Editing SSC/BipolytropeGeneralizationVersion2

__FORCETOC__
=Bipolytrope Generalization=

==Setup==

In a [[SSCpt1/Virial#Free_Energy_Expression|more general context]], we have discussed a Gibbs-like free-energy function of the generic form,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots
</math>
</div> 
Here we are interested in examining the free energy of isolated, nonrotating, spherically symmetric bipolytropes, so we can drop the term that accounts for the influence of an external pressure and we can drop the kinetic energy term.  But we need to consider separately the contributions to the reservoir of thermodynamic energy by the core and envelope.  In particular, we will assume that compressions/expansions occur adiabatically, but that the core and the envelope evolve along separate adiabats &#8212; <math>\gamma_c</math> and <math>\gamma_e</math>, respectively.  

===Review of Isolated Polytrope===
If we were configuring isolated polytropes &#8212; instead of ''bi''polytropes &#8212; the free-energy expression would be, simply,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_A \, ,
</math>
</div> 
and, following the detailed steps presented in our [[SSCpt1/Virial#Implementation|introductory discussion of the free energy of spherically symmetric, configurations]], properly normalized expressions for the two contributing energy terms would be,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \int_0^{\chi = R_\mathrm{edge}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* 
</math>
  </td>
</tr>


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

<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_g}-1)}  \int_0^{\chi=R_\mathrm{edge}^*}  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>~\frac{1}{({\gamma_g}-1)} \biggl( \frac{4\pi}{3} \biggr)^{1-\gamma}
\biggl\{ \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{\gamma}
\int_0^{1}  3x^2 \biggl[ \frac{P(x)}{P_c} \biggr]  dx \biggr\}_\mathrm{eq} \cdot \chi^{3-3\gamma} \, ,</math>
  </td>
</tr>

</table>
</div>
where, 
<div 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} \int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx \, .</math>
  </td>
</tr>
</table>
</div>
We note that, because <math>~M_r(x)/M_\mathrm{tot} = 1</math> in the limit, <math>~x \rightarrow 1</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq}   </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \int_0^{1}  3x^2 \biggl( \frac{\rho(x)}{\rho_c} \biggr)  dx \biggr]^{-1} \, ,</math>
  </td>
</tr>
</table>
</div>
or, if desired, the central-to-mean density ratio in one or both energy terms could be replaced by a term involving the normalized central pressure and the dimensionless equilibrium radius, <math>~\chi_\mathrm{eq}</math>, via the relation,
<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}
</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>

===Bipolytrope===
When considering an isolated, spherically symmetric ''bi''polytropic configuration, each energy term will be made up of separate contributions coming from the core and envelope, that is,
<div align="center">
<math>
\mathfrak{G} = (W_\mathrm{grav} + \mathfrak{S}_A)_\mathrm{core} + (W_\mathrm{grav} + \mathfrak{S}_A)_\mathrm{env}  \, .
</math>
</div> 

