Editing SSC/Structure/BiPolytropes/Analytic51/Pt4 (section)

===Expression for Free Energy===
In order to construct the free energy function, we need mathematical expressions for the gravitational potential energy, <math>W</math>, and for the thermal energy content, <math>S</math>, of the models; and it will be natural to break both energy expressions into separate components derived for the <math>n_c=5</math> core and for the <math>n_e = 1</math> envelope.  Consistent with the above equilibrium model derivations, we will work with dimensionless variables.  Specifically, we define,

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>~W^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{W}{[K_c^5/G^3]^{1/2}}</math>
  </td>

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

  <td align="right">
<math>~S^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{S}{[K_c^5/G^3]^{1/2}} \, .</math>
  </td>
</tr>
</table>
</div>

Drawing on the various functional expressions that are provided in the above derivations, including the [[SSC/Structure/BiPolytropes/Analytic51/Pt2#Parameter_Values|Table of Parameters]], integrals over the material in the core give us,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~S^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{3}{2} \int_0^{r_i} \biggl(\frac{P^*}{\rho^*}\biggr)_\mathrm{core} (4\pi \rho^*)_\mathrm{core} (r^*)^2 dr^*</math>
  </td>

  <td align="center" rowspan="5" width="8%">
&nbsp;
  </td>
  <td align="center" rowspan="5">
[[File:Mathematica01.png|275px|center|Mathematica Integral]]
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 6\pi \biggl( \frac{3}{2\pi} \biggr)^{3/2} \int_0^{\xi_i} \biggl(1+\frac{1}{3}\xi^2\biggr)^{-3} \xi^2 d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 6\pi \biggl( \frac{3^2}{2\pi} \biggr)^{3/2} \int_0^{x_i} \biggl(1+x^2\biggr)^{-3} x^2 dx</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \biggl( \frac{3^8}{2^7\pi} \biggr)^{1/2} \biggl[ \frac{x_i}{(1+x_i^2)} - \frac{2x_i}{(1+x_i^2)^2} + \tan^{-1}(x_i) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl( \frac{3^8}{2^5\pi} \biggr)^{1/2} \biggl[ x_i (x_i^4 - 1 )(1+x_i^2)^{-3}   + \tan^{-1}(x_i) \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where, in order to streamline the integral for Mathematica, we have used the substitution, <math>~x \equiv \xi/\sqrt{3}</math>; and,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>~W^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \int_0^{r_i} (4\pi M_r^* \rho^*)_\mathrm{core} (r^*) dr^*</math>
  </td>

  <td align="center" rowspan="5" width="4%">
&nbsp;
  </td>
  <td align="center" rowspan="5">
[[File:Mathematica02.png|275px|center|Mathematica Integral]]
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - 4\pi \int_0^{\xi_i} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \biggl( \frac{3}{2\pi} \biggr) \xi d\xi</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{2^3\cdot 3^8}{\pi } \biggr)^{1/2}\int_0^{x_i}  \biggl( 1 + x^2 \biggr)^{-4} x^4 dx</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{2^3\cdot 3^8}{\pi } \biggr)^{1/2}
\biggl[ 3\tan^{-1}(x_i) + \frac{x_i(3x_i^4 -8x_i^2 -3)}{(1+x_i^2)^3} \biggr] \biggl( \frac{1}{2^4 \cdot 3} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2}
\biggl[ x_i \biggl(x_i^4 - \frac{8}{3} x_i^2 -1 \biggr) (1 + x_i^2)^{-3}  + \tan^{-1}(x_i) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
(Apology:  The parameter <math>x_i</math> introduced here is identical to the parameter <math>~\ell_i</math> that was introduced earlier in the context of [[SSC/Structure/BiPolytropes/Analytic51/Pt3#Limiting_Mass|our discussion of the "Limiting Mass"]] of these models.  Sorry for the unnecessary duplication of parameters and possible confusion!)  

<div id="Buchdahl1978">
<table border="1" align="center" width="85%" cellpadding="8">
<tr><td align="left">
While our aim, here, has been to determine an expression for the gravitational potential energy of a ''truncated'' <math>n = 5</math> polytropic sphere, our derived expression can also give the gravitational potential of an ''isolated'' <math>n = 5</math> polytrope by evaluating the expression in the limit <math>x_i \rightarrow \infty</math>.  In this limit, the first term inside the square brackets goes to zero, while the second term,
<div align="center">
<math>\lim_{x_i \to \infty}\tan^{-1}(x_i) = \frac{\pi}{2} \, .</math>
</div>

We see, therefore, that,
<div align="center">
<math>
W^* \biggr|_\mathrm{tot} =
\lim_{x_i \to \infty}W^* = - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2}\frac{\pi}{2} 
= - \biggl( \frac{3^8 \pi}{2^7} \biggr)^{1/2} \, .
</math>
</div>

Taking into account our adopted energy normalization, this can be rewritten with the dimensions of energy as,

<div align="center" id="twoSplusWcore">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>W_\mathrm{grav} \biggr|_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \biggl( \frac{3^8 \pi}{2^7} \biggr)^{1/2} \biggl( \frac{K_c^5}{G^3} \biggr)^{1/2} 
= - \biggl( \frac{3^8 \pi}{2^7} \biggr)^{1/2} \biggl[ \frac{\pi}{2^3\cdot 3^7} \biggr]^{1/2} \frac{GM^2 }{a_5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \biggl( \frac{3 \pi^2}{2^{10}} \biggr)^{1/2} \frac{GM^2}{a_5} \, ,
</math>
  </td>
</tr>
</table>
</div>

where we have elected to write the total gravitational potential energy in terms of the natural scale length for <math>~n = 5</math> polytropes, which, [[SSC/Structure/Polytropes/Analytic#Primary_E-Type_Solution_2|as documented elsewhere]], is, 

