Editing Appendix/Ramblings/Photosphere (section)

===Better Temperature Estimate===

Now let's assume that the common envelope puffs up adiabatically, in which case the temperature of the photosphere (and envelope) should be,
<div align="center">
<math>T_{ph} \approx T_{sh} \biggl( \frac{\rho_{ph}} {\rho_{sh}} \biggr)^{\gamma - 1}
\approx T_{sh} \biggl( \frac{R_{ph}} {R_a} \biggr)^{3(1-\gamma)}</math> .
</div>
So, by analogy with the above initial derivation, if we demand that,
<div align="center">
<math>
4\pi R_{ph}^2 \sigma T_{ph}^4 = L_{acc} \, ,
</math>
</div>
then,
<div align="center">
<math>
\biggl[ \frac{R_{ph}}{R_a} \biggr]^2 = \frac{L_{acc}}{\pi a_{rad}c R_a^2 T^4_{sh}} \biggl[\frac{R_{ph}}{R_a} \biggr]^{12(\gamma-1)} 
\approx \biggl[ \frac{20}{\pi^4} \biggr] \biggl[  \frac{\mu_e^2 M_{Ch}}{ m_3 M_a} \biggr]^2 \biggl[ \frac{\dot{M}}{M_a} \cdot \frac{R_a}{c }  \biggr] f^{-4} \biggl[\frac{R_{ph}}{R_a} \biggr]^{12(\gamma-1)}
</math>
</div>
<div align="center">
<math>
\Rightarrow~~~~~ \biggl[ \frac{R_{ph}}{R_a} \biggr]^{2(6\gamma - 7)} \approx 
\biggl[ \frac{20}{\pi^4} \biggr]^{-1} \biggl[  \frac{\mu_e^2 M_{Ch}}{ m_3 M_a} \biggr]^{-2} \biggl[ \frac{\dot{M}}{M_a} \cdot \frac{R_a}{c }  \biggr]^{-1} f^{4}
</math>
</div>
<div align="center">
<math>
\Rightarrow~~~~~ \frac{R_{ph}}{R_a}  \approx \biggl\{
\biggl[ \frac{20}{\pi^4} \biggr]^{-1} \biggl[  \frac{\mu_e^2 M_{Ch}}{ m_3 M_a} \biggr]^{-2} \biggl[ \frac{\dot{M}}{M_a} \cdot \frac{R_a}{c }  \biggr]^{-1} f^{4}
\biggr\}^{1/[2(6\gamma-7)]} \, .
</math>
</div>
Hence, for a <math>\gamma=5/3</math> envelope,
<div align="center">
<math>
\frac{R_{ph}}{R_a}  \approx \biggl\{
\biggl[ \frac{20}{\pi^4} \biggr]^{-1} \biggl[  \frac{\mu_e^2 M_{Ch}}{ m_3 M_a} \biggr]^{-2} \biggl[ \frac{\dot{M}}{M_a} \cdot \frac{R_a}{c }  \biggr]^{-1} f^{4}
\biggr\}^{1/6} \, .
</math>
</div>
Now, based on the accompanying discussion of the [[SSC/Structure/WhiteDwarfs#Chandrasekhar_mass|Chandrasekhar mass]],
<div align="center">
<math>\frac{ m_3 M_\odot}{\mu_e^2 M_{Ch}}=0.353 \, .</math>
</div>
Hence,
<div align="center">
<math>
\frac{R_{ph}}{R_a}  \approx 0.920 \biggl[ \frac{\dot{M}}{M_a} \cdot \frac{R_a}{c }  \biggr]^{-1/6} f^{2/3}  \approx \biggl[ \frac{\dot{M}}{M_a} \cdot \frac{R_a}{c }  \biggr]^{-1/6}  \, .
</math>
</div>
This relation behaves in a very different way from the initially derived relation.  For small mass-transfer rates, this relation predicts that the photospheric radius will be quite large relative to the radius of the accretor.