====Partitioning the Mass====
The core will be principally defined in terms of two dimensionless parameters &#8212; <math>~q</math> and <math>~\nu</math> &#8212; which are, respectively, the core's radius relative to the bipolytrope's total radius, and the core's mass relative to the total mass of the bipolytropic configuration, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>q</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>x_i = \frac{r_i}{R_\mathrm{edge}} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\nu</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{M_\mathrm{core}}{M_\mathrm{tot}} \, .</math>
  </td>
</tr>
</table>
</div>
Given the separate (equilibrium) density profiles of the core and the envelope while sticking to the notation used in our [[SSCpt1/Virial#Normalizations|introductory discussion]], we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>(\mathrm{For}~0 \leq r^* \leq r_i^*)</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>M_r </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) R_\mathrm{norm}^3 \rho_\mathrm{norm} \int_0^{r^*}  3 (r^*)^2 \rho_\mathrm{core}^* dr^* 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{tot} \cdot \chi^3 \int_0^{x}  3 \biggl[ \frac{\rho_\mathrm{core}(x)}{{\bar\rho}_\mathrm{core}} \biggr] 
\biggl[ \frac{M_\mathrm{core}/(x_i R_\mathrm{edge})^3}{M_\mathrm{tot}/R^3_\mathrm{norm}} \biggr] x^2 dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{tot} \biggl( \frac{\nu}{q^3} \biggr) \int_0^{x}  3
\biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{core}  
x^2 dx \, ; 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(\mathrm{For}~r_i^* \leq r^* \leq R_\mathrm{edge}^*)</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~M_r </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> M_\mathrm{core} + 
\biggl( \frac{4\pi}{3} \biggr) R_\mathrm{norm}^3 \rho_\mathrm{norm} \int_{r_i^*}^{r^*}  3 (r^*)^2 \rho_\mathrm{env}^* dr^* 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>M_\mathrm{core} +
M_\mathrm{tot} \cdot \chi^3 \int_{x_i}^{x}  3 \biggl[ \frac{\rho_\mathrm{env}(x)}{{\bar\rho}_\mathrm{env}} \biggr] 
\biggl\{ \frac{M_\mathrm{env}/[ (1-x_i^3) R_\mathrm{edge}^3]}{M_\mathrm{tot}/R^3_\mathrm{norm}} \biggr\} x^2 dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{tot} \biggl\{\nu + \biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{x_i}^{x}  3
\biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{env}  
x^2 dx \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

'''EXAMPLES'''
* [[SSC/Structure/BiPolytropes/FreeEnergy00#Mass_Profile|Mass profile of bipolytrope]] with <math>(n_c, n_e) = (0, 0)</math>.
* [[SSC/Structure/BiPolytropes/FreeEnergy51#Mass_Profile|Mass profile of bipolytrope]] with <math>(n_c, n_e) = (5, 1)</math>.

====Separate Contributions to Gravitational Potential Energy====

Given the separate (equilibrium) density and <math>~M_r</math> profiles of the core and the envelope while sticking to the notation used in our [[SSCpt1/Virial#Normalizations|introductory discussion]], we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~W_\mathrm{grav}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- E_\mathrm{norm} \int_0^{r_i^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- E_\mathrm{norm} \cdot \chi^{-1} \biggl( \frac{\nu}{q^3} \biggr)
\int_0^{x_i} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{core}  \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{core}  dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~W_\mathrm{grav}\biggr|_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- E_\mathrm{norm} \int_{r_i^*}^{\chi = R_\mathrm{edge}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- E_\mathrm{norm} \cdot \chi^{-1} \biggl( \frac{1-\nu}{1-q^3} \biggr)
\int_{x_i}^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{env}  \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{env}  dx 
</math>
  </td>
</tr>

</table>
</div>


'''EXAMPLES'''
* [[SSC/Structure/BiPolytropes/FreeEnergy00#Gravitational_Potential_Energy|Gravitational Potential Energy of bipolytrope]] with <math>(n_c, n_e) = (0, 0)</math>.
* [[SSC/Structure/BiPolytropes/FreeEnergy51#Gravitational_Potential_Energy|Gravitational Potential Energy of bipolytrope]] with <math>(n_c, n_e) = (5, 1)</math>.

====Separate Thermodynamic Energy Reservoirs====
In our [[SSCpt1/Virial#Looking_Ahead_to_Bipolytropes|introductory discussion of the free energy function for spherically symmetric configurations]], we developed expressions that define the separate contributions to the thermodynamic energy reservoir that pertain to the core and envelope of bipolytropic configurations.  In that discussion we pointed out that, in general 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 is therefore 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>

<div align="center">
<table border="1" cellpadding="5" width="75%">
<tr><td align="left">
<div align="center">
<font color="red" size="+1"> KEY QUESTION:  How should piecewise pressure be normalized? </font>
</div>
Here we will operate under two constraints when relating the central pressure, <math>~P_0</math>, to the ''core'' pressure at the interface, <math>~P_{ic}</math>.  First, the pressure in the core will drop monotonically from its central value according to the following very general prescription,
<div align="center">
<math>~P_\mathrm{core}(x) = P_0 [1 - p_c(x)] \, .</math>
</div>
Second, it is the pressure ''at the interface'', <math>~P_{ic} = P_\mathrm{core}(q)</math>, that will be determined by the virial equilibrium condition.  There are two possible ways to determine the central pressure from knowledge of <math>~P_{ic}</math>, but which is the physically correct method to embrace?

