Editing Appendix/Ramblings/Radiation/CodeUnits (section)

==Opacities==
How should an opacity coefficient be introduced into Dominic's rad-hydrocode?  Let's examine the simplest case of free-free absorption, <math>\kappa_\mathrm{T}</math> (''i.e.,'' [http://en.wikipedia.org/wiki/Thomson_scattering Thompson scattering]):
<div align="center">
<math>
\kappa_\mathrm{T}|_\mathrm{cgs} = \frac{\sigma_\mathrm{T}}{m_p} \biggl[\frac{1}{2}(1+X) \biggl(\frac{m_p}{m_u}\biggr)\biggr] = 0.2003101 (1+X)~\mathrm{cm}^2~\mathrm{g}^{-1},
</math>
</div>
where, {{Math/C_ThompsonCrossSection}}, {{Math/C_AtomicMassUnit}}, and {{Math/C_ProtonMass}} are all physical constants defined in an [[Appendix/VariablesTemplates|accompanying appendix]].  Therefore, for '''Case A''' (which assumes pure helium, so <math>X = 0)</math>, the value of this free-free (Thompson) opacity in code units is,
<div align="center">
<math>\kappa_\mathrm{T}|_\mathrm{code} = 0.2003101~\mathrm{cm}^2~\mathrm{g}^{-1} \biggl[ \frac{\kappa_\mathrm{cgs}}{\kappa_\mathrm{code}} \biggr]^{-1} = 8.434\times 10^{12}</math> .
</div>

When Thompson scattering dominates the opacity, the mean-free-path of a photon is,
<div align="center">
<math>
\ell_\mathrm{mfp} = \frac{1}{\kappa_\mathrm{T}\rho}.
</math>
</div>
This means that, in Dominic's rad-hydrocode, <math>\ell_\mathrm{mfp}</math> will be less than or equal to the size of one radial grid zone, <math>(\Delta R)_\mathrm{Nic\_code}</math>, whenever,
<div align="center">
<math>
[\kappa_\mathrm{T}]_\mathrm{code} ~\rho_\mathrm{code} \ge \frac{1}{(\Delta R)_\mathrm{Nic\_code}} = \frac{128}{\pi}
</math>
<br />&nbsp; <br />
<math>
\Rightarrow ~~~~~\rho_\mathrm{code} \ge 4.83\times 10^{-12} .
</math>
</div>

It is perhaps more instructive to write this last expression in a form that will permit us to determine how this threshold value of <math>\rho_\mathrm{code}</math> depends on the chosen set of scaling parameters.  Specifically, we can write,
<div align="center">
<math>
\rho_\mathrm{code} \ge \rho_\mathrm{threshold} \equiv \frac{128}{\pi (0.200 ~\mathrm{cm}^2~\mathrm{g}^{-1}) } \biggl[ \biggl(\frac{\ell_\mathrm{cgs}}{\ell_\mathrm{code}} \biggr)^2 \biggl(\frac{m_\mathrm{cgs}}{m_\mathrm{code}} \biggr)^{-1} \biggr] = 5.164\times 10^{-21} \biggl[ \frac{\tilde{c}^4 \tilde{a}^{1/2}}{\tilde{r}^2 \bar{\mu}^2 \tilde{g}^{1/2}} \biggr] .
</math>
</div>
To check this relation, note that when '''Case A''' parameter values are used, the combination of factors inside the last set of square brackets gives <math>9.367\times 10^{8}</math>, which produces the same value for <math>\rho_\mathrm{threshold}</math> (in code units) as before.