Editing Appendix/Ramblings/Radiation/InitialTemperatures (section)

===Temperature===
Currently Dominic is ignoring the effects of electron degeneracy pressure, so in applying the above <math>T(\rho)</math> solution to his models we can set <math>F(\chi) = 0</math> in the definition of <math>a_0</math>. Doing this, we find that,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\theta 
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{3 \tilde{r}\rho_\mathrm{code}}{4 \tilde{a}} \biggr]^{1/3} ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\lambda
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl(\frac{256~ \tilde{a} }{81~ \tilde{r}^4} \biggr)K_\mathrm{code}^3 \rho_\mathrm{code}  \biggr]^{1/3} .
</math>
  </td>
</tr>

</table>

For purposes of discussion, we will define <math>\rho_1</math> as the value of <math>\rho_\mathrm{code}</math> when <math>\lambda = 1</math>, that is,
<div align="center">
<math>\rho_1 \equiv  \frac{81~ \tilde{r}^4}{256~ \tilde{a} K_\mathrm{code}^3} .
</math>
</div>

As the [[SR/Ptot_QuarticSolution#Limiting_Regimes|accompanying discussion]] points out, the limiting behavior of the quartic solution is as follows:  

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
For <math>\rho_\mathrm{code} \ll \rho_1 </math>
  </td>
  <td align="right">
&nbsp; &nbsp; <math>\cdots </math> &nbsp; &nbsp;
  </td>
  <td align="left">
<math>T_\mathrm{code} \approx \frac{a_0}{a_1}</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>\biggl( \frac{K_\mathrm{code}}{\tilde{r}} \biggr) \rho_\mathrm{code}^{2/3}</math>
  </td>
</tr>

<tr>
  <td align="right">
For <math>\rho_\mathrm{code} \gg \rho_1 </math>
  </td>
  <td align="right">
&nbsp; &nbsp; <math>\cdots </math> &nbsp; &nbsp;
  </td>
  <td align="left">
<math>T_\mathrm{code} \approx \biggl( \frac{a_0}{a_4} \biggr)^{1/4}</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{3K_\mathrm{code}}{\tilde{a}} \biggr)\rho_\mathrm{code}^{5/3} \biggr]^{1/4}
</math>
  </td>
</tr>
</table>

Now let's plug in numerical values for the two stars in Dominic's ''Case A,'' <math>q_0 = 0.7</math> model evolution, as drawn from the [[Appendix/Ramblings/Radiation/SummaryScalings#TableProperties|accompanying ''properties'' table]].

<table align="center" border="1" cellpadding="5" width="60%">
<tr>
  <td align="center" colspan="7">
<font color="red">Case A:</font> &nbsp; <math>\tilde{g} = 1</math>; <math>\tilde{c} = 198</math>; <math>\tilde{r} = 0.44</math>; <math>\tilde{a} = 0.044</math>; <math>\bar\mu = 4/3</math>;
  </td>
</tr>

<tr>
  <td align="center">
'''Star'''
  </td>
  <td align="center">
<math>K_\mathrm{code}</math>
  </td>
  <td align="center">
<math>\theta</math>
  </td>

  <td align="center">
<math>\lambda</math>
  </td>
  <td align="center">
<math>\rho_1</math>
  </td>
  <td align="center">
<math>T_\mathrm{code}</math> (for <math>\rho_\mathrm{code} \ll \rho_1</math>)
  </td>
  <td align="center">
<math>P_\mathrm{rad}/P_\mathrm{gas}</math> (for <math>\rho_\mathrm{code} \ll \rho_1</math>)
  </td>
</tr>

<tr>
  <td align="center">
Accretor
  </td>
  <td align="center">
<math>0.2571</math>
  </td>
  <td align="center">
<math>1.957 \rho_\mathrm{code}^{1/3}</math>
  </td>

  <td align="center">
<math>0.3980\rho_\mathrm{code}^{1/3}</math>
  </td>
  <td align="center">
<math>15.86</math>
  </td>
  <td align="center">
<math>0.5843~\rho_\mathrm{code}^{2/3}</math>
  </td>
  <td align="center">
<math>6.65\times 10^{-3}~\rho_\mathrm{code}</math>
  </td>
</tr>

<tr>
  <td align="center">
Donor
  </td>
  <td align="center">
<math>0.2364</math>
  </td>
  <td align="center">
<math>1.957 \rho_\mathrm{code}^{1/3}</math>
  </td>

  <td align="center">
<math>0.3660\rho_\mathrm{code}^{1/3}</math>
  </td>
  <td align="center">
<math>20.4</math>
  </td>
  <td align="center">
<math>0.5373~\rho_\mathrm{code}^{2/3}</math>
  </td>
  <td align="center">
<math>5.17\times 10^{-3}~\rho_\mathrm{code}</math>
  </td>
</tr>

</table>