Editing Apps/RotatingPolytropes (section)

====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,

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  <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,

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<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,

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  <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),
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  <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,
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<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>