Rayleigh Scattering of Isolated Species

( Species == ions, atoms, molecules )

• See Polarization for the low wavenumber (frequency / speed of light ) approximation used for Rayleigh scattering.

Scattering is due to the polarization of species. The polarization can be summed from the behavior of individual resonances and damping factors (related to resonance bandwidth), which I have not yet been able to find. For mostly-isolated atoms in high vacuum, Beers line broadening will