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=BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (3/2, 3)=
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<font size="-1">[[H_BookTiledMenu#MoreModels|<b>Milne</b>]]<br /> (1930)</font>
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Here we lay out the procedure for constructing a [[SSC/Structure/BiPolytropes#BiPolytropes|bipolytrope]] in which the core has an <math>~n_c=\tfrac{3}{2}</math> polytropic index and the envelope has an <math>~n_e=3</math> polytropic index.  We will build our discussion around the work of [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930, MNRAS, 91, 4)] who, as we shall see, justified these two indexes on physical grounds.  While this system cannot be described by closed-form, analytic expressions, it is of particular interest because &#8212; as far as we have been able to determine &#8212; its examination by Milne represents the first "composite polytrope" to be discussed in the astrophysics literature.    

In deriving the properties of this model, we will follow the [[SSC/Structure/BiPolytropes#Solution_Steps|general solution steps for constructing a bipolytrope]] that are outlined in a separate chapter of this H_Book.  That group of general solution steps was drawn largely from chapter IV, &sect;28 of Chandrasekhar's book titled, "An Introduction to the Study of Stellar Structure" [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]].   At the end of that chapter (specifically, p. 182), Chandrasekhar acknowledges that "[Milne's] method is largely used in &sect; 28."  It seems fitting, therefore, that we highlight the features of the ''specific'' bipolytropic configuration that [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)] chose to build.

{{ SGFworkInProgress }}


==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>


With this construction in mind, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] 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, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] 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, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] 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 [[#Step_8:__Throughout_the_envelope|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, [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] 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, below, 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 core structure that will be incorporated into our derivation of the bipolytrope's properties, below.

This is precisely the approach taken by [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)].  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>

==The Point-Source Model==

According to Chapter IX.3 (p. 332) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], in the so-called "point-source" model, <font color="darkgreen">"&hellip; it is assumed that the entire source of energy is liberated at the center of the star; analytically, the assumption is that <math>L_r =~\mathrm{constant}~= L</math>."</font>

===Handling Radiation Transport===

Here we begin with the [[PGE/FirstLawOfThermodynamics#Example_B|familiar expression for the radiation flux]],  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\vec{F}_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{c}{3\rho\kappa_R} \nabla (a_\mathrm{rad}T^4) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>-\chi_\mathrm{rad} \nabla T \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. I, &sect;2, p. 17, Eq. (2.17)
  </td>
  <td align="left" colspan="2">and &nbsp; &nbsp;[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 57, Eq. (67)
  </td>
</tr>
</table>
</div>
where [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] refers to 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\chi_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{4c a_\mathrm{rad} T^3}{3\kappa \rho} \, ,
</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 57, Eq. (68)
</div>
as the coefficient of ''radiative'' conductivity.  When modeling spherically symmetric configurations, the radiation flux has only a radial component, that is, <math>\vec{F}_\mathrm{rad} = \hat{e}_r(F_r)</math>.  And, as pointed out in the context of Eq. (170) on p. 214 of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] <font color="darkgreen">&hellip; the quantity <math>L_r \equiv 4\pi r^2 F_r</math>, which is the net amount of energy crossing a spherical surface of radius <math>r</math>, is generally introduced instead of <math>F_r</math>.</font>  We therefore have,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right"><math>\frac{L_r}{4\pi r^2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
- \frac{c}{3\rho\kappa_R} \frac{d}{dr} (a_\mathrm{rad}T^4)
  </math></td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\frac{dT}{dr}
  </math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- \frac{3}{4ca_\mathrm{rad}} \frac{\rho\kappa_R}{T^3} \frac{L_r}{4\pi r^2}</math></td>
</tr>
</table>
[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter V, Eq. (171)<br />
[<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], &sect;6, Eq. (6-4a)<br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;9.1, Eq. (9.6)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;7.1, Eq. (7.8)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], &sect;5.2, Eq. (5.15)
</div>

