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

===Envelope===

Now, inside the ''envelope'' of his composite polytrope, {{ Milne30 }} considered that the effects of electron degeneracy pressure could be ignored and, accordingly, employed throughout the envelope the expression,
<div align="center">
<math>\frac{P_\mathrm{gas}}{P_\mathrm{rad}} \biggr|_\mathrm{env} = \frac{\beta}{1-\beta} \, , </math>
</div>
or (see Milne's equation 24),
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<math>\biggl( \frac{\mathfrak{\Re}}{\mu_e}\biggr) \rho = \frac{1}{3}a_\mathrm{rad}T^3 \biggl( \frac{\beta}{1-\beta} \biggr) \, .</math>
</div>

If the parameter, <math>~\beta</math>, is constant throughout the envelope &#8212; which Milne assumes &#8212; then this last expression can be interpreted as defining a <math>~T(\rho)</math> function throughout the envelope of the form,
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<math>T = \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \rho^{1/3} \, .</math>
</div>
Now, returning to the definition of <math>~\beta</math> while ignoring the effects of degeneracy pressure, we recognize that the total pressure in the envelope can be written in the form of a ''modified'' ideal gas relation, namely,
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<math>~\beta P = P_\mathrm{gas} + \cancelto{0}{P_\mathrm{deg}} = \biggl(\frac{\Re}{\mu_e}\biggr) \rho T \, ,</math>
</div>
with the ''specific'' <math>~T(\rho)</math> behavior just derived.   This allows us to write the envelope's total pressure as,
<div align="center">
<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}{\beta}\biggl(\frac{\Re}{\mu_e}\biggr) \rho 
\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \rho^{1/3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \rho^{1 + 1/3} \, ,</math>
  </td>
</tr>
</table>
</div>

which can be immediately associated with a polytropic relation of the form,
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<math>~P = K_e \rho^{1 + 1/n_e} \, ,</math>
</div>
with,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~n_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~K_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \, .</math>
  </td>
</tr>
</table>
</div>
So, from the solution, <math>~\phi(\eta)</math>, to the Lane-Emden equation of index <math>~n=3</math>, we will be able to determine that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_e \phi^3 \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a_3 \eta \, ,</math>
  </td>
</tr>
</table>
</div>
where &#8212; see our [[SSC/Structure/Polytropes#Lane-Emden_Equation|general introduction to the Lane-Emden equation]] &#8212;
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_3^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{K_e}{\pi G}\biggr) \rho_e^{-2/3} \, .</math>
  </td>
</tr>
</table>
</div>
This is the envelope structure that will be incorporated into our derivation of the bipolytrope's properties, below. 

In contrast to this approach, {{ Milne30 }} chose to relate the solution to the envelope's <math>~n=3</math> Lane-Emden equation directly to the temperature via the expression,


<div align="center">
<math>T = \lambda \phi \, ,</math>
</div>
and deduced that the corresponding radial scale-factor is (see Milne's equation 27),
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a^2_\mathrm{Milne}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\lambda^2} \biggl(\frac{\Re}{\mu_e}\biggr)^{2} \frac{(1-\beta)}{\beta^2} \biggl(\frac{3}{\pi a_\mathrm{rad}G}\biggr)\, .</math>
  </td>
</tr>
</table>
</div>
In order to demonstrate the relationship between our radial scale-factor <math>~(a_3)</math> and Milne's, we note that,

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

<tr>
  <td align="right">
<math>~\phi^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{T}{\lambda}\biggr)^3 = \frac{\rho}{\rho_e}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~~\lambda^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_e \biggl(\frac{T^3}{\rho}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_e \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~~\lambda^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_e^{-2/3} \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{-2/3} \, .</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a^2_\mathrm{Milne}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_e^{-2/3} \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{-2/3}\biggl(\frac{\Re}{\mu_e}\biggr)^{2} \frac{(1-\beta)}{\beta^2} \biggl(\frac{3}{\pi a_\mathrm{rad}G}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_e^{-2/3} \biggl( \frac{1}{\pi G} \biggr) \biggl[ \biggl(\frac{\Re}{\mu_e}\biggr)^{4} \frac{(1-\beta)}{\beta^{4}} \biggl(\frac{3}{a_\mathrm{rad}}\biggr) \biggr]^{1/3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_e^{-2/3} \biggl( \frac{K_e}{\pi G} \biggr)  \, .</math>
  </td>
</tr>
</table>
</div>
It is clear, therefore, that the two radial scale-factors are the same.  In preparation for our [[SSC/Structure/BiPolytropes/Analytic1.53/Pt3#Step_8:_Throughout_the_envelope_(ηi_≤_η_≤_ξs)|further discussion of the structure of this bipolytrope's envelope, below]],  it is useful to highlight the following two expressions that have been developed here in the process of showing the correspondence between our work and that of Milne:

<div align="center" id="HighlightedExpressions">
<table border="1" cellpadding="5" align="center" width="50%">
<tr><td align="center">A Pair of Highlighted Relations</td></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\rho_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda^3 \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda^3 \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^{-1} \frac{\beta}{(1-\beta)} \cdot \frac{1}{3} a_\mathrm{rad} \biggr]</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~K_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} </math>
  </td>
</tr>
</table>
</div>

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