<font color="darkgreen">'''Case A:'''</font>  Addition &hellip;
<div align="center">
<math>~P_0 = P_{ic} + \Delta P \, ,</math>&nbsp;&nbsp;&nbsp;&nbsp; where &nbsp;&nbsp;&nbsp;&nbsp;<math>\Delta P \equiv P_0 p_c(q) \, ,</math>
</div>
in which case, we should write that,
<div align="center">
<math>~P_\mathrm{core}(x) = ( P_{ic} +  \Delta P ) - P_0 p_c(x)  \, .</math>
</div>


<font color="darkgreen">'''Case B:'''</font>  Multiplication &hellip;
<div align="center">
<math>~P_{ic} = P_0 [1-p_c(q)] ~~~~\Rightarrow~~~~~ P_0 = \frac{P_{ic}}{[1-p_c(q)]} \, ,</math>
</div>
in which case, we should write that,
<div align="center">
<math>~P_\mathrm{core}(x) = P_{ic} \biggl[ \frac{ 1-p_c(x) }{ 1-p_c(q) } \biggr]  \, .</math>
</div>

Prior to August 2014, we have been naively implementing "Case A,"  effectively assuming that the quantity, <math>~\Delta P</math> (as well as <math>~P_{ic}</math>), is held fixed as we search for the equilibrium value of <math>~P_0</math>.  See, for example, the comment dated <font color="red">12 February 2014</font> in connection with my [[SSC/VirialStability#Discussion_with_Kundan_Kadam|discussions with Kundan Kadam]], or even [[SSC/BipolytropeGeneralization#DerivationTable|the "new derivation" summarized in the table below]], where we have set,
<div align="center">
<math>~\Delta P = \Pi q^2 \, .</math>
</div>

But we now suspect that "Case B" is the proper approach to embrace because, once the parameter <math>~q</math> has been specified, it allows for the function, <math>~P_\mathrm{core}(x)</math>, to scale with the system size in exactly the same way as the interface pressure scales with size.

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


With these generic expressions for the pressure profile in hand, the separate components of the thermodynamic energy reservoir derived in our [[SSCpt1/Virial#Looking_Ahead_to_Bipolytropes|introductory discussion]] are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}</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">
<math>~\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
  </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>


'''EXAMPLES'''
* [[SSC/Structure/BiPolytropes/FreeEnergy00#Thermodynamic_Energy_Reservoir|Thermodynamic Energy Reservoir of bipolytrope]] with <math>(n_c, n_e) = (0, 0)</math>.
* [[SSC/Structure/BiPolytropes/FreeEnergy51#Thermodynamic_Energy_Reservoir|Thermodynamic Energy Reservoir of bipolytrope]] with <math>(n_c, n_e) = (5, 1)</math>.

==Generalized Free-Energy Expression==
Bringing all of these expressions together, the normalized free-energy function for bipolytropes is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{core}
+ \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core}
+ \biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env}
+ \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-3\mathcal{A} \chi^{-1} - \frac{\mathcal{B}_\mathrm{core}}{(1-\gamma_c)} \chi^{3-3\gamma_c} - \frac{\mathcal{B}_\mathrm{env}}{(1-\gamma_e)} \chi^{3-3\gamma_e} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\nu}{q^3} \biggr) \int_0^{q} \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{core}  \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{core}  x dx 
~+
\biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{q}^{1} \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{env}  \biggl[ \frac{\rho(x)}{\bar\rho} \biggr]_\mathrm{env}  x dx \, ,
</math>
  </td>
</tr>