<table border="1" align="center" cellpadding="8" width="60%"><tr><td align="left">
Dimensional Analysis:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\frac{\kappa_R L}{ca_\mathrm{rad}}</math></td>
  <td align="center"><math>\sim</math></td>
  <td align="left"><math>
\frac{rT^4}{\rho} \sim m^{-1}\ell^4 ( {^\circ}K)^4 \, .
  </math></td>
</tr>
</table>
NOTE:  This is consistent with the opacity, <math>\kappa_R \sim (\ell^2 m^{-1})</math>.
</td></tr></table>

===Harrison's Approach===
Following [https://ui.adsabs.harvard.edu/abs/1946ApJ...103..193H/abstract M. Hall Harrison (1946, ApJ, Vol. 103, 193)], we seek to solve this last expression in concert with solutions to a pair of additional key governing relations for spherically symmetric equilibrium configurations, namely,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="center">{{Math/EQ_SShydrostaticBalance01}}</td>
  <td align="center">; &nbsp; &nbsp; &nbsp;</td>
  <td align="center">{{Math/EQ_SSmassConservation01}}</td>
</tr>

<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1946ApJ...103..193H/abstract M. Hall Harrison (1946)], p. 196, Eq. (20)
  </td>
</tr>
</table>

while adopting (see [[SR/IdealGas#Consequential_Ideal_Gas_Relations|related discussion]])

<div align="center">
<span id="IdealGas:FormA"><font color="#770000">'''Form A'''</font></span><br />
of the Ideal Gas Equation of State,

{{ Template:Math/EQ_EOSideal0A }}
</div>
and adopting [https://en.wikipedia.org/wiki/Kramers'_opacity_law Kramers' opacity law], that is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\kappa_R ~~ \rightarrow ~~ \kappa_\mathrm{Kramers}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\kappa_0 \rho T^{-7 / 2} \, .</math></td>
</tr>
</table>

<table border="1" align="center" cellpadding="8" width="60%"><tr><td align="left">
Dimensional Analysis:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\biggl[ \frac{\kappa_0 L}{ca_\mathrm{rad}} \biggr]</math></td>
  <td align="center"><math>\sim</math></td>
  <td align="left"><math>
\frac{rT^{15/2}}{\rho^2} \sim m^{-2}\ell^7 ( {^\circ}K)^{15/2}
  </math></td>
</tr>

<tr>
  <td align="right"><math>\frac{\mathfrak{R}}{\mu}</math></td>
  <td align="center"><math>\sim</math></td>
  <td align="left"><math>
t^{-2}\ell^2 ( {^\circ}K)^{-1}
  </math></td>
</tr>
</table>

----

We note as well that the leading coefficient in Kramers' opacity is,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right"><math>\kappa_0</math></td>
  <td align="center"><math>\approx</math></td>
  <td align="left"><math>
5 \times 10^{22} ~\mathrm{cm}^5~\mathrm{g}^{-2} ({^\circ}K)^{7 / 2} \, .
  </math></td>
</tr>
</table>
[<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], &sect;3.3, Eq. (3-170)<br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;17.2, Eq. (17.5)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;4.4.2, Eq. (4.35)
</div>

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

===Does a Polytropic Relation Work?===

Let's examine whether a point-source model can be represented by a polytropic relation.

====Adopting Temperature (instead of enthalpy)====
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>K\rho^{1 + 1/n}  ~~~~\Rightarrow~~~~  \rho = \biggl(\frac{P}{K}\biggr)^{n/(n+1)} \, ;</math>
</tr>

<tr>
  <td align="right"><math>\biggl(\frac{\mathfrak{R}}{\mu}\biggr)\rho T</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>K\rho^{1 + 1/n} ~~~~\Rightarrow~~~~ T = \biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)\rho^{1/n} \, ;</math>
</tr>

