Editing Apps/RotatingPolytropes (section)

===Milne's Additional Specificity===

====Selected Equation of State====

In his examination of the effects of (uniform) rotation on the equilibrium structure of an otherwise spherically symmetric star, [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)] focused on models in which the pressure,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_\mathrm{gas} + P_\mathrm{rad} \, .</math>
  </td>
</tr>
</table>

When a dimensionless parameter, <math>~\beta</math>, is used to quantify the ratio of the gas pressure to the total pressure &#8212; that is, if we set

<div align="center">
<math>~\beta \equiv \frac{P_\mathrm{gas}}{P_\mathrm{tot}} ~~~~\Rightarrow ~~~~ \frac{P_\mathrm{rad}}{P_\mathrm{tot}} = (1-\beta) \, ,</math>
</div>

then, as we have [[SSC/Structure/BiPolytropes/Analytic1.5_3#Envelope|detailed in our separate discussion]] of [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne's (1930)] early work on bipolytropic stellar models, the pressure-density relation and the temperature-density relation become, respectively,
<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>~ 
\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>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{T^3}{\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr] \, .</math>
  </td>
</tr>
</table>

Now, when building a realistic stellar model, one must expect that, in general, the parameter <math>~\beta</math> will vary with position throughout the model.  But ''if'' the assumption is made that <math>~\beta</math> has the same value throughout the equilibrium configuration, then  we are effectively adopting a polytropic equation of state with,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~K</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>

With this realization, [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar's (1933)] work should be considered a ''generalization'' of [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne's (1923)] modeling effort.  Also, the results of the latter's work should match Chandrasekhar's results for the specific case of a rotating <math>~n=3</math> polytrope.

====Radiative Equilibrium====

[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)] made an effort to ensure that his equilibrium models were not only in hydrostatic balance but that they also were in "radiative equilibrium;"  see especially his &sect;II.9.  Drawing on ''[[PGE/FirstLawOfThermodynamics#Example_B|Example B]]'' from our introductory discussion of [[PGE/FirstLawOfThermodynamics#Nonadiabatic_Environments|nonadiabatic environments]], Milne accomplished this by, effectively, adopting the steady-state specific-entropy expression,

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

<tr>
  <td align="right">
<math>~\rho T \cancelto{0}{\frac{ds_\mathrm{tot}}{dt}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho \epsilon_\mathrm{nuc} - \nabla \cdot \vec{F}_\mathrm{rad} \, .
</math>
  </td>
</tr>
</table>

In this expression, <math>~\epsilon_\mathrm{nuc}(\rho,T)</math> specifies the rate at which (specific) energy is released via thermonuclear reactions, and

<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>
</tr>
</table>

Given that the energy per unit volume in the radiation field is, <math>~E_\mathrm{rad} = a_\mathrm{rad} T^4</math>, this "radiative equilibrium" condition may be rewritten as,

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

<tr>
  <td align="right">
<math>~
\nabla \cdot \biggl[ \frac{1}{3\rho\kappa_R} \nabla E_\mathrm{rad} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\rho \epsilon_\mathrm{nuc} }{c} \, .</math>
  </td>
</tr>
</table>
This is identical to equation (7) of [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)] except:  (a) His divergence and gradient operators appear as they would in Cartesian coordinates; and (b) an extra factor of <math>~4\pi</math> appears in the term on the right-hand side of his expression.  We attribute the extra factor of <math>~4\pi</math> to slightly different definitions of the energy derived from nuclear reactions; specifically, we suspect that, <math>~\epsilon_\mathrm{nuc} = 4\pi \epsilon_\mathrm{Milne}</math>, because immediately following his equation (5) &#8212; near the top of his p. 123 &#8212; we find the sentence &hellip; "<font color="green">Now suppose that <math>~4\pi\epsilon</math> is the energy evolved per unit mass per second at</font> [a given location]."

