Editing Apps/RotatingPolytropes (section)

====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.