Editing Appendix/Ramblings/Radiation/CodeUnits (section)

==Ratio of Gas Pressure to Radiation Pressure==

Let's define the following pressure ratios:

<div align="center">
<math>
\Gamma \equiv \frac{P_\mathrm{gas}}{P_\mathrm{rad}} = \frac{3 (\Re/\bar\mu)}{a_\mathrm{rad}} \biggl[ \frac{\rho}{T^3} \biggr]_\mathrm{cgs} ,
</math>
</div>
and,
<div align="center">
<math>
\beta \equiv \frac{P_\mathrm{gas}}{(P_\mathrm{gas}+P_\mathrm{rad})} = \frac{\Gamma}{1 + \Gamma} .
</math>
</div>

Following Dominic's definition of code units, above:  <math>T^3</math> should be normalized by <math>[ c^6/(\Re/\bar\mu )^3]</math>; the mass density should be normalized by the quantity <math>[a_\mathrm{rad} c^6/(\Re/\bar\mu)^4]</math>; and <math>(\Re/\bar\mu)</math> should be replaced by <math>\tilde{r}</math>.  Hence, the ratio of gas pressure to radiation pressure can be written as,
<div align="center">
<math>
\Gamma = \frac{3(\Re/\bar\mu)}{a_\mathrm{rad}} \biggl[\frac{\rho}{T^3}\biggr]_\mathrm{code} \biggl[\frac{c^6 a_\mathrm{rad}}{(\Re/\bar\mu)^4}
\biggl( \frac{\tilde{r}^4}{\tilde{c}^6 \tilde{a}} \biggr)\biggr] \biggl[\frac{(\Re/\bar\mu)^3}{c^6}
\biggl( \frac{\tilde{c}^6 }{\tilde{r}^3} \biggr)\biggr]
= 3\biggl[ \frac{\rho_\mathrm{code}}{T_\mathrm{code}^3} \biggr] \frac{\tilde{r}}{\tilde{a}} .
</math>
</div>

We might, in addition, ask what the central temperature is in an <math>n=3/2</math> polytrope.  Well, if the gas pressure dominates (''i.e.,'' if <math>\Gamma \gg 1</math>),
<div align="center">
<math>
P_\mathrm{cgs} \approx \frac{\Re}{\bar\mu} \rho_\mathrm{cgs} T_\mathrm{cgs};
</math>
</div>
and in code units,
<div align="center">
<math>
P_\mathrm{code} = \kappa_\mathrm{code} \rho_\mathrm{code}^{5/3} .
</math>
</div>

Hence,

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
T_\mathrm{code}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} T_\mathrm{cgs} \approx \biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} \frac{1}{(\Re/\bar\mu)} \biggl(\frac{P_\mathrm{cgs}}{\rho_\mathrm{cgs}}\biggr)

</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\Re/\bar\mu)} \biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} \biggl(\frac{P_\mathrm{cgs}/P_\mathrm{code}}{\rho_\mathrm{cgs}/\rho_\mathrm{code}}\biggr) \biggl(\frac{P_\mathrm{code}}{\rho_\mathrm{code}}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\Re/\bar\mu)} \biggl( \frac{T_\mathrm{cgs}}{T_\mathrm{code}} \biggr)^{-1} \biggl( \frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^{2}\biggl( \frac{t_\mathrm{cgs}}{t_\mathrm{code}} \biggr)^{-2} \kappa_\mathrm{code}\rho_\mathrm{code}^{2/3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{\tilde{r}} ~\kappa_\mathrm{code}\rho_\mathrm{code}^{2/3} .
</math>
  </td>
</tr>

</table>

This, in turn, tells us that at the center of a polytropic star,
<div align="center">
<math>
\Gamma \approx \frac{3\tilde{r}^4}{\tilde{a}}~\kappa_\mathrm{code}^{-3} \rho_\mathrm{code}^{-1} .
</math>
</div>

This derivation will need to be modified to handle the more general case when <math>\Gamma</math> is not necessarily large.