<tr>
  <td align="right"><math>T^{n}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)^{n}\rho  
~~~~\Rightarrow~~~~  
T^n = \biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)^{n}\biggl(\frac{P}{K}\biggr)^{n/(n+1)} 
~~~~\Rightarrow~~~~  
T^{(n+1)} = \biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)^{(n+1)}\biggl(\frac{P}{K}\biggr) 
\, .</math>
</tr>
</table>
Hydrostatic balance is governed by the single 2<sup>nd</sup> order ODE,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\frac{1}{r^2} \frac{d}{dr} \biggl[ \frac{r^2}{\rho} \frac{dP}{dr} \biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- 4\pi G\rho \, .</math>
</tr>
</table>
Normally in order to arrive at the Lane-Emden equation, <math>P</math> is converted to <math>\rho</math>; here, let's convert both <math>P</math> and <math>\rho</math> to <math>T</math>.  First, on the RHS we have,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\rho</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mathfrak{R}/\mu}{K}\biggr)^n T^n \, ;</math>
</tr>
</table>
and second, on the LHS we have,

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

<tr>
  <td align="right"><math>\frac{1}{\rho} \frac{dP}{dr}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{K}{\mathfrak{R}/\mu}\biggr)^n T^{-n} \cdot \frac{d}{dr}\biggl[
K^{-n}(\mathfrak{R}/\mu)^{(n+1)} T^{(n+1)}
\biggr]
</math>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\mathfrak{R}}{\mu}\biggr) T^{-n} \cdot \frac{d}{dr}\biggl[T^{(n+1)}
\biggr]
</math>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(n+1)\biggl(\frac{\mathfrak{R}}{\mu}\biggr) \frac{dT}{dr}
</math>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(n+1)\biggl(\frac{\mathfrak{R}}{\mu}\biggr) \biggl\{
- \frac{3}{4ca_\mathrm{rad}} \frac{\rho\kappa_R}{T^3} \frac{L}{4\pi r^2}
\biggr\}
</math>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- (n+1)\biggl(\frac{\mathfrak{R}}{\mu}\biggr) \biggl[
\frac{3L}{16\pi ca_\mathrm{rad}} 
\biggr] \biggl[ \frac{\rho}{r^2T^3} \biggr]\kappa_0 \rho T^{-7 / 2} 
</math>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{r^2}{\rho} \frac{dP}{dr}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- (n+1)\biggl(\frac{\mathfrak{R}}{\mu}\biggr) \biggl[
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} 
\biggr] \rho^2 T^{-13 / 2} 
</math>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- (n+1)K^{-2n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{2n+1} \biggl[
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} 
\biggr] T^{(4n-13 )/ 2} \, .
</math>
</tr>
</table>
So, the hydrostatic-balance condition becomes,

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

<tr>
  <td align="right">
<math>
- (n+1)K^{-2n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{2n+1} \biggl[
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} 
\biggr] \cdot \frac{1}{r^2}\frac{dT^{(4n-13 )/ 2}}{dr} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- 4\pi G\biggl(\frac{\mathfrak{R}/\mu}{K}\biggr)^n T^n </math></td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{(n+1)}{4\pi G K^{2n}}\biggl(\frac{\mathfrak{R}/\mu}{K}\biggr)^{-n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{2n+1} \biggl[
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} 
\biggr] \cdot \frac{1}{r^2}\frac{dT^{(4n-13 )/ 2}}{dr} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> T^n </math></td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\frac{\mathcal{A}}{r^2}\frac{dT^{(4n-13 )/ 2}}{dr} 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> T^n </math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ {\mathcal{A}}^{-1}r^2 dr </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
T^{-n} \cdot dT^{(4n-13 )/ 2} \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\mathcal{A}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\frac{(n+1)}{4\pi G K^n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{n+1} \biggl[
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} 
\biggr] \, .  
</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="8" width="60%"><tr><td align="left">
Dimensional Analysis:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\biggl[ \frac{\kappa_0 L}{ca_\mathrm{rad}} \biggr]</math></td>
  <td align="center"><math>\sim</math></td>
  <td align="left"><math>
m^{-2}\ell^7 ( {^\circ}K)^{15/2}
  </math></td>
</tr>

<tr>
  <td align="right"><math>\frac{\mathfrak{R}}{\mu}</math></td>
  <td align="center"><math>\sim</math></td>
  <td align="left"><math>
t^{-2}\ell^2 ( {^\circ}K)^{-1}
  </math></td>
</tr>

