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

==Milne's (1930) Choice of Equations of State==

As has been detailed in our [[SR#Equation_of_State|introductory discussion of analytically expressible equations of state]] and as is summarized in the following table, often the total gas pressure can be expressed as the sum of three separate components:  a component of ideal gas pressure, a component of radiation pressure, and a component due to a degenerate electron gas.  As a result, the total pressure is given by the expression,
<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>~P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad} \, .</math>
  </td>
</tr>
</table>
</div>

<table width="95%" align="center" border=1 cellpadding=5>
<tr>
<td align="center" width="25%"><font color="darkblue">Ideal Gas</font></td>
<td align="center"><font color="darkblue">Degenerate Electron Gas</font></td>
<td align="center" width="25%"><font color="darkblue">Radiation</font></td>
</tr>
<tr>
<td align="center">
{{Math/EQ_EOSideal0A}}
</td>
<td align="center">
{{Math/EQ_ZTFG01}}
</td>
<td align="center">
{{Math/EQ_EOSradiation01}}
</td>
</tr>

</table>


<span id="BetaDefinition">With this construction</span> in mind, {{ Milne30 }} also introduced the parameter, <math>~\beta</math>, to define the ratio of gas pressure (meaning, ideal-gas plus degeneracy pressure) to total pressure, that is,
<div align="center">
<math>\beta \equiv \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P}  \, ,</math>
</div>
in which case, also,
<div align="center">
<math>\frac{P_\mathrm{rad}}{P}  = 1-\beta </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\frac{P_\mathrm{gas} + P_\mathrm{deg}}{P_\mathrm{rad}}  = \frac{\beta}{1-\beta} \, .</math>
</div>


(We also have referenced this [[SR#The_Parameter.2C_.CE.B2|parameter, &beta;]], in the context of a broader discussion of equations of state and modeling [[SR#Time-Dependent_Problems|time-dependent flows]].)
 

===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),
<div align="center">
<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,
<div align="center">
<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,
<div align="center">
<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,
<div align="center">
<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),
<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>~\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>

===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,
<div align="center">
<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,

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

<tr>
  <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,
<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_0 \theta^{3/2} \, ,</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_{3/2} \xi \, ,</math>
  </td>
</tr>
</table>
</div>
where &#8212; see our [[SSC/Structure/Polytropes#Lane-Emden_Equation|general introduction to the Lane-Emden equation]] &#8212;
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(a_{3/2})^2</math>
  </td>
  <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,
<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}{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,
<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}{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,
<div align="center">
<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>.