from the time-varying configuration size.  Making use of the dimensionless ''internal'' coordinates,
<div align="center">
<math>x \equiv \frac{r}{R_\mathrm{limit}} \, ,~~~~w \equiv \frac{\varpi}{R_\mathrm{limit}} \, ,
~~~~\zeta \equiv \frac{z}{R_\mathrm{limit}} \, ,
</math>
</div>
that always run from zero to one, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>\rightarrow~</math>
  </td>
  <td align="left">
<math>
x \biggl( \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \biggr) = x \chi \, ;
</math>
&nbsp;&nbsp;&nbsp;and, likewise, &nbsp;&nbsp;&nbsp;
<math>
~~~~\varpi^* ~\rightarrow~ w \chi \, ;
~~~~z^* ~\rightarrow~ \zeta \chi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>\rightarrow~</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\rho(x)}{\bar\rho} \biggr] \biggl( \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr)
= \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] \biggl( \frac{M_\mathrm{limit}/R_\mathrm{limit}^3}{M_\mathrm{tot}/R_\mathrm{norm}^3} \biggr)
= \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] \biggl(  \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \chi^{-3}  
= \frac{\rho_c}{\bar\rho} \biggl[ \frac{\rho(x)}{\rho_c} \biggr] \biggl(  \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \chi^{-3} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>\rightarrow~</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{P(x)}{P_c} \biggr] \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl( \frac{K\rho_c^\gamma}{P_\mathrm{norm}} \biggr)
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl( \frac{\rho_c}{\bar\rho} \biggr)^\gamma 
\biggl[ \frac{(3M_\mathrm{limit}/4\pi R_\mathrm{limit}^3)^\gamma}{K^{-1}P_\mathrm{norm}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^\gamma 
\biggl[ \frac{K M_\mathrm{tot}^\gamma}{P_\mathrm{norm} R_\mathrm{norm}^{3\gamma}} \biggr] \biggl(  \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\biggl( \frac{R_\mathrm{limit}}{R_\mathrm{norm}}  \biggr)^{-3\gamma}
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^\gamma \biggl(  \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\chi^{-3\gamma} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\dot\varphi}{\dot\varphi_c}</math>
  </td>
  <td align="center">
<math>\rightarrow~</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{limit}} \biggr] \biggl( \frac{\dot\varphi_\mathrm{limit}}{\dot\varphi_c}\biggr)
= \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{limit}} \biggr] \biggl( \frac{R_\mathrm{limit}}{R_\mathrm{eq}}\biggr)^{-2}
= \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{limit}} \biggr] \chi_\mathrm{eq}^{2} \chi^{-2} \, .
</math>
  </td>
</tr>
</table>
</div>

=====Summary of Normalized Expressions=====
Hence, our normalized expressions become,
<div align="center">
<table border="1" cellpadding="8">
<tr><th align="center">
Normalized Expressions
</th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{P_e V}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)  \chi^3 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
- \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]  \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^2_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\int_0^{1} 5x \biggl\{\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx\biggr\}  \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\int_0^{1}  3x^2 \biggl[ \frac{P(x)}{P_c} \biggr]  dx \biggr\} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^{1} 
\biggl\{ \ln \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] -3\ln \biggl[ \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr]  \biggr\} 
3 x^2 \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] dx </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-3 \ln \chi  + \mathrm{constant} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{T_\mathrm{rot}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^6 \pi} \biggr) \biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \biggl( \frac{\rho_c}{\bar\rho}  \biggr)_\mathrm{eq} 
\int_0^{1} \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{edge}} \biggr]^2 w^3  dw
\int_{-\sqrt{1 - w^2}}^{\sqrt{1 - w^2}}  \biggl[ \frac{\rho(w,\zeta)}{\rho_c} \biggr]   d\zeta \, .
</math>
  </td>
</tr>
</table>

</td></tr>
<tr><td align="left>
<font color="red">NOTE to self (21 September 2014)<b></b></font>:  The expressions for <math>\mathfrak{S}_I</math> and <math>T_\mathrm{rot}</math> may not properly account for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>.
