Editing SSC/Structure/BiPolytropes/FreeEnergy51/Pt2

__FORCETOC__
=Free Energy of BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)=
<table border="1" align="center" width="100%" colspan="8">
  <td align="center" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiiPolytropes/FreeEnergy51|Part I:&nbsp; Mass Profile]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"3 width="33%"><br />[[SSC/Structure/BiPolytropes/FreeEnergy51/Pt2|Part II:&nbsp; Gravitational Potential Energy]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/FreeEnergy51/Pt3|Part III:&nbsp; Thermal Energy Reservoir]]
&nbsp;
  </td>
</tr>
</table>

==Gravitational Potential Energy==

===The Core===

Borrowing from our derivation, above, of the mass distribution in this type of bipolytrope, the expression for the gravitational potential energy in the core that has been [[SSC/BipolytropeGeneralizationVersion2#Separate_Contributions_to_Gravitational_Potential_Energy|outlined in our accompanying overview]] may be written as,
<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} \cdot \chi^{-1} \biggl[ \frac{\nu}{q^3} \biggl(\frac{\rho_0}{\bar\rho} \biggr)_\mathrm{core} \biggr]_\mathrm{eq}
\int_0^{q} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{core}  \biggl[ \frac{\rho(x)}{\rho_0} \biggr]_\mathrm{core}  dx 
</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} \biggl( 1 + a_\xi q^2 \biggr)^{3/2}  \biggr]_\mathrm{eq}
\int_0^{q} 3x \biggl\{ 
\nu \biggl( \frac{x^3}{q^3} \biggr) \biggl[ \frac{ 1 + a_\xi x^2 }{ 1 + a_\xi q^2 } \biggr]^{-3/2} 
\biggr\} 
\biggl( 1 + a_\xi x^2 \biggr)^{-5/2}  dx 
</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[ 3\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + a_\xi q^2 \biggr)^{3}  \biggr]_\mathrm{eq}
\int_0^{q} x^4 \biggl( 1 + a_\xi x^2 \biggr)^{-4}  dx 
</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[ 3\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + a_\xi q^2 \biggr)^{3}  \biggr]_\mathrm{eq}
\biggl\{
\frac{a_\xi^{1/2} q(3a_\xi^2 q^4 - 8a_\xi q^2 - 3) + 3(a_\xi q^2 +1)^3 \tan^{-1}(a_\xi^{1/2} q)}{48 a_\xi^{5/2}(a_\xi q^2 + 1)^3}
\biggr\} 
</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[\biggl(\frac{3}{2^4}\biggr) a_\xi^{-5/2}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + a_\xi q^2 \biggr)^{3}  \biggr]_\mathrm{eq}
\biggl[
a_\xi^{1/2} q(a_\xi^2 q^4 - \frac{8}{3}a_\xi q^2 - 1) (a_\xi q^2 +1)^{-3} + \tan^{-1}(a_\xi^{1/2} q)
\biggr] \, .
</math>
  </td>
</tr>

</table>
</div>

<table border="1" cellpadding="5" align="center" width="90%">
<tr><td align="left">

MORE USEFUL:

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

<tr>
  <td align="right">
<math>~- \chi \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
\biggl[\biggl(\frac{3}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5}\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3}  \biggr]_\mathrm{eq}
\biggl[
\ell_i (\ell_i^4 - \frac{8}{3}\ell_i^2 - 1) (\ell_i^2 +1)^{-3} + \tan^{-1}\ell_i
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>\frac{3}{5}
\biggl[\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3}  \biggr]_\mathrm{eq}
\biggl(\frac{5}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5} \biggl[
\ell_i (\ell_i^4 - \frac{8}{3}\ell_i^2 - 1) (\ell_i^2 +1)^{-3} + \tan^{-1}\ell_i
\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

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

But, also from our above discussion of the mass profile, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~a_\xi^{-5/2} \biggl( \frac{\nu}{q^3} \biggr)^2 (1 + a_\xi q^2)^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_\mathrm{eq} \biggl( \frac{2^3 \cdot 3^6}{\pi} \biggr)^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \frac{\chi_\mathrm{eq}}{\chi} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2}  
\biggl[
a_\xi^{1/2} q(a_\xi^2 q^4 - \frac{8}{3}a_\xi q^2 - 1) (a_\xi q^2 +1)^{-3} + \tan^{-1}(a_\xi^{1/2} q)
\biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
After making the substitution, <math>~(a_\xi^{1/2} q) \rightarrow x_i</math>, this expression agrees with a result for the dimensionless energy, <math>~W^*_\mathrm{core}</math>, [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|derived by Tohline in the context of detailed force-balanced bipolytropes]].