====Compare Radiative Equilibrium with Mechanical Equilibrium====

Given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tfrac{1}{3} E_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_\mathrm{rad} = (1-\beta) P_\mathrm{tot} \, ,</math>
  </td>
</tr>
</table>
the radiative equilibrium condition can be rewritten as,

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

<tr>
  <td align="right">
<math>~
\nabla \cdot \biggl\{ \frac{1}{\rho\kappa_R} \nabla \biggl[(1-\beta)P_\mathrm{tot}\biggr] \biggr\} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\rho \epsilon_\mathrm{nuc} }{c} \, .</math>
  </td>
</tr>
</table>

If we now express the differential operators in terms of [[AxisymmetricConfigurations/PGE#Spherical_Coordinate_Base|spherical coordinates]] and (for the time being) assume that <math>~\kappa_R</math> and <math>~\beta</math> are both independent of position, this becomes,

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

<tr>
  <td align="right">
<math>~- \frac{\rho \epsilon_\mathrm{nuc}\kappa_R }{c~(1-\beta)} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\nabla \cdot \biggl\{  \biggl[{\hat{e}}_r \biggl[ \frac{1}{\rho } \frac{\partial P_\mathrm{tot}}{\partial r}\biggr] + {\hat{e}}_\theta \biggl[\frac{1}{\rho r} \frac{\partial P_\mathrm{tot}}{\partial \theta}\biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{\rho } \frac{\partial P_\mathrm{tot}}{\partial r}\biggr] 
+ \frac{1}{r\sin\theta} \frac{\partial}{\partial\theta}\biggl[\frac{\sin\theta}{\rho r} \frac{\partial P_\mathrm{tot}}{\partial \theta}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{\rho } \frac{\partial P_\mathrm{tot}}{\partial r}\biggr] 
+ \frac{1}{r} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)}{\rho r} \frac{\partial P_\mathrm{tot}}{\partial \mu}\biggr] \, .
</math>
  </td>
</tr>
</table>

Next, following [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne's (1923)] lead, let's assume that the pressure, <math>~P</math>, that appears in Chandrasekhar's "fundamental equation" is only slightly different from <math>~P_\mathrm{tot}</math> &#8212; that is, let's write,

<div align="center">
<math>~P = P_\mathrm{tot} + \delta P \, ,</math>
</div>

then subtract the derived radiative equilibrium relation from Chandrasekhar's "fundamental equation."  Doing this, we obtain,

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

<tr>
  <td align="right">
<math>~2\omega_0^2 -4\pi G\rho + \frac{\rho \epsilon_\mathrm{nuc}\kappa_R }{c~(1-\beta)} </math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial (\delta P)}{\partial r} \biggr]  
+ \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial (\delta P)}{\partial\mu}  \biggr] \, .
</math>
  </td>
</tr>
</table>
Notice that, if we specifically choose the value of <math>~\beta</math> such that (see [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne's] &sect;I.6),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{\kappa_R (\epsilon_\mathrm{nuc}/4\pi)}{c~G} \, ,</math>
  </td>
</tr>
</table>
then the left-hand side of this relation simplifies considerably.  Specifically, we end up with,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial U}{\partial r} \biggr]  
+ \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial U}{\partial\mu}  \biggr] \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], p. 125, Eq. (20)
  </td>
</tr>
</table>
where we have adopted the variable notation used by [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], viz., <math>~\delta P \rightarrow \omega_0^2 U \, .</math>  And, simultaneously,  the condition for radiative equilibrium takes the form,

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

<tr>
  <td align="right">
<math>~- \frac{\rho \epsilon_\mathrm{nuc}\kappa_R }{c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{3\rho } \frac{\partial E_\mathrm{rad}}{\partial r}\biggr] 
+ \frac{1}{r} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)}{3 \rho r} \frac{\partial E_\mathrm{rad}}{\partial \mu}\biggr] 
</math>
  </td>
  <td align="center">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ - 4\pi G \rho (1-\beta) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{3\rho } \frac{\partial (a_\mathrm{rad} T^4)}{\partial r}\biggr] 
+ \frac{1}{r} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)}{3 \rho r} \frac{\partial (a_\mathrm{rad} T^4)}{\partial \mu}\biggr] 
</math>
  </td>
  <td align="center">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ - \frac{3\pi G \rho (1-\beta)}{a_\mathrm{rad} } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2 T^3}{\rho } \frac{\partial T}{\partial r}\biggr] 
+ \frac{1}{r^2} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)T^3}{ \rho} \frac{\partial T}{\partial \mu}\biggr] \, .
</math>
  </td>
  <td align="right">
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 25 July 2019:  Milne's Equation (19) does not include the factor of a_rad (the radiation constant) that is shown here in the denominator of the left-hand-side; this appears to be a typesetting error, as Milne's expression is even dimensionally incorrect without this factor.]]
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], p. 125, Eq. (19)
  </td>
  <td align="center">&nbsp;</td>
</tr>
</table>