</td></tr>
</table>
</div>


It should be emphasized that the coefficient involving the density ratio, <math>(\rho_c/\bar\rho)</math>, that lies outside of the integral in most of these expressions depends only on the internal structure, and not the overall size, of the configuration.  It can therefore be evaluated at any time.  We usually will choose to evaluate this coefficient in an equilibrium state, that is, when <math>R_\mathrm{limit} \rightarrow R_\mathrm{eq}</math>.  Accordingly, the subscript "eq" has been attached to this coefficient.  The inverse of this density ratio can be obtained from the integral expression for <math>M_r</math> by recognizing that <math>M_r \rightarrow M_\mathrm{limit}</math> when the upper limit on the integral <math>x \rightarrow 1</math>.  Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl(\frac{\rho_c}{\bar\rho} \biggr)^{-1}_\mathrm{eq}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^{1}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]_\mathrm{eq}  dx \, .</math>
  </td>
</tr>
</table>
</div>
This coefficient also may be rewritten in terms of the central pressure in the equilibrium state; specifically, using a sequence of steps similar to the ones that were used, above, in rewriting <math>P^*</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi^{3\gamma} \biggr]_\mathrm{eq} \, .</math>
  </td>
</tr>
</table>
</div>

=====Looking Ahead to Bipolytropes=====
<div id="BiPolytrope">
<table border="1" align="center" width="90%" cellpadding="20">
<tr><td align="left">
<b><font color="purple">ASIDE:</font></b> When we discuss the free energy of bipolytropic configurations, we will need to divide the expression for <math>\mathfrak{S}_A/E_\mathrm{norm}</math> into two parts &#8212; one accounting for the reservoir of thermodynamic energy in the bipolytrope's "core" and one accounting for the reservoir of thermodynamic energy in the bipolytrope's "envelope."  It is useful to develop this two-part expression here, while the definition of <math>\mathfrak{S}_A</math> is fresh in our minds and to show how the two-part expression reduces to the simpler expression for <math>\mathfrak{S}_A/E_\mathrm{norm}</math>, just derived, when there is no distinction drawn between the properties of the core and the envelope. 

In what follows, we will use the subscript ''core'' (or "c") when referencing physical properties of the bipolytrope's core and the subscript ''env'' (or "e") for the envelope; and, as above, we will use <math>x \equiv r/R_\mathrm{edge}</math> to denote the dimensionless radial location within a configuration of radius, <math>R_\mathrm{edge}</math>.  The dimensionless radial coordinate, <math>q \equiv x_i = r_i/R_\mathrm{edge}</math>, will identify the radial ''interface'' where the core meets the envelope; that is, <math>q</math> will identify both the outer edge of the core and the inner edge of the envelope.  In general, separate expressions will define the run of pressure through the core and through the envelope.  We can assume that, for the core, the pressure drops monotonically from a value of <math>P_0</math> at the center of the configuration according to an expression of the form,
<div align="center">
<math>P_\mathrm{core}(x) = P_0 [1 - p_c(x)]</math> &nbsp; &nbsp;&nbsp; for &nbsp; &nbsp;&nbsp; <math>0 \leq x \leq q \, ,</math>
</div>
and that, for the envelope, the pressure drops monotonically from a value of <math>P_{ie}</math> at the interface according to an expression of the form,
<div align="center">
<math>P_\mathrm{env}(x) = P_{ie} [1 - p_e(x)]</math> &nbsp; &nbsp;&nbsp; for &nbsp; &nbsp;&nbsp; <math>q \leq x \leq 1 \, ,</math>
</div>
where <math>p_c(x)</math> and <math>p_e(x)</math> are both dimensionless functions that will depend on the equations of state that are chosen for the core and envelope, respectively.  By prescription, the pressure in the envelope must drop to zero at the surface of the bipolytropic configuration, hence, we should expect that <math>p_e(1) = 1</math>.  Furthermore, by prescription, the pressure in the core will drop to a value, <math>P_{ic}</math>, at the interface, so we can write,
<div align="center">
<math>P_{ic} = P_0 [1 - p_c(q)] \, .</math>
</div>

In equilibrium &#8212; that is, when <math>R_\mathrm{edge} = R_\mathrm{eq}</math> &#8212; we will demand that the pressure at the interface be the same, whether it is referenced in the core or in the envelope, that is, we will demand that <math>P_{ic} = P_{ie} \, .</math>  It will therefore prove to be strategically advantageous to rewrite the expression for the run of pressure through the core in terms of the pressure at the interface rather than in terms of the central pressure; specifically,
<div align="center">
<math>P_\mathrm{core}(x) = P_{ic} \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] \, .</math> 
</div>
Referencing these prescriptions for <math>P_\mathrm{core}(x)</math> and <math>P_\mathrm{env}(x)</math>, the two-part expression for the reservoir of thermodynamic energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{({\gamma_c}-1)}  \int_0^{r_i/R_\mathrm{norm}}  4\pi (r^*)^2 P^*_\mathrm{core} dr^*
+ \frac{1}{({\gamma_e}-1)}  \int_{r_i/R_\mathrm{norm}}^\chi  4\pi (r^*)^2 P^*_\mathrm{env} dr^* 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi \chi^3 }{({\gamma_c}-1)} \biggl[ \frac{P_{ic}}{P_\mathrm{norm}} \biggr] \int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx
+ \frac{4\pi \chi^3 }{({\gamma_e}-1)} \biggl[ \frac{P_{ie}}{P_\mathrm{norm}} \biggr]  \int_q^1  \biggl[1 - p_e(x) \biggr]  x^2 dx \, .