===The Envelope===

Again, borrowing from our derivation, above, of the mass distribution in this type of bipolytrope, the expression for the gravitational potential energy in the envelope that has been [[SSC/BipolytropeGeneralizationVersion2#Separate_Contributions_to_Gravitational_Potential_Energy|outlined in our accompanying overview]] may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \chi^{-1} \biggl[ \biggl( \frac{1-\nu}{1-q^3} \biggr) \biggl( \frac{\rho_0}{\bar\rho}\biggr)_\mathrm{env} \biggr]_\mathrm{eq}
\int_{q}^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]_\mathrm{env}  \biggl[ \frac{\rho(x)}{\rho_0} \biggr]_\mathrm{env}  dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \chi^{-1} \biggl[ \frac{\nu}{q^3 \theta_i^3} \biggr]
\int_{q}^{1} 3x \biggl\{ \frac{\nu}{C_1} \biggl[ \sin(B-b_\eta x) + xb_\eta \cos(B - b_\eta x) \biggr]\biggr\}  
\biggl\{ A \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i^5 \biggl[ \frac{\sin(b_\eta x - B)}{b_\eta x} \biggr] \biggr\}  dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \chi^{-1} \biggl( \frac{\mu_e}{\mu_c} \biggr)  \biggl[ \frac{3\nu^2 A \theta_i^2}{b_\eta q^3 } \biggr] \frac{1}{C_1} 
\int_{q}^{1} [ xb_\eta \cos(b_\eta x-B) - \sin(b_\eta x- B)  ]\sin(b_\eta x - B)   dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \chi^{-1} \biggl( \frac{\mu_e}{\mu_c} \biggr)  \biggl[ \frac{3\nu^2 A \theta_i^2}{b_\eta q^3 } \biggr] 
\biggl[ - 3A\theta_i^2 \biggl( \frac{\mu_e}{\mu_c} \biggr) (b_\eta q)^{-3} \biggr]
\int_{q}^{1} [ xb_\eta \cos(b_\eta x-B) - \sin(b_\eta x- B)  ]\sin(b_\eta x - B)   dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \chi^{-1} \biggl( \frac{\mu_e}{\mu_c} \biggr)^2  \biggl[ \frac{3^2 A^2 }{b_\eta^4 } \biggr] \biggl( \frac{\nu^2 \theta_i^4}{q^6} \biggr)
\int_{q}^{1} [\sin(b_\eta x- B) - xb_\eta \cos(b_\eta x-B) ]\sin(b_\eta x - B)   dx \, .
</math>
  </td>
</tr>

</table>
</div>
But we also know, [[SSC/Structure/BiiPolytropes/FreeEnergy51#EquilibriumRadius|from above]], that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_\mathrm{eq} = b_\eta^{-5} \biggl( \frac{3^4  \pi}{2^3} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^5 \frac{\nu^2 \theta_i^4}{q^6}</math>
  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp;<math>~~~~\Rightarrow ~~~~</math>&nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>
~\frac{\nu^2 \theta_i^4}{q^6} = \chi_\mathrm{eq} b_\eta^{5} \biggl( \frac{2^3}{3^4  \pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5}   \, .
</math>
  </td>
</tr>
</table>
</div>
So we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \frac{\chi_\mathrm{eq}}{\chi} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3}  \biggl( \frac{2^3}{\pi} \biggr)^{1/2} A^2 b_\eta
\int_{q}^{1} [\sin(b_\eta x- B) - xb_\eta \cos(b_\eta x-B) ]\sin(b_\eta x - B)   dx \, .
</math>
  </td>
</tr>

</table>
</div>


The integral can be broken into two separate parts:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\int_{q}^{1} \sin^2(b_\eta x - B) dx
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{4b_\eta} \biggl\{2 b_\eta x - \sin[2(b_\eta x-B)] \biggr\}_q^1 \, ,
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
- \int_{q}^{1} xb_\eta \cos(b_\eta x-B)\sin(b_\eta x - B) dx
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{8b_\eta} \biggl\{2b_\eta x \cos[2(b_\eta x - B)]  - \sin[2(b_\eta x-B)]  \biggr\}_q^1 \, .</math>
  </td>
</tr>
</table>
</div>
(Note:  We have dropped integration constants that might result from carrying out an indefinite integral because such constants would disappear upon application of our specified limits of integration.)   When added together, they give,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\int_q^1 \ldots ~dx</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{8b_\eta} \biggl\{2b_\eta x \cos[2(b_\eta x - B)] - 3\sin[2(b_\eta x-B)]  + 4 b_\eta x \biggr\}_q^1
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{8b_\eta} \biggl\{2b_\eta x \biggl[ 1 - 2\sin^2(b_\eta x - B) \biggr] - 3\sin[2(b_\eta x-B)] +  4 b_\eta x \biggr\}_q^1
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{1}{8b_\eta} \biggl[6 b_\eta x  - 3\sin[2(b_\eta x-B)] - 4b_\eta x \sin^2(b_\eta x - B)  \biggr]_q^1 \, .
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\biggl( \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>
- \frac{\chi_\mathrm{eq}}{\chi} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3}  \biggl( \frac{1}{2^3\pi} \biggr)^{1/2} A^2 
\biggl[6 b_\eta x  - 3\sin[2(b_\eta x-B)] - 4b_\eta x \sin^2(b_\eta x - B)  \biggr]_q^1 \, .
</math>
  </td>
</tr>

</table>
</div>

This expression matches in detail the expression for the gravitational potential energy of the envelope derived in the context of our [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|derivation of detailed force-balanced models of this bipolytrope]].

=See Also=

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