Editing Appendix/Ramblings/Radiation/CodeUnits (section)

===Logic Used by Dominic Marcello===
At our group meeting on 21 July 2010, Dominic explained how he had established the values of various coupling constants in his first, long, <math>q_0 = 0.7</math> simulations.  In place of the physical constants, 
{{Math/C_GravitationalConstant}}, {{Math/C_SpeedOfLight}}, {{Math/C_GasConstant}}, and {{Math/C_RadiationConstant}}, Dominic used the following code-unit
values &#8212; hereafter referred to as '''Case A''':
*<math>\tilde{g} = 1</math>
*<math>\tilde{c} = 198</math>
*<math>\tilde{r} = 0.44</math>
*<math>\tilde{a} = 0.044</math>
This means that any temperature in the simulation that has a value <math>T_\mathrm{code}</math> in code units must represent an actual physical temperature <math>T_\mathrm{cgs}</math> in cgs units (''i.e.,'' measured in Kelvins) of,
<div align="center">
<math>
T_\mathrm{cgs} = \biggl[ \biggl(\frac{c^2}{\Re}\biggr)\biggl(\frac{\tilde{c}^2}{\tilde{r}}\biggr)^{-1} \biggr] T_\mathrm{code} ;
</math>
</div>
any length-scale in the simulation that has a value <math>\ell_\mathrm{code}</math> must represent an actual physical length <math>\ell_\mathrm{cgs}</math> in cgs units of,
<div align="center">
<math>
\ell_\mathrm{cgs} = \biggl[\biggl(\frac{\Re^4}{c^4 G a_\mathrm{rad}}\biggr)\biggl(\frac{\tilde{r}^4}{\tilde{c}^4 \tilde{g}\tilde{a}}\biggr)^{-1}\biggr]^{1/2} \ell_\mathrm{code} ;
</math>
</div>
any time in the simulation that has a value <math>t_\mathrm{code}</math> must represent an actual physical time <math>t_\mathrm{cgs}</math> in cgs units of,
<div align="center">
<math>
t_\mathrm{cgs} = \biggl[\biggl(\frac{\Re^4 }{c^6 G a_\mathrm{rad}}\biggr)\biggl(\frac{\tilde{r}^4}{\tilde{c}^6 \tilde{g}\tilde{a}}\biggr)^{-1}\biggr]^{1/2} t_\mathrm{code} ;
</math>
</div>
and, finally, 
any mass in the simulation that has a value <math>m_\mathrm{code}</math> must represent an actual physical mass <math>m_\mathrm{cgs}</math> in cgs units of,
<div align="center">
<math>
m_\mathrm{cgs} = \biggl[\biggl(\frac{\Re^4}{G^3 a_\mathrm{rad}}\biggr)\biggl(\frac{\tilde{r}^4}{\tilde{g}^3 \tilde{a}}\biggr)^{-1}\biggr]^{1/2} m_\mathrm{code} .
</math>
</div>


Now, the SCF-code-generated polytropic binary that Wes Even gave to Dominic had the following properties, in dimensionless code units:
* <math>[M_\mathrm{total}]_\mathrm{code} = 0.85</math>;
* <math>[R_\mathrm{Accretor}]_\mathrm{code} = 0.4</math>; and
* <math>[P_\mathrm{orbit}]_\mathrm{code} = 31</math>.
According to Dominic's calculations this means that his simulation represents a real binary system with the following properties:
* <math>[M_\mathrm{total}]_\mathrm{cgs} = 0.1 M_\odot</math>;
* <math>[R_\mathrm{Accretor}]_\mathrm{cgs} = 0.56 R_\odot</math>; and
* <math>[P_\mathrm{orbit}]_\mathrm{cgs} = 28~\mathrm{minutes}</math>.
Conversely &#8212; assuming pure helium, that is, a mean molecular weight {{Math/MP_MeanMolecularWeight}} of 2 &#8212; since the Thompson cross-section is <math>[\sigma_T]_\mathrm{cgs} = 0.2~\mathrm{cm}^2~\mathrm{g}^{-1}</math>,  Dominic determined that, in the code, he needed to set the Thompson cross-section value to <math>[\sigma_T]_\mathrm{code} = 8\times 10^{12}</math>.
Finally, Dominic pointed out that the characteristic size of a grid cell in the code is <math>[\Delta z]_\mathrm{code} = 0.025</math>.  Hence, if only the Thompson cross-section is relevant, the mean-free-path of a photon will equal the size of one grid cell if,
<div align="center">
<math>
\biggl[\frac{1}{\sigma_T\rho}\biggr]_\mathrm{code} = [\Delta z]_\mathrm{code}
</math><br/><br />

<math>
\Rightarrow ~~~~~ [\rho]_\mathrm{code} = \biggl[\frac{1}{\sigma_T(\Delta z)}\biggr]_\mathrm{code} = \frac{1}{2\times 10^{11}} .
</math>
</div>