</math>
  </td>
</tr>
</table>
</div>
As is implied by the subscripts on the adiabatic exponents that appear in the leading factor of each of the two terms, we are assuming that, as the bipolytropic system expands or contracts, the thermodynamic properties of the material in the envelope will vary as prescribed by an adiabat of index, <math>\gamma_e</math>, while the thermodynamic properties of material in the core will vary as prescribed by a, generally different, adiabat of index, <math>\gamma_c</math>.  Therefore, as the radius of the bipolytropic configuration, <math>R_\mathrm{edge}</math>, is varied, the density of each fluid element will vary and, in the core, the pressure of each fluid element will vary as <math>P \propto \rho^{\gamma_c}</math> while, in the envelope, the pressure of each fluid element will vary as <math>P \propto \rho^{\gamma_e}</math>.  If we furthermore assume that the mass in the core and the mass in the envelope remain constant during a phase of contraction or expansion, the density of each fluid element will vary as <math>R_\mathrm{edge}^{-3}</math>, whether the material is associated with the core or with the envelope.  Therefore, using the subscript, "eq," to identify the value of thermodynamic quantities when the system is in an equilibrium state and, accordingly, <math>R_\mathrm{edge} = R_\mathrm{eq}</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_c} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_c} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_e} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e} \, .</math>
  </td>
</tr>
</table>
</div>
In particular, for any <math>R_\mathrm{edge}</math>, material associated with the core that lies at the interface will have a pressure given by the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>P_{ic}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(P_{ic})_\mathrm{eq} \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_c} 
= (P_{ic})_\mathrm{eq} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{+3\gamma_c}\biggl( \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)^{-3\gamma_c} 
= (P_{ic})_\mathrm{eq} \chi_\mathrm{eq}^{+3\gamma_c} \chi^{-3\gamma_c} 
\, ,</math>
  </td>
</tr>
</table>
</div>
while material associated with the envelope that lies at the interface will have a pressure given by the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>P_{ie}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(P_{ie})_\mathrm{eq} \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e} 
= (P_{ie})_\mathrm{eq} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{+3\gamma_e}\biggl( \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)^{-3\gamma_e} 
= (P_{ie})_\mathrm{eq} \chi_\mathrm{eq}^{+3\gamma_e} \chi^{-3\gamma_e} 
\, .</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi }{({\gamma_c}-1)}  \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} 
\int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ ~\frac{4\pi }{({\gamma_e}-1)}  \biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_e} 
\int_q^1  \biggl[1 - p_e(x) \biggr]  x^2 dx \, .
</math>
  </td>
</tr>
</table>
</div>
----


Now, let's see how this expression simplifies if <math>P_{ie} = P_{ic}</math> and <math>\gamma_e = \gamma_c</math> and, hence, the properties of the envelope are indistinguishable from the properties of the core.  We note, first, that in this limit, <math>P_\mathrm{core}(x)</math> and <math>P_\mathrm{env}(x)</math> must be identical functions of <math>x</math>, that is, it must be the case that <math>p_e(x)</math> is related to <math>p_c(x)</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>1 - p_e(x) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1 - p_c(x)}{1-p_c(q)} \, .</math>
  </td>
</tr>
</table>
</div>

We therefore obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi }{({\gamma_c}-1)}  \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} 
\biggl\{ \int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx + \int_q^1  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi }{({\gamma_c}-1)}  \biggl[ \frac{P_0 \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} 
\biggl\{ \int_0^1  \biggl[1 - p_c(x)\biggr]  x^2 dx  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi }{({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma}
\int_0^{1}  \biggl[ \frac{P(x)}{P_c} \biggr] x^2 dx \biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
as desired.

</td></tr>
</table>
</div>