<tr>
  <td align="right">polytropic <math>K</math></td>
  <td align="center"><math>\sim</math></td>
  <td align="left"><math>
t^{-2} \ell^{2 + 3/n} m^{-1 / n}
  </math></td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\mathcal{A}</math></td>
  <td align="center"><math>\sim</math></td>
  <td align="left"><math>
G^{-1} K^{-n} \biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{n+1} \biggl[ m^{-2}\ell^7 ( {^\circ}K)^{15/2} \biggr]
  </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\sim</math></td>
  <td align="left"><math>
\biggl[\ell^{-3} m t^{2}\biggr] \biggl[ t^{-2} \ell^{2 + 3/n} m^{-1 / n} \biggr]^{-n} 
\biggl[ t^{-2}\ell^2 ( {^\circ}K)^{-1}\biggr]^{n+1} \biggl[ m^{-2}\ell^7 ( {^\circ}K)^{15/2} \biggr]
  </math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\sim</math></td>
  <td align="left"><math>
\biggl[\ell^{-3} \biggr] \biggl[ \ell^{2 + 3/n}  \biggr]^{-n} 
\biggl[ \ell^2 \biggr]^{n+1} \biggl[ \ell^7\biggr] ( {^\circ}K)^{13/2-n} 
\sim
\ell^3( {^\circ}K)^{13/2-n} 
\, .
  </math></td>
</tr>
</table>

Now, the [[SSC/Structure/Polytropes#Lane-Emden_Equation|characteristic length scale for polytropic configurations]] is given by the expression,

<div align="center">
<math>~
a_\mathrm{n}^2 \equiv \biggl[\frac{(n+1)K}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr] \, .
</math>
</div>
If we divide <math>\mathcal{A}</math> by <math>a_n^3</math>, the resulting expression should give us the characteristic temperature of the envelope.  Specifically, we find that,

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

<tr>
  <td align="right"><math>\frac{\mathcal{A}}{a_n^3}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{(n+1)}{4\pi G K^n}\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{n+1} \biggl[
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} 
\biggr] 
\times
\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{-3 / 2}   
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(n+1)^{- 1 / 2}
\rho_c^{3(n-1)/2n}   
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{n+1} \biggl[
\frac{3\kappa_0 L}{16\pi ca_\mathrm{rad}} 
\biggr] (4\pi G)^{ 1 / 2} K^{-n - 3 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\sim</math></td>
  <td align="left">
<math>
\biggl[m \ell^{-3}\biggr]^{3(n-1)/2n}   
\biggl[ t^{-2}\ell^2 ( {^\circ}K)^{-1}\biggr]^{n+1} \biggl[
m^{-2}\ell^7 ( {^\circ}K)^{15/2} 
\biggr] \biggl[ \ell^3 m^{-1} t^{-2} \biggr]^{ 1 / 2} 
\biggl[ t^{-2} \ell^{2 + 3/n} m^{-1 / n} \biggr]^{-n - 3 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\sim</math></td>
  <td align="left">
<math>
t^{-2n - 2 - 1 + 2n + 3}
m^{ 3(n-1)/2n - 2 - 1 / 2 + 1 + 3/2n}
\ell^{ -9(n-1)/2n + 2n + 2 + 7 + 3 / 2}
\biggl[ \ell^{(2n + 3)/n} \biggr]^{-(2n + 3) / 2}
( {^\circ}K)^{13/2 - n }  
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\sim</math></td>
  <td align="left">
<math>
\ell^{ (9 + 4n^2 + 12n)/2n }
\ell^{-(2n + 3)^2 / 2n}
( {^\circ}K)^{13/2 - n }  
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>\sim</math></td>
  <td align="left">
<math>
( {^\circ}K)^{(13-2n)/2 }  \, .
</math>
  </td>
</tr>
</table>