<div align="center" id="twoSplusWcore">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>a_{5}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ \frac{3K}{2\pi G} \biggr]^{1/2} \rho_c^{-2/5} 
= \biggl[ \frac{3K}{2\pi G} \biggr]^{1/2} \biggl[ \frac{\pi M^2 G^3}{2\cdot 3^4} \biggr] K^{-3}
= GM^2 \biggl[ \frac{\pi G^3}{2^3\cdot 3^7 K^5} \biggr]^{1/2} \, .
</math>
  </td>
</tr>
</table>
</div>

As can be seen from the following, boxed-in equation excerpt, our derived expression for the total gravitational potential energy of an ''isolated'' <math>n=5</math> polytrope exactly matches the result derived by {{ Buchdahl78full }}.  The primary purpose of Buchdahl's ''short communication'' was to point out that, despite the fact that its radius extends to infinity, "the gravitational potential energy of [an ''isolated''] polytrope of index 5 is finite."

<div align="center">
<table border="1" align="center" cellpadding="8" width="75%">
<tr><td align="center">
Equation excerpt from p. 116 of<br />{{ Buchdahl78figure }}
<!--[http://adsabs.harvard.edu/abs/1978AuJPh..31..115B H. A. Buchdahl (1978, Astralian J. Phys., 31, 115)]-->
</td></tr>
<tr><td align="center">
<!-- [[File:Buchdahl1978.png|350px|center|Buchdahl (1978, Australian J. Phys., 31, 115)]] -->
<!-- [[Image:AAAwaiting01.png|350px|center|Buchdahl (1978, Australian J. Phys., 31, 115)]] -->
<math>\Omega = -(\pi\sqrt{3}/32)GM^2/\alpha \, .</math>
</td></tr>
<tr><td align="left">Note that a comparison between Buchdahl's derived expression and our expression in the limit <math>x_i \rightarrow \infty</math> requires the parameter substitutions,
<div align="center"><math>\Omega \rightarrow W_\mathrm{grav}|_\mathrm{tot}</math> &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>\alpha \rightarrow a_5</math></div>
</td></tr>
</table>
</div>

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

Notice that these two terms combine to give, for the core,

<div align="center" id="twoSplusWcore">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>\biggl( 2S + W \biggr)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{2 \cdot 3^6}{\pi } \biggr)^{1/2} \frac{x_i^3}{(1 + x_i^2)^3} 
= \biggl( \frac{2}{\pi } \biggr)^{1/2} 3^{3/2} \xi_i^3 \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-3}\, .</math>
  </td>
</tr>
</table>
</div>

Similarly, integrals over the material in the envelope give us,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>S^*_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{2} \int_{r_i}^R \biggl(\frac{P^*}{\rho^*}\biggr)_\mathrm{env} (4\pi \rho^*)_\mathrm{env} (r^*)^2 dr^*</math>
  </td>

  <td align="center" rowspan="5" width="8%">
&nbsp;
  </td>
  <td align="center" rowspan="5">
[[File:Mathematica03.png|300px|center|Mathematica Integral]]
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>6\pi \int_{\eta_i}^{\eta_s} [\theta^{6}_i \phi^{2}] \biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2} \biggr]^3 \eta^2 d\eta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3^2}{2\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2 \int_{\eta_i}^{\eta_s} [\sin(\eta - B)]^2 d\eta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3^2}{2^5\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2 
\biggl\{  2(\eta - B) - \sin[2(\eta-B)] \biggr\}_{\eta_i}^{\eta_s} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{1}{2^5\pi} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2 
\biggl\{  6(\eta - B) - 3\sin[2(\eta-B)] \biggr\}_{\eta_i}^{\eta_s} \, ; </math>
  </td>
</tr>
</table>
</div>

and,

<div align="center">
<table border="0" cellpadding="4">
<tr>
  <td align="right">
<math>W^*_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \int_{r_i}^{R} (4\pi M_r^* \rho^*)_\mathrm{env} (r^*) dr^*</math>
  </td>

  <td align="center" rowspan="6" width="4%">
&nbsp;
  </td>
  <td align="center" rowspan="6">
[[File:Mathematica04.png|300px|center|Mathematica Integral]]
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 4\pi \int_{\eta_i}^{\eta_s} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{5}_i \phi
\biggl[ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2} \biggr]^2 \eta d\eta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\biggl( \frac{2^3}{\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} 
\int_{\eta_i}^{\eta_s}  \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) \phi  \eta d\eta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\biggl( \frac{2^3}{\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2
\int_{\eta_i}^{\eta_s}  [ \sin(\eta-B) - \eta\cos(\eta-B) ] \sin(\eta - B)  d\eta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\biggl( \frac{1}{2^3\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2
\biggl\{ - 3\sin[2(\eta - B)] +2\eta \cos[2(\eta - B)] + 4(\eta - B) + 2B \biggr\}_{\eta_i}^{\eta_s}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\biggl( \frac{1}{2^3\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2
\biggl\{6(\eta-B) - 3\sin[2(\eta - B)] -4\eta\sin^2(\eta-B) + 4B \biggr\}_{\eta_i}^{\eta_s} \, .
</math>
  </td>
</tr>
</table>
</div>

In this case, the two terms combine to give, for the envelope,

<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>\biggl( 2S + W \biggr)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{1}{2^3\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2
\biggl[4\eta\sin^2(\eta-B) + 4B \biggr]_{\eta_i}^{\eta_s}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{2}{\pi} \biggr)^{1/2}  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} A^2
\biggl[\eta_s\sin^2(\eta_s-B) - \eta_i\sin^2(\eta_i-B) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>