Editing SSC/VirialStability (section)

=BiPolytrope Structural Relations=
[Following a discussion that Tohline had with Kundan Kadam on <font color="red">3 July 2013</font>, we have decided to carry out a virial equilibrium and stability analysis of nonrotating bipolytropes.]  

We will adopt the following approach:
* Properties of the core <math>\cdots</math> 
** Uniform density, <math>\rho_c</math>;
** Polytropic constant, <math>K_c</math>, and polytropic index, <math>n_c</math>;
** Surface of the core at <math>r_i</math>;
* Properties of the envelope <math>\cdots</math> 
** Uniform density, <math>\rho_e</math>;
** Polytropic constant, <math>K_e</math>, and polytropic index, <math>n_e</math>;
** Base of the core at <math>r_i</math> and surface at <math>R</math>.

Use the dimensionless radius,
<div align="center">
<math>\xi \equiv \frac{r}{r_i}</math>.
</div>
Then, <math>\xi_i = 1</math> and <math>\xi_s \equiv R/r_i</math>.

==Expressions for Mass==
Inside the core, the expression for the mass interior to any radius, <math>0 \le \xi \le 1</math>, is,
<div align="center">
<math>M_\xi = \frac{4\pi}{3} \rho_c r_i^3 \xi^3</math> .
</div>
The expression for the mass interior to any position within the envelope, <math>1 \le \xi \le \xi_s</math>, is,
<div align="center">
<math>M_\xi = \frac{4\pi}{3} r_i^3 \biggl[\rho_c  + \rho_e(\xi^3 - 1) \biggr]</math> .
</div>
Hence, the mass of the core, the mass of the envelope, and the total mass are, respectively,
<div align="center">
<math>
M_\mathrm{core} = \frac{4\pi}{3} \rho_c r_i^3
</math> ;

<math>
M_\mathrm{env} = \frac{4\pi}{3} r_i^3 \biggl[\rho_e (\xi_s^3 - 1) \biggr] 
</math> ;

<math>
M_\mathrm{tot} = \frac{4\pi}{3} r_i^3 \biggl[\rho_c  + \rho_e(\xi_s^3 - 1) \biggr] 
</math> .
</div>

Following the work of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)] &#8212; see [[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|our accompanying discussion]]  &#8212; we are seeking equilibrium configurations in the <math>\nu - q</math> plane where,
<table align="center" border="0" cellpadding="10">
<tr>
  <td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math> &nbsp; &nbsp; and &nbsp; &nbsp;
<math>q \equiv \frac{r_i}{R} = \frac{1}{\xi_s}</math>.
  </td>
</tr>
</table>
So we can combine the above expressions to obtain,
<div align="center">
<math>\frac{\rho_e}{\rho_c} =  \frac{M_\mathrm{env}}{M_\mathrm{core}} (\xi_s^3 - 1)^{-1} 
= \biggl[ \frac{1-\nu}{\nu}\biggr] (\xi_s^3 - 1)^{-1}
= \frac{q^3}{\nu}\biggl( \frac{1 - \nu}{1- q^3} \biggr) \, ,
</math>
</div>
or,
<div align="center">
<math>\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^3} - 1\biggr) \biggr]^{-1}</math> .
</div>

<table align="center" border="3" cellpadding="10" width="65%">
<tr>
  <td align="left">
It is worth noting that exactly the same result arises from an examination of the analytically definable, structural properties of [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|bipolytropes having <math>n_c = 5</math> and <math>n_e = 1</math>]].  That is, the ratio of the average density in the envelope to the average density in the core is,
<div align="center">
<math>
\frac{\bar{\rho}_e}{\bar{\rho}_c} = 
\frac{(M^*_\mathrm{tot} - M^*_\mathrm{core}) / [(R^*)^3 - (r^*_\mathrm{core})^3]}{M^*_\mathrm{core} / (r^*_\mathrm{core})^3 } =
\frac{q^3 (1- \nu)}{\nu (1-q^3)} \, .
</math>
</div>
This is, of course, at it should be.
  </td>
</tr>
</table>


It is worth noting that, because <math>\bar\rho \equiv 3M_\mathrm{tot}/(4\pi R^3) \,</math>, we can write,
<div align="center">
<math>\frac{\rho_c}{\bar\rho} = \frac{\nu}{q^3} \, ,</math> &nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp; 
<math>\frac{\rho_e}{\bar\rho} = \frac{1-\nu}{1-q^3} \, ,</math>
</div>
which is consistent with the above expression for the ratio, <math>\rho_e/\rho_c \, .</math>  The following figure shows how <math>\nu \,</math> varies with <math>q \,</math> for various choices of the mass density ratio, <math>\rho_e/\rho_c \,</math>.  It illustrates that, for a given core-to-total mass ratio, <math>\nu \,</math>, the relative location of the interface radius, <math>q \,</math>, can vary between zero and one, but each value of <math>q \,</math> reflects a different ratio of envelope-to-core mass density.

