Editing Apps/RotatingPolytropes (section)

====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]."