Editing SSC/FreeEnergy/PolytropesEmbedded/Pt3A (section)

==Order of Magnitude Derivation==
Let's begin by providing very rough, approximate expressions for each of these four terms, assuming that <math>~n_c = 5</math> and <math>~n_e = 1</math>.  
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<math>~W_\mathrm{grav}\biggr|_\mathrm{core}</math>
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<math>~\approx</math>
  </td>
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<math>~- \mathfrak{a}_c \biggl[ \frac{GM_\mathrm{tot} M_c}{(R_i/2)} \biggr] 
= - 2\mathfrak{a}_c \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{\nu}{q}\biggr) \biggr] \, ;</math>
  </td>
</tr>

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<math>~W_\mathrm{grav}\biggr|_\mathrm{env}</math>
  </td>
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<math>~\approx</math>
  </td>
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<math>~- \mathfrak{a}_e \biggl[ \frac{GM_\mathrm{tot} M_e}{(R_i+R)/2} \biggr] 
= - 2\mathfrak{a}_e \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] \, ;</math>
  </td>
</tr>

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<math>~\mathfrak{S}_\mathrm{therm}\biggr|_\mathrm{core} = U_\mathrm{int}\biggr|_\mathrm{core} </math>
  </td>
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<math>~\approx</math>
  </td>
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<math>~\mathfrak{b}_c \cdot n_cK_c M_c ({\bar\rho}_c)^{1/n_c}
= 5\mathfrak{b}_c \cdot K_c M_\mathrm{tot}\nu \biggl[ \frac{3M_c}{4\pi R_i^3} \biggr]^{1/5} 
</math>
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</tr>

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&nbsp;
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<math>~=</math>
  </td>
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<math>~\mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} K_c (M_\mathrm{tot}\nu)^{6/5} (Rq)^{-3/5}  
\, ;</math>
  </td>
</tr>

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  <td align="right">
<math>~\mathfrak{S}_\mathrm{therm}\biggr|_\mathrm{env} = U_\mathrm{int}\biggr|_\mathrm{env} </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\mathfrak{b}_e \cdot n_eK_e M_\mathrm{env} ({\bar\rho}_e)^{1/n_e}
= \mathfrak{b}_e \cdot K_e M_\mathrm{tot}(1-\nu) \biggl[ \frac{3M_\mathrm{env}}{4\pi (R^3-R_i^3)} \biggr]
</math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~
\mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr) K_e [M_\mathrm{tot}(1-\nu)]^2 [R^3(1-q^3)]^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
In writing this last expression, it has been necessary to (temporarily) introduce a sixth physical parameter, namely, the polytropic constant that characterizes the envelope material, <math>~K_e</math>.  But this constant can be expressed in terms of <math>~K_c</math> via a relation that ensures continuity of pressure across the interface while taking into account the drop in mean molecular weight across the interface, that is,
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<math>~K_e ({\bar\rho}_e)^{(n_e+1)/n_e}</math>
  </td>
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<math>~\approx</math>
  </td>
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<math>~K_c ({\bar\rho}_c)^{(n_c+1)/n_c}</math>
  </td>
</tr>

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<math>~\Rightarrow ~~~~ K_e \biggl[\biggl( \frac{\mu_e}{\mu_c} \biggr) {\bar\rho}_c\biggr]^{2}</math>
  </td>
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<math>~\approx</math>
  </td>
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<math>~K_c ({\bar\rho}_c)^{6/5}</math>
  </td>
</tr>

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  <td align="right">
<math>~\Rightarrow ~~~~ \frac{K_e}{K_c} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{2}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
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<math>~\biggl[ \frac{3M_\mathrm{tot}\nu}{4\pi (Rq)^3} \biggr]^{-4/5} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, the fourth energy term may be rewritten in the form,

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  <td align="right">
<math>~\mathfrak{S}_\mathrm{therm}\biggr|_\mathrm{env} = U_\mathrm{int}\biggr|_\mathrm{env} </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr) \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
K_c\biggl[ \frac{3M_\mathrm{tot}\nu}{4\pi (Rq)^3} \biggr]^{-4/5}  [M_\mathrm{tot}(1-\nu)]^2 [R^3(1-q^3)]^{-1} 
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~
\mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
K_c M_\mathrm{tot}^{6/5}R^{-3/5}\biggl[ \frac{q^3}{\nu} \biggr]^{4/5}  \frac{(1-\nu)^2}{(1-q^3)} \, .
</math>
  </td>
</tr>
</table>
</div>

Putting all the terms together gives,
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<math>~\mathfrak{G}</math>
  </td>
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<math>~\approx</math>
  </td>
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<math>~
- 2\mathfrak{a}_c \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{\nu}{q}\biggr) \biggr] 
- 2\mathfrak{a}_e \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggl(\frac{1-\nu}{1+q}\biggr) \biggr] 
+ \mathfrak{b}_c \biggl( \frac{3\cdot 5^5}{2^2\pi} \biggr)^{1/5} K_c (M_\mathrm{tot}\nu)^{6/5} (Rq)^{-3/5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
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<math>~
+ \mathfrak{b}_e \biggl( \frac{3}{2^2\pi } \biggr)^{1/5} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
K_c M_\mathrm{tot}^{6/5}R^{-3/5}\biggl[ \frac{q^3}{\nu} \biggr]^{4/5}  \frac{(1-\nu)^2}{(1-q^3)} 
</math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~
- 2 \mathcal{A}_\mathrm{biP} \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggr] 
+ \mathcal{B}_\mathrm{biP} K_c \biggl[\frac{(\nu M_\mathrm{tot})^{2}}{ qR} \biggr]^{3/5}
</math>
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</tr>

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<math>~\Rightarrow ~~~~ \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
  </td>
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<math>~=</math>
  </td>
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<math>~
- 2 \mathcal{A}_\mathrm{biP} \biggl[ \frac{GM_\mathrm{tot}^2 }{R} \biggr] \biggl(\frac{G^3}{K_c^5}\biggr)^{1/2}
+ \mathcal{B}_\mathrm{biP} \biggl(\frac{\nu^2}{q}\biggr)^{3/5} K_c \biggl[\frac{M_\mathrm{tot}^{2}}{ R} \biggr]^{3/5}\biggl(\frac{G^3}{K_c^5}\biggr)^{1/2}
</math>
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</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~
- 2 \mathcal{A}_\mathrm{biP} \biggl[ \frac{R_\mathrm{norm}}{R} \biggr] 
+ \mathcal{B}_\mathrm{biP} \biggl(\frac{\nu^2}{q}\biggr)^{3/5} \biggl[\frac{R_\mathrm{norm}}{ R} \biggr]^{3/5} \, ,
</math>
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</table>
</div>

where,
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<math>~\mathcal{A}_\mathrm{biP}</math>
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<math>~\equiv</math>
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<math>~\biggl[ \mathfrak{a}_c\biggl(\frac{\nu}{q}\biggr)  + \mathfrak{a}_e  \biggl(\frac{1-\nu}{1+q}\biggr)  \biggr] \, ,</math>
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  <td align="right">
<math>~\mathcal{B}_\mathrm{biP}</math>
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  <td align="center">
<math>~\equiv</math>
  </td>
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<math>~\biggl( \frac{3}{2^2\pi} \biggr)^{1/5} \biggl[5\mathfrak{b}_c  
+ \mathfrak{b}_e \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \frac{q^3(1-\nu)^2}{\nu^2(1-q^3)} \biggr] \, .</math>
  </td>
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</table>
</div>