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

Quite generally we can write,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\frac{1}{\alpha}~\frac{dT^\alpha}{dr}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
T^{\alpha - 1} \frac{dT}{dr} \, .
</math>
  </td>
</tr>
</table>
Rewriting the hydrostatic-balance condition, we find that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>{\mathcal{A}}^{-1}r^2 </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
T^{-n} \cdot \frac{d}{dr}\biggl[T^{(4n-13 )/ 2}\biggr]
=
T^{-n} \biggl(\frac{4n-13}{2}  \biggr) T^{(4n-15 )/ 2} \cdot \frac{dT}{dr}
=
\biggl(\frac{4n-13}{2}  \biggr) T^{(2n-15 )/ 2} \cdot \frac{dT}{dr} \, .
</math>
  </td>
</tr>
</table>
Associating the exponents,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\alpha - 1</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{2n-15}{2} ~~~\Rightarrow ~~~ \alpha = \frac{2n-13}{2} \, , 
</math>
  </td>
</tr>
</table>
we can write,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>{\mathcal{A}}^{-1}r^2 </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{4n-13}{2}  \biggr) \biggl(\frac{2}{2n-13}  \biggr)\frac{d}{dr} \biggl[ T^{(2n-13)/2 } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ 3d(r^3)</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\mathcal{A} \biggl(\frac{4n-13}{2n-13}  \biggr) d\biggl[ T^{(2n-13)/2 } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ 0</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
d\biggl[ \mathcal{A} \biggl(\frac{13-4n}{13 - 2n}  \biggr) T^{(2n-13)/2 } - 3r^3\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~</math> constant</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\mathcal{A}}{a_n^3} \biggl(\frac{13-4n}{13 - 2n}  \biggr) T^{(2n-13)/2 } - 3\biggl( \frac{r}{a_n}\biggr) ^3  \, .
</math>
  </td>
</tr>
</table>

====Adopting Enthalpy (instead of Temperature)====

For [[SR#Barotropic_Structure|polytropic configurations]] the enthalpy, <math>H</math>, can easily be adopted in place of temperature via the relation,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\biggl(\frac{\mathfrak{R}}{\mu}\biggr) T</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{H}{(n+1)} \, .</math></td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right"><math>P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>K\rho^{1 + 1/n}\, ;</math></td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>H</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>K\rho^{1/n}\, ;</math></td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>H^{n+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>K^n P\, .</math></td>
</table>
And the radiation-transport equation can be rewritten in the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>\frac{L_r}{4\pi r^2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
- \frac{ca_\mathrm{rad}}{3\rho\kappa_R (n+1)^4}  
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-4} 
\frac{dH^4}{dr}
</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
- \frac{ca_\mathrm{rad}}{3\kappa_0 (n+1)^4}  
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-4} \biggl[\rho^{-2} T^{7 / 2}\biggr]
\frac{dH^4}{dr}
</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
- \frac{ca_\mathrm{rad}}{3\kappa_0 (n+1)^4}  
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-4} 
\biggl\{\biggl(\frac{K}{H}\biggr)^{2n}  \biggl[ \biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-1} \frac{H}{(n+1)}\biggr]^{7 / 2}\biggr\}
4 H^3 \frac{dH}{dr}
</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
- \frac{4ca_\mathrm{rad} K^{2n}}{3\kappa_0 (n+1)^{15 / 2}}  
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{-15 / 2} 
H^{(13 - 4n) / 2}
\frac{dH}{dr}
</math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ r^2 \frac{dH}{dr} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
- \biggl[ \frac{3\kappa_0 L (n+1)^{15 / 2}}{16\pi ca_\mathrm{rad} K^{2n}}  
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{15 / 2}  \biggr]
H^{(4n - 13 ) / 2} \ .
</math></td>
</tr>
</table>

In terms of the enthalpy, the hydrostatic-balance expression becomes,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>- 4\pi G\biggl[\frac{H^n}{K^n}\biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{1}{r^2} \frac{d}{dr} \biggl\{r^2\biggl[\frac{K^n}{H^n}\biggr] 
\frac{d}{dr} \biggl[K^{-n} H^{n+1}  \biggr]
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ - 4\pi GH^n</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{K^n}{r^2} \frac{d}{dr} \biggl\{
\biggl[\frac{r^2}{H^n}\biggr] 
\frac{d}{dr} \biggl[H^{n+1}  \biggr]
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{(n+1)K^n}{r^2} \frac{d}{dr} \biggl\{
r^2 \frac{dH}{dr} 
\biggr\} \, .</math>
  </td>
</tr>
</table>
Combining these two equations gives,

