Editing SSC/Structure/BiPolytropes/Analytic1.53 (section)

===Core===

In contrast to the envelope, {{ Milne30 }} assumed that the (non-relativistic; "NR") electron degeneracy pressure dominates over the ideal-gas pressure in the core.  That is, he assumed that, throughout the core of his composite polytropic configuration,
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<math>~\beta P = \cancelto{0}{P_\mathrm{gas}} + P_\mathrm{deg}\biggr|_\mathrm{NR}  \,.</math>
</div>

As we have [[SR#Nonrelativistic_ZTF_Gas|demonstrated elsewhere]], the non-relativistic expression for the degeneracy pressure is,

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

<tr>
  <td align="right">
<math>~P_\mathrm{deg}\biggr|_\mathrm{NR}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^2 \cdot 5}\biggl( \frac{3}{\pi} \biggr)^{2/3} \biggl( \frac{h^2}{m_e} \biggr) \biggl[ \frac{\rho}{(\mu_{e^-}) m_p}\biggr]^{5/3} \, ,</math>
  </td>
</tr>
</table>
</div>

which can be associated with a polytropic relation of the form,
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<math>~P_\mathrm{deg}\biggr|_\mathrm{NR} = K_c \rho^{1 + 1/n_c} \, ,</math>
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that is, a total pressure of the form,
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<math>~\beta P = K_c \rho^{1 + 1/n_c} \, ,</math>
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with,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~n_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3}{2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~K_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^2 \cdot 5}\biggl( \frac{3}{\pi} \biggr)^{2/3} \biggl( \frac{h^2}{m_e} \biggr) \biggl[ \frac{1}{(\mu_{e^-}) m_p}\biggr]^{5/3} \, .</math>
  </td>
</tr>
</table>
</div>
(Note that, here only, we have used the parameter, <math>\mu_{e^-}</math>, to denote the molecular weight of electrons, instead of just <math>\mu_e</math>, in order not to confuse it with the mean molecular weight assigned to the envelope material.)  So, from the solution, <math>\theta(\xi)</math>, to the Lane-Emden equation of index <math>n=\tfrac{3}{2}</math>, we will be able to determine that,
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  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0 \theta^{3/2} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
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<tr>
  <td align="right">
<math>~r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a_{3/2} \xi \, ,</math>
  </td>
</tr>
</table>
</div>
where &#8212; see our [[SSC/Structure/Polytropes#Lane-Emden_Equation|general introduction to the Lane-Emden equation]] &#8212;
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(a_{3/2})^2</math>
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  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{5K_c}{2^3\pi G}\biggr) \rho_0^{-1/3} \, .</math>
  </td>
</tr>
</table>
</div>
This is the [[SSC/Structure/BiPolytropes/Analytic1.53/Pt3#Step_4:_Throughout_the_core_(0_≤_ξ_≤_ξi)|core structure that will be incorporated into our derivation]] of the bipolytrope's properties.

This is precisely the approach taken by {{ Milne30 }}.  Just before his equation (43), Milne states that, "the equation of state when the electrons alone are degenerate can be shown" to be,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5}\biggl( \frac{3}{4\pi}\biggr)^{2/3} \frac{h^2}{(2m_H)^{5/3} m_e q_e^{2/3} }~ \rho^{5/3} \, ,</math>
  </td>
</tr>
</table>
</div>
which, upon regrouping terms gives,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{5}\biggl( \frac{3}{\pi}\biggr)^{2/3}\biggl( \frac{1}{2^4 q_e^2 }\biggr)^{1/3} \frac{h^2}{m_e  }\biggl( \frac{\rho}{2 m_H} \biggr)^{5/3} \, .</math>
  </td>
</tr>
</table>
</div>
Recognizing that Milne set <math>~q_e = 2</math>, as "the statistical weight of an electron," and that he adopted a molecular weight of the electrons, <math>\mu_{e^-}=2</math>, this expression for the equation of state exactly matches our expression for <math>P_\mathrm{deg}|_\mathrm{NR}</math>.  Our enlistment of an <math>n_c  = \tfrac{3}{2}</math> polytropic equation of state for the core is therefore also perfectly aligned with Milne's treatment of the core; in particular, according to Milne, at each radial location throughout the core the total pressure can be obtained from the expression,
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<math>P = \frac{K}{\beta} ~\rho^{5/3} \, ,</math>
</div>
with Milne's coefficient, <math>K</math>, having the same definition as our coefficient, <math>K_c</math>.