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

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


<table border="1" cellpadding="5" align="center" width="60%">
<tr>
  <td align="left">

<font color="darkblue">'''ASIDE''':</font>  In some of our [[SSC/BipolytropeGeneralization#Old_Stuff|older derivations]], the function names <math>s_\mathrm{core}</math> and <math>s_\mathrm{env}</math> were introduced as a shorthand notation.  When referenced to our present, broad treatment of the free-energy function for bipolytropes, we note that,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>q^3 s_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^q  3\biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(1-q^3) s_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_q^1  3\biggl[1 - p_e(x) \biggr]  x^2 dx \, .</math>
  </td>
</tr>
</table>
</div>

We also previously adopted the coefficient notation <math>\mathcal{B}</math> (with no subscript) for what is now called <math>\mathcal{B}_\mathrm{core}</math>, and we used <math>\mathcal{C}</math> for what is now labeled <math>\mathcal{B}_\mathrm{env}</math>.

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

==Extrema and Virial Equilibrium==
Extrema arise in the free-energy function wherever,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial \mathfrak{G}^*}{\partial \chi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
</div>
that is, when,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~3\mathcal{A}\chi^{-2} - 3\mathcal{B}_\mathrm{core} \chi^{2-3\gamma_c} - 3\mathcal{B}_\mathrm{env} \chi^{2-3\gamma_e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
Values of the dimensionless variable, <math>~\chi</math>, that provide solutions to this algebraic equation identify the size of equilibrium configurations and will henceforth be labeled with the "eq" subscript, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\chi ~~~ \rightarrow ~~~ \chi_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, .</math>
  </td>
</tr>
</table>
</div>

===Virial Theorem===
We can rewrite the equilibrium condition as,
<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>~- \chi_\mathrm{eq}^{-1} 
\biggl[ -3\mathcal{A}\chi_\mathrm{eq}^{-1} - 
3\mathcal{B}_\mathrm{core} \chi_\mathrm{eq}^{3-3\gamma_c} - 3\mathcal{B}_\mathrm{env} \chi_\mathrm{eq}^{3-3\gamma_e} \biggr]_\mathrm{eq}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\chi_\mathrm{eq}^{-1} }{E_\mathrm{norm}}
\biggl[ ( W_\mathrm{grav})_\mathrm{core} + ( W_\mathrm{grav} )_\mathrm{env} 
+ 3 (\gamma_c-1) ( \mathfrak{S}_A )_\mathrm{core}
+ 3 (\gamma_e-1) ( \mathfrak{S}_A)_\mathrm{env}
\biggr]_\mathrm{eq} \, .
</math>
  </td>
</tr>
</table>
</div>
Drawing from our [[VE#Relationship_to_the_System.27s_Internal_Energy|introductory discussion of the reservoir of thermodynamic energy]], we note that, for adiabatic systems, <math>~\mathfrak{G}_A</math> is equivalent to the internal energy of the system and therefore its relationship to the thermal energy, <math>~S_\mathrm{therm}</math>, is,
<div align="center">
<math>\mathfrak{G}_A = \frac{2}{3(\gamma-1)} S_\mathrm{therm} \, .</math>
</div>
(This applies separately for the core and the envelope.)  We therefore recognize that our derived expression for equilibrium systems is none other than the virial theorem applied to bipolytropic configurations, specifically, in equilibrium,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
( W_\mathrm{grav})_\mathrm{core} + ( W_\mathrm{grav} )_\mathrm{env} + 2 [
( S_\mathrm{therm} )_\mathrm{core} + ( S_\mathrm{therm} )_\mathrm{env} ] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>
</div>

<div id="ExampleVirialTheorem">
<table border="1" align="center" width="90%" cellpadding="20">
<tr><th align="center">
Example Bipolytrope Virial Theorem
</th></tr>
<tr><td align="left">
Virial theorem for <math>(n_c, n_e) = (0, 0) </math> bipolytrope:


In equilibrium, we will demand that <math>P_{ie} = P_{ic}</math> and we will set <math>\chi \rightarrow \chi_\mathrm{eq}</math>.  Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{S_\mathrm{therm}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr)_\mathrm{core} + \biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr)_\mathrm{env}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3(\gamma_c-1)}{2}\biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{core} 
+ \frac{3(\gamma_e-1)}{2} \biggl( \frac{\mathfrak{S}_A}{E_\mathrm{norm}} \biggr)_\mathrm{env}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\pi \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3-3\gamma_c} 
\biggl\{ \biggl( \frac{P_0}{P_{ic}} \biggr) \biggl[ q^3 - \biggl( \frac{3b_\xi}{5} \biggr) q^5 \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+~
2\pi \biggl[ \frac{P_{ic} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3-3\gamma_e} 
\biggl\{ (1-q^3) +  b_\xi \biggl(\frac{P_0}{P_{ic} } \biggr) \biggl[\frac{2}{5} q^5 \mathfrak{F}
\biggr]  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\pi \biggl[ \frac{P_{0}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3} 
\biggl\{ \biggl[ q^3 - \biggl( \frac{3b_\xi}{5} \biggr) q^5 \biggr] + \frac{P_{ic} }{P_0} (1-q^3) +\frac{2}{5} b_\xi q^5 \mathfrak{F} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\pi  \biggl[ \frac{P_{0}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{3} 
\biggl[ q^3 - \biggl( \frac{3b_\xi}{5} \biggr) q^5  + (1- b_\xi q^2) (1-q^3) +\frac{2}{5} b_\xi q^5 \mathfrak{F} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\pi  \biggl[ \frac{P_{0} R_\mathrm{edge}^4}{P_\mathrm{norm} R_\mathrm{norm}^4} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{-1}
\biggl[ 1 - \biggl( \frac{3b_\xi}{5} \biggr) q^5  - b_\xi (q^2-q^5) +\frac{2}{5} b_\xi q^5 \mathfrak{F} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\pi  \biggl[ \frac{P_{0} R_\mathrm{edge}^4}{GM_\mathrm{tot}^2} \biggr]_\mathrm{eq} \chi_\mathrm{eq}^{-1}
\biggl[ 1 + b_\xi \biggl(\frac{2}{5} q^5 \mathfrak{F} - \frac{3}{5}q^5 + q^5 - q^2 \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\pi  \biggl[ \frac{1}{b_\xi} \biggl( \frac{3}{2^3 \pi} \biggr) \frac{\nu^2}{q^6} \biggr] \chi_\mathrm{eq}^{-1}
\biggl\{ 1 + b_\xi \biggl[ \frac{2}{5} q^5 (\mathfrak{F} + 1) - q^2 \biggr] \biggr\}
</math>
  </td>
</tr>
</table>
</div>

The virial theorem states that, in equilibrium,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{2S_\mathrm{therm}}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \, ,</math>
  </td>
</tr>
</table>
</div>
which, in turn, implies,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl[ \frac{1}{b_\xi} \biggl( \frac{3}{2} \biggr) \frac{\nu^2}{q^6} \biggr] \chi_\mathrm{eq}^{-1}
\biggl\{ 1 + b_\xi \biggl[ \frac{2}{5} q^5 (\mathfrak{F} + 1) - q^2 \biggr] \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3}{5}\biggl(\frac{\nu^2}{q} \biggr) f(\nu,q) \chi_\mathrm{eq}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~
\frac{1}{b_\xi} + \biggl[ \frac{2}{5} q^5 (\mathfrak{F} + 1) - q^2 \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{2}{5}\cdot q^5 f
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~
\frac{1}{b_\xi}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
q^2 + \frac{2}{5}q^5 (f - 1 - \mathfrak{F} ) \, . 
</math>
  </td>
</tr>
</table>
</div>
Now, a bit of algebra shows that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{2}{5}q^5 (f - 1 - \mathfrak{F} )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[
2q^2(1-q) + \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-3q^2 + 2q^3) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Hence, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{b_\xi} =
\biggl( \frac{2^3 \pi}{3} \biggr) \frac{P_0 R_\mathrm{edge}^4}{G M_\mathrm{tot}^2 } \biggl( \frac{q^3}{\nu}\biggr)^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~q^2 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[
2q^2(1-q) + \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-3q^2 + 2q^3) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~
\frac{P_0 R_\mathrm{edge}^4}{G M_\mathrm{tot}^2 } 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{3}{2^3 \pi} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2 \biggl\{ q^2 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl[
2q^2(1-q) + \biggl( \frac{\rho_e}{\rho_c} \biggr) (1-3q^2 + 2q^3) \biggr] \biggr\} \, .</math>
  </td>
</tr>
</table>
</div>

This exactly matches the equilibrium relation that was derived from our [[SSC/Structure/BiPolytropes/Analytic00#CentralPressure|detailed force-balance analysis of]] <math>(n_c, n_e) = (0, 0)</math> bipolytropes.

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

===More Utilitarian Form===
Multiplying the equilibrium condition through by <math>~(\chi^2/3)</math> &#8212; and appending the "eq" suffix to <math>~\chi</math>, throughout &#8212; gives,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{B}_\mathrm{core} \chi_\mathrm{eq}^{4-3\gamma_c} + \mathcal{B}_\mathrm{env} \chi_\mathrm{eq}^{4-3\gamma_e} \, .</math>
  </td>
</tr>
</table>
</div>
Inserting the generic definitions of the coefficients <math>~\mathcal{B}_\mathrm{core}</math> and <math>~\mathcal{B}_\mathrm{env}</math> &#8212; expressed in [[#Generalized_Free-Energy_Expression|shorthand notation as referenced above]] &#8212; and demanding that the interface pressures be identical, gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ \frac{4\pi}{3} \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} 
q^3 s_\mathrm{core} \biggr\} \chi_\mathrm{eq}^{4-3\gamma_c} 
+ \biggl\{ \frac{4\pi}{3} \biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} (1-q^3) s_\mathrm{env} \biggr\} \chi_\mathrm{eq}^{4-3\gamma_e} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \biggl[ \frac{P_i \chi^4}{P_\mathrm{norm}} \biggr]_\mathrm{eq} [
q^3 s_\mathrm{core} + (1-q^3) s_\mathrm{env} ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4\pi}{3} \biggl[ \frac{P_i R_\mathrm{edge}^4}{GM_\mathrm{tot}^2} \biggr]_\mathrm{eq} [
q^3 s_\mathrm{core} + (1-q^3) s_\mathrm{env} ] \, .
</math>
  </td>
</tr>
</table>
</div>


'''EXAMPLES'''
* [[SSC/Structure/BiPolytropes/FreeEnergy00#Virial_Theorem|Virial Equilibrium Bipolytrope Configurations]] with <math>~(n_c, n_e) = (0, 0)</math>.
* [[SSC/Structure/BiPolytropes/FreeEnergy51#Virial_Theorem|Virial Equilibrium Bipolytrope Configurations]] with <math>~(n_c, n_e) = (5, 1)</math>.

=Related Discussions=
* [[SSC/Structure/BiPolytropes/FreeEnergy00|Free-energy determination of equilibrium configurations for BiPolytropes]] with <math>n_c = 0</math> and <math>n_e=0</math>.
* [[SSC/Structure/BiPolytropes/FreeEnergy51#Free_Energy_of_BiPolytrope_with|Free-energy determination of equilibrium configurations for BiPolytropes]] with <math>~n_c = 5</math> and <math>~n_e=1</math>.
* [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|Analytic solution of Detailed-Force-Balance BiPolytrope]] with <math>n_c = 0</math> and <math>n_e=0</math>.
* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|Analytic solution of Detailed-Force-Balance BiPolytrope]] with <math>n_c = 5</math> and <math>~n_e=1</math>.
* [[SSC/BipolytropeGeneralization|Old ''Bipolytrope Generalization'' derivations]].



=See Also=
<ul>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
</ul>

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