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

<tr>
  <td align="right"><math>H^n</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{(n+1)K^n}{4\pi G r^2} \frac{d}{dr} \biggl\{
\biggl[ \frac{3\kappa_0 L (n+1)^{15 / 2}}{16\pi ca_\mathrm{rad} K^{2n}}  
\biggl(\frac{\mathfrak{R}}{\mu}\biggr)^{15 / 2}  \biggr]
H^{(4n - 13 ) / 2} 
\biggr\} \, .</math>
  </td>
</tr>
</table>

===Power-Law Density Distribution===

In an [[SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|accompanying discussion]], we have demonstrated that power-law density distributions can provide analytic solutions of the Lane-Emden equation, although the associated boundary conditions do not naturally conform to the boundary conditions that are suitable to astrophysical configurations.  We have just shown that the point-source envelope configuration appears to admit a power-law temperature solution; specifically, setting the integration constant to zero, our result gives,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\frac{\mathcal{A}}{a_n^3} \biggl(\frac{13-4n}{13 - 2n}  \biggr) \biggl[ \frac{T}{T_\mathrm{norm}}\biggr]^{(2n-13)/2 } </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3\xi^3 \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>
\xi 
</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\frac{r}{a_n} \, , 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>
T_\mathrm{norm}
</math>
  </td>
  <td align="center"><math>\equiv</math></td>
  <td align="left">
<math>
\biggl[\frac{\mathcal{A}}{a_n^3}\biggr]^{2/(13-2n)}
\, .
</math>
  </td>
</tr>
</table>

==Our Derivation==
===Steps 2 &amp; 3===

Throughout the core, the properties of this bipolytrope can be expressed in terms of the Lane-Emden function, <math>~\theta(\xi)</math>, which derives from a solution of the 2<sup>nd</sup>-order ODE,
<div align="center">
<math>
\frac{1}{\xi^2} \frac{d}{d\xi} \biggl[ \xi^2 \frac{d\theta}{d\xi}\biggr] = - \theta^{3/2} \, ,
</math>
</div>
subject to the boundary conditions,
<div align="center">
<math>~\theta = 1</math> &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>~\frac{d\theta}{d\xi} = 0</math>
&nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; <math>~\xi = 0</math>.
</div>

The first zero of the function <math>~\theta(\xi)</math> and, hence, the surface of the corresponding isolated <math>~n=\tfrac{3}{2}</math> polytrope is located at <math>~\xi_s = 3.65375</math> (see Table 4 in chapter IV on p. 96 of [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]]).  Hence, the interface between the core and the envelope can be positioned anywhere within the range, <math>~0 < \xi_i < \xi_s = 3.65375</math>.

===Step 4:  Throughout the core (0 &le; &xi; &le; &xi;<sub>i</sub>)===
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="3">
Specify:  <math>~K_c</math> and <math>~\rho_0 ~\Rightarrow</math>
  </td>
  <td colspan="2">
&nbsp;
  </td>
</tr>
<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0 \theta^{n_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0 \theta^{3/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K_c \rho_0^{5/3} \theta^{5/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{5K_c}{8\pi G} \biggr]^{1/2} \rho_0^{-1/6} \xi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~M_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^2(2 \pi )^{1/2}}  \biggl[ \frac{5K_c}{G} \biggr]^{3/2} 
\rho_0^{1/2} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
  </td>
</tr>
</table>

</div>

By comparison, the expressions that [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne (1930)] derived for the run of <math>~\rho</math>, <math>~r</math>, and <math>~M_r</math> throughout the core are presented in his paper as, respectively, equations (90), (88), and (87).   In an effort to facilitate this comparison, Milne's expressions &#8212; which also specifically apply to the outer edge of the core, whose identity is associated with primed variable names in Milne's notation &#8212; are reprinted as extracted images in the following boxed-in table. 

