Editing SSC/FreeEnergy/Powerpoint (section)

====Basic Properties====
For bipolytropes, in general, let <math>~n =n_c</math> and <math>~j = n_e</math>.  The statement of virial equilibrium is, then,

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<math>~\frac{ b}{n_c c}\cdot x^{(n_c-3)/n_c }_\mathrm{eq} -  \frac{a}{3c} + \frac{1}{n_e}\cdot  x^{(n_e-3)/n_e}_\mathrm{eq}  </math>
  </td>
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<math>~=</math>
  </td>
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<math>~  0 \, .</math>
  </td>
</tr>
</table>
</div>
And the critical equilibrium configuration has,
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<math>~[x_\mathrm{eq}^{(n_e-3)/n_e}]_\mathrm{crit} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a}{3^2c}\biggl[ \frac{n_e^2(n_c-3)}{n_c-n_e} \biggr] 
\, .
</math>
  </td>
</tr>
</table>
</div>

Here we choose to set <math>~n_c = 5</math> and <math>~n_e = 1</math>.  Hence, these two conditions become, respectively,

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<math>~\frac{ b}{5 c}\cdot x^{2/5 }_\mathrm{eq} + x^{-2}_\mathrm{eq}  </math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{a}{3c}  \, ,</math>
  </td>
</tr>
</table>
</div>
and,
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<math>~[x_\mathrm{eq}]_\mathrm{crit} </math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2\cdot 3^2c}{a} \biggr]^{1/2} 
\, .
</math>
  </td>
</tr>
</table>
</div>

<!-- EQUATIONS USEFUL in POWERPOINT PRESENTATION
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<math>~\theta_5(\xi) ~~~~\theta_1(\eta)</math>


<math>~\mu_e/\mu_c</math>


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<math>~\ell_i</math>
  </td>
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<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\xi_i}{\sqrt{3}}</math>
  </td>
</tr>
</table>
</div>

-->

Comparing the general free-energy expression at the beginning of this chapter with the free-energy expression provided via our [[SSC/FreeEnergy/PolytropesEmbedded#BiPolytropeFreeEnergy|accompanying summary discussion of five-one bipolytropes]], we find that,

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<math>~a</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(3\mathfrak{L}_i  +12\mathfrak{K}_i) x_\mathrm{eq} 
\biggl[  \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr]   </math>
  </td>
</tr>

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  <td align="right">
<math>~b</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(5 \mathfrak{L}_i) x_\mathrm{eq}^{3/5} \biggl[  \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr]  </math>
  </td>
</tr>

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  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~(4\mathfrak{K}_i) x_\mathrm{eq}^{3} \biggl[  \biggl( \frac{3^6}{2^5\pi} \biggr)^{1/2} \frac{\ell_i^3}{(1+\ell_i^2)^3} \biggr]  </math>
  </td>
</tr>
</table>
</div>

where,

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<math>~\mathfrak{L}_i</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{(\ell_i^4-1)}{\ell_i^2} + \frac{(1+\ell_i^2)^3}{\ell_i^3} \cdot \tan^{-1}\ell_i \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\mathfrak{K}_i</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
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<math>~\frac{\Lambda_i}{\eta_i}  + \frac{(1+\Lambda_i^2)}{\eta_i} \biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i\biggr] \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\Lambda_i</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
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<math>~\frac{1}{\eta_i} - \ell_i \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\eta_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl[\frac{\ell_i }{(1+\ell_i^2)}\biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
and,
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<math>~x_\mathrm{eq} = \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
  </td>
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<math>~=</math>
  </td>
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<math>~\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{\nu^2}{q} \cdot \frac{(1+\ell_i^2)^3}{3^3\ell_i^5} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{(1+\ell_i^2)^3}{3^3\ell_i^3} \biggl[ \frac{1 }{(1+\Lambda_i^2)} \biggr]
\biggl\{ 1 + \frac{1}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]  \biggr\}^{-1}</math>
  </td>
</tr>
</table>
</div>


<!-- MORE EXPRESSIONS FOR POWERPOINT PRESENTATION 
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  <td align="right">
<math>~x_\mathrm{eq}</math>
  </td>
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<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\pi}{2^3}\biggr)^{1/2} \frac{(1+\ell_i^2)^3}{3^3\ell_i^3} \biggl[ \frac{1 }{(1+\Lambda_i^2)} \biggr]
\biggl\{ 1 + \frac{1}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]  \biggr\}^{-1}</math>
  </td>
</tr>

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  <td align="right">
<math>~\Lambda_i</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{\eta_i} - \ell_i </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\eta_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl[\frac{\ell_i }{(1+\ell_i^2)}\biggr] </math>
  </td>
</tr>

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  <td align="right">
<math>~R_0 = R_\mathrm{norm}</math>
  </td>
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<math>~=</math>
  </td>
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<math>~
\biggl( \frac{G}{K_c} \biggr)^{5/2} M_\mathrm{tot}^2
</math>
  </td>
</tr>
</table>
</div>
-->

In rewriting this last expression, we have made use of the two relations,
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<math>~q \equiv \frac{r_i}{R}</math>
  </td>
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<math>~=</math>
  </td>
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<math>~\biggl\{ 1 + \frac{1}{\eta_i}\biggl[\frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr] \biggr\}^{-1} \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\ell_i q}{(1+\Lambda_i^2)^{1/2}} \, .
</math>
  </td>
</tr>
</table>
</div>

Consistent with our [[SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives|generic discussion of the stability of bipolytropes]] and the ''specific'' discussion of [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|the stability of bipolytropes having]] <math>~(n_c, n_e) = (5, 1)</math>, it can straightforwardly be shown that <math>~\partial \mathfrak{G}/\partial \chi = 0</math> is satisfied by setting <math>~\Chi = 1</math>; that is, the equilibrium condition is,

Furthermore, the equilibrium configuration is unstable whenever <math>~\partial^2 \mathfrak{G}/\partial \chi^2 < 0</math>, that is, the transition from stable to unstable configurations whenever,

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  <td align="right">
<math>~\frac{ \mathfrak{L}_i}{\mathfrak{K}_i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~20 \, .</math>
  </td>
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</table>
</div>
[[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|Table 1 of an accompanying chapter]] &#8212; and the red-dashed curve in the figure adjacent to that table &#8212; identifies some key properties of the model that marks the transition from stable to unstable configurations along equilibrium sequences that have various values of the mean-molecular weight ratio, <math>~\mu_e/\mu_c</math>.