<div align="center">
<table border="1">
<tr>
  <td align="center">
[[File:NuVersusQ.png|600px|center|Nu versus Q]]
  </td>
</tr>
</table>
</div>

==Energy Expressions==

The gravitational potential energy of the bipolytropic configuration is obtained by integrating over the following differential energy contribution,
<div align="center">
<math>dW = - \biggl( \frac{GM_r}{r} \biggr) dm</math> .
</div>

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

<tr>
  <td align="right">
<math>W = W_\mathrm{core} + W_\mathrm{env}</math>
  </td>
  <td align="left">
<math>
= - G \biggl\{ \int_0^{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_c dr + \int^R_{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_e dr \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - G \biggl\{ \int_0^1 \biggl( \frac{4\pi }{3} \rho_c r_i^3 \xi^3 \biggr) 4\pi r_i^2 \rho_c \xi d\xi 
+ \int_1^{\xi_s} \frac{4\pi}{3} \rho_c r_i^3 \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] 4\pi r_i^2 \rho_e \xi d\xi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \int_0^1  \xi^4 d\xi 
+ \int_1^{\xi_s} \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] \biggl( \frac{\rho_e}{\rho_c} \biggr) \xi d\xi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \frac{1}{5} 
+ \biggl( \frac{\rho_e}{\rho_c} \biggr) \int_1^{\xi_s} \xi d\xi 
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \int_1^{\xi_s}  (\xi^3 - 1) \xi d\xi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - \biggl( \frac{GM^2_\mathrm{tot}}{R} \biggr) 3\nu^2 \xi_s \biggl\{ \frac{1}{5} 
+ \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \frac{1}{5}(\xi_s^5 - 1) - \frac{1}{2}(\xi_s^2-1) \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - \biggl( \frac{3GM^2_\mathrm{tot}}{5R_0} \biggr) \biggl( \frac{R}{R_0}\biggr)^{-1} \nu^2 \xi_s f(\nu,q) \, ,
</math>
  </td>
</tr>

</table>
where <math>R_0</math> is an, as yet unspecified, normalization radius, and
<div align="center">
<math>
f(\nu,q) \equiv 1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \, .
</math>
</div>

I like the form of this expression.  The leading term, which scales as <math>R^{-1}</math>, encapsulates the behavior of the gravitational potential energy for a given choice of the internal structure, namely, a given choice of <math>\xi_s</math>, <math>\nu</math>, and density ratio <math>(\rho_e/\rho_c)</math>.  Actually, only two internal structural parameters need to be specified &#8212; <math>\nu</math> and <math>\xi_s</math> (or, <math>q</math>).  From these two, the expression shown above allows the determination of <math>(\rho_e/\rho_c)</math>.

<table border="3" cellpadding="10" align="center">
<tr>
  <td align="left">

'''Sanity Check:'''  Uniform Density Configuration
<div align="center">
<math>
M_\mathrm{core} = \frac{4\pi}{3} r_i^3 \rho_c \, ;
</math>

<math>
M_\mathrm{env} = \frac{4\pi}{3} r_i^3 \rho_e (\xi_s^3 -1) \, ;
</math>

<math>
M_\mathrm{tot} = \frac{4\pi}{3} r_i^3 \rho_c \biggl[ 1 + \frac{\rho_e}{\rho_c} (\xi_s^3 -1) \biggr] \, .
</math>
</div>
If <math>\rho_e/\rho_c = 1</math>, then,
<div align="center">
<math>
\frac{M_\mathrm{env}}{M_\mathrm{core}} = \biggl(\frac{1}{\nu} - 1\biggr) = (\xi_s^3 -1) = \biggl(\frac{1}{q^3}-1\biggr) 
~~~~\Rightarrow ~~~~ \nu = q^3 \, .
</math>
</div>

The gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>W\biggr|_{\rho_e/\rho_c = 1}</math>
  </td>
  <td align="left">
<math>
= - \biggl( \frac{GM^2_\mathrm{tot}}{R} \biggr) 3\nu^2 \xi_s \biggl\{ \frac{1}{5} 
+ \frac{1}{2}  (\xi_s^2 - 1)
+ \biggl[ \frac{1}{5}(\xi_s^5 - 1) - \frac{1}{2}(\xi_s^2-1) \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - \frac{3}{5} \biggl( \frac{GM^2_\mathrm{tot}}{R} \biggr)\nu^2 \xi_s \biggl\{ \frac{1}{5} 
+ \biggl[ \frac{1}{5}(\xi_s^5 - 1)  \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - \frac{3}{5} \biggl( \frac{GM^2_\mathrm{tot}}{R} \biggr)\nu^2 \xi_s^6 
= - \frac{3}{5} \biggl( \frac{GM^2_\mathrm{tot}}{R} \biggr) \biggl( \frac{\nu}{q^3}\biggr)^2 =  - \frac{3}{5} \biggl( \frac{GM^2_\mathrm{tot}}{R} \biggr)  \, .
</math>
  </td>
</tr>
</table>
</div>

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

Drawing on expressions developed in our [[VE#Virial_Equation|introductory discussion of the virial equation]], the internal energy of the bipolytropic configuration is,
<table border="0" align="center">
<tr>
  <td align="right">
<math>
U = U_\mathrm{core} + U_\mathrm{env}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ M_\mathrm{core} \biggl[ (1 - \delta_{\infty n_c}) n_c K_c \rho_c^{1/n_c}  + \delta_{\infty n_c} c_s^2 \ln(\rho_c/\rho_0) \biggr] 
+ n_e M_\mathrm{env} K_e \rho_e^{1/n_e}  \biggr\} \, ,
</math>
  </td>
</tr>
</table>
where <math>\rho_0</math> is an, as yet unspecified, normalization density, and we have allowed for either an isothermal (<math>\delta_{\infty n_c} = 1</math>) or an adiabatic (<math>\delta_{\infty n_c} = 0</math>) core.