<div align="center">
<table border="2" cellpadding="10">
<tr>
  <th align="center">
Equations extracted<sup>&dagger;</sup> from [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M E. A. Milne (1930)]<p></p>
"''The Analysis of Stellar Structure''"<p></p>
MNRAS, vol. 91, pp. 4 - 55 &copy; Royal Astronomical Society
  </th>
</tr>
<tr>
  <td>
[[File:CoreRelations01.png|500px|center|Milne (1930)]]
  </td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Equations displayed here, as a single digital image, with presentation order &amp; layout modified from the original publication.</td></tr>
</table>
</div>

It is clear that the agreement between our derivation and Milne's is exact, once it is realized that Milne has used <math>~\psi(\eta)</math> to represent the Lane_Emden function for the <math>~n_c = \tfrac{3}{2}</math> core, whereas we have represented this function by <math>~\theta(\xi)</math>; and Milne has identified the configuration's central density as <math>~\lambda_2</math>, whereas we have used the notation, <math>~\rho_0</math>.

===Step 5: Interface Conditions===

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td colspan="3">
&nbsp;
  </td>
  <td align="left" colspan="2">
Setting <math>~n_c=\tfrac{3}{2}</math>, <math>~n_e=3</math>, and <math>~\phi_i = 1 ~~~~\Rightarrow</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\rho_e}{\rho_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \phi_i^{-n_e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{3/2}_i </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{K_e}{K_c} \biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\rho_0^{1/n_c - 1/n_e}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} \theta^{1 - n_c/n_e}_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\rho_0^{1/3}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-4/3} \theta^{1/2}_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\eta_i}{\xi_i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{n_c + 1}{n_e+1} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \theta_i^{(n_c-1)/2} \phi_i^{(1-n_e)/2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{5}{8}\biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \theta_i^{1/4}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{n_c + 1}{n_e + 1} \biggr]^{1/2} \theta_i^{- (n_c + 1)/2} \phi_i^{(n_e+1)/2} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{5}{8}\biggr)^{1/2} \theta_i^{- 5/4} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>
  </td>
</tr>
</table>
</div>

===Step 8:  Throughout the envelope (&eta;<sub>i</sub> &le; &eta; &le; &xi;<sub>s</sub>)===
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="5">
&nbsp;
  </td>
  <td align="left" colspan="2">
Knowing:  <math>~K_e/K_c</math> and <math>~\rho_e/\rho_0</math> from Step 5 &nbsp; <math>\Rightarrow</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_e \phi^{n_e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0 \biggl(\frac{\rho_e}{\rho_0}\biggr) \phi^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0 \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{3/2}_i \phi^3</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K_e \rho_e^{1+1/n_e} \phi^{n_e + 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{4/3} \biggl(\frac{K_e }{K_c}\biggr) \biggl(\frac{\rho_e}{\rho_0}\biggr)^{4/3} \phi^{4}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{5/3}  \theta^{5/2}_i \phi^{4}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{1/2} \rho_e^{(1-n_e)/(2n_e)} \eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K_c}{\pi G} \biggr]^{1/2} \rho_0^{-1/3} \biggl( \frac{K_e}{K_c}\biggr)^{1/2} \biggl( \frac{\rho_e}{\rho_0} \biggr)^{-1/3} \eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K_c}{\pi G} \biggr]^{1/2} \rho_0^{-1/6} \biggl( \frac{\mu_e}{\mu_c}\biggr)^{-1} \theta_i^{-1/4} \eta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~M_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \biggl[ \frac{(n_e + 1)K_e}{4\pi G} \biggr]^{3/2} \rho_e^{(3-n_e)/(2n_e)} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \biggl[ \frac{K_c}{\pi G} \biggr]^{3/2} \biggl( \frac{K_e}{K_c}\biggr)^{3/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2^4}{\pi} \biggr)^{1/2} \biggl[ \frac{K_c}{G} \biggr]^{3/2} \rho_0^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr)^{-2} \theta_i^{3/4}
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
</tr>
</table>

</div>


Instead of working completely across this table in order to relate the envelope's density, radial coordinate, and mass to properties of the core, it is worth pausing to insert into the leftmost set of relations the expressions for <math>\rho_e</math> and <math>K_e</math> that were [[SSC/Structure/BiPolytropes/Analytic1.53#HighlightedExpressions|derived above]].  In doing this, we obtain, 
<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>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^{-1} \frac{\beta}{(1-\beta)} \cdot \frac{1}{3} a_\mathrm{rad} \biggr] \lambda^3 \phi^3 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{K_e}{\pi G} \biggr]^{1/2} \rho_e^{-1/3} \eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(\pi G)^{1/2}} \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/6} 
\biggl\{ \lambda^3 \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^{-1} \frac{\beta}{(1-\beta)} \cdot \frac{1}{3} a_\mathrm{rad} \biggr] \biggr\}^{-1/3} \eta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(\pi G)^{1/2}} \biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/6} 
\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^2\frac{(1-\beta)^2}{\beta^2} \biggl( \frac{3}{a_\mathrm{rad} }\biggr)^2 \biggr] ^{1/6} \frac{\eta}{\lambda} </math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4 \biggl(  \frac{3}{\pi a_\mathrm{rad}G^3}  \biggr)^{1/2} \biggl(\frac{\Re}{\mu_e}\biggr)^2 \frac{(1-\beta)^{1/2}}{\beta^2}
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>

By comparison, the expressions that {{ Milne30 }} derived for the run of <math>~\rho</math>, <math>~r</math>, and <math>~M_r</math> throughout the envelope are presented in his paper as, respectively, equations (89), (86), and (85).   In an effort to facilitate this comparison, Milne's expressions &#8212; which also specifically apply to the base of the envelope, whose identity is associated with primed variable names in Milne's notation &#8212; are reprinted as extracted images in the following boxed-in table. 

<div align="center">
<table border="2" cellpadding="10">
<tr>
  <th align="center">
Equations extracted<sup>&dagger;</sup> from <br />{{ Milne30figure }}
  </th>
</tr>
<tr>
  <td>
[[File:EnvelopeRelations01.png|500px|center|Milne (1930)]]
  </td>
</tr>
<tr><td align="left">
<sup>&dagger;</sup>Equations displayed here, as a single digital image, with presentation order &amp; layout modified from the original publication.
</td></tr>
</table>
</div>

The agreement between our derivation and Milne's is exact, once it is realized that Milne has used <math>\theta(\xi)</math> to represent the Lane_Emden function for the <math>n_e = 3</math> envelope, whereas we have represented this function by <math>\phi(\eta)</math>; and in place of Milne's coefficient, <math>\lambda_1</math>, we have simply written, <math>\lambda</math>.

=See Also=
<ul>
  <li>[https://ui.adsabs.harvard.edu/abs/1944ApJ...100..343H/abstract M. Hall Harrison (1944, ApJ, 100, 343 - 346)], ''The Generalized Cowling Model'' &#8212; Bibliographic Code: 1944ApJ...100..343H<br />(3<sup>rd</sup> paragraph on p. 343): <font color="darkgreen">"We shall consider a composite model made up of a central core described by the Lane-Emden function of index <math>n=\tfrac{3}{2}</math> and a point-source envelope."</font></li>
  <li>[https://ui.adsabs.harvard.edu/abs/1946ApJ...103..193H/abstract M. Hall Harrison (1946, ApJ, 103, 193 - 206)], ''Stellar Models with Partially Degenerate Isothermal Cores and Point-Source Envelopes'' &#8212; Bibliographic Code: 1946ApJ...103..193H</li>
  <li>[https://ui.adsabs.harvard.edu/abs/1947ApJ...105..322H/abstract M. Hall Harrison (1947, ApJ, 105, 322 - 326)], ''Stellar Models with Isothermal Cores and Point-Source Envelopes'' &#8212; Bibliographic Code: 1947ApJ...105..322H</li>
  <li>[https://ui.adsabs.harvard.edu/abs/1960PASJ...12...21S/abstract K. Suda &amp; Z. Hitotuyanagi (1960, PASJapan, 12, 21 - 27)], ''Stellar Models with Partially Degenerate Isothermal Cores''</li>
</ul>


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