Editing Appendix/Ramblings/Radiation/InitialTemperatures (section)

===Other Physical Variables===

The ratio of radiation pressure to gas pressure (see the last column of the above table) is calculated via the relation,
<div align="center">
<math>
\frac{1}{\Gamma} = \frac{P_\mathrm{rad}}{P_\mathrm{gas}} = \biggl( \frac{\tilde{a}}{3\tilde{r}} \biggr) \frac{T_\mathrm{code}^3}{\rho_\mathrm{code}} .
</math>
</div>
Also note that,
<div align="center">
<math>
\beta \equiv \frac{P_\mathrm{gas}}{P_\mathrm{total}} = \frac{1}{1+P_\mathrm{rad}/P_\mathrm{gas}} .
</math>
</div>

In order to avoid establishing stellar structures that are convectively unstable, Dominic also needs to choose an ''evolutionary'' ratio of specific heats, <math>\gamma</math>, such that its value is everywhere greater than a critical value, <math>\gamma_c</math>, established at the center of the accretor. From Equation (131) in Chapter II of [[Appendix/References#C67|Chandrasekhar (1967)]] we see that <math>\gamma_c</math> depends on each star's central value of <math>\beta</math> &#8212; that is, it depends on <math>\beta_c</math> &#8212; and on each star's ''structural'' <math>\Gamma_1 \equiv d\ln P/d\ln \rho</math> (which is <math>5/3</math> for our two {{Math/MP_PolytropicIndex}} <math>=3/2</math> polytropic stars) in the following way:
<div align="center">
<math>
\gamma_c = \biggl[ \frac{12(1-\beta_c)(\Gamma_1 - \beta_c) -\beta_c(\Gamma_1 - \beta_c) - (4-3\beta_c)^2}{12(1-\beta_c)(\Gamma_1 - \beta_c)- (4-3\beta_c)^2}  \biggr].
</math>
</div>

Plugging <math>\rho_\mathrm{code}^\mathrm{max}</math> into these expressions lets us tabulate various properties at the center of both stars.

<table align="center" border="1" cellpadding="5" width="75%">
<tr>
  <td align="center" colspan="10">
<b><font color="darkblue">Central Stellar Values</font></b>
  </td>
</tr>
<tr>
  <td colspan="2">
&nbsp;
  </td>
  <td align="center" colspan="3">
Approximations
  </td>
  <td align="center" colspan="5">
From Quartic Solution
  </td>
</tr>

<tr>
  <td align="center">
'''Star'''
  </td>
  <td align="center">
<math>\rho^\mathrm{max}_\mathrm{code}</math>
  </td>
  <td align="center">
<math>T_\mathrm{code}^\mathrm{max}</math> 
  </td>
  <td align="center">
<math>\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr|_c</math> 
  </td>
  <td align="center">
<math>\beta_c</math> 
  </td>

  <td align="center">
<math>T_\mathrm{code}^\mathrm{max}</math> 
  </td>
  <td align="center">
<math>\frac{P_\mathrm{rad}}{P_\mathrm{gas}}\biggr|_c</math> 
  </td>
  <td align="center">
<math>\beta_c</math> 
  </td>
  <td align="center">
<math>\gamma_c</math> 
  </td>
  <td align="center">
<math>T_\mathrm{cgs}^\mathrm{max}</math> 
  </td>
</tr>

<tr>
  <td align="center">
Accretor
  </td>
  <td align="center">
<math>1.0000</math>
  </td>
  <td align="center">
<math>0.5843</math>
  </td>
  <td align="center">
<math>6.65 \times 10^{-3}</math>
  </td>
  <td align="center">
<math>0.99339</math>
  </td>

  <td align="center" colspan="1">
0.5805
  </td>
  <td align="center" colspan="1">
<math>6.522\times 10^{-3}</math>
  </td>
  <td align="center" colspan="1">
0.99352
  </td>
  <td align="center" colspan="1">
1.67765
  </td>
  <td align="center" colspan="1">
<math>9.39\times 10^{7}~\mathrm{K}</math>
  </td>
</tr>

<tr>
  <td align="center">
Donor
  </td>
  <td align="center">
<math>0.6077</math>
  </td>
  <td align="center">
<math>0.3854</math>
  </td>
  <td align="center">
<math>3.14\times 10^{-3}</math>
  </td>
  <td align="center">
<math>0.99687</math>
  </td>

  <td align="center" colspan="1">
0.3842
  </td>
  <td align="center" colspan="1">
<math>3.111\times 10^{-3}</math>
  </td>
  <td align="center" colspan="1">
0.99690
  </td>
  <td align="center" colspan="1">
1.67188
  </td>
  <td align="center" colspan="1">
<math>6.22\times 10^{7}~\mathrm{K}</math>
  </td>
</tr>

</table>

The central values of <math>P_\mathrm{rad}/P_\mathrm{gas}</math> obtained via the quartic solution match exactly the values that Dominic read straight from the rad-hydrocode, namely, <math>6.5\times 10^{-3}</math> (for the accretor) and <math>3.1\times 10^{-3}</math> (for the donor).  (See email from Dominic to Joel dated 8/4/2010.)  In this email, Dominic also stated that he chose <math>\gamma = 1.67114094</math>; I'm not quite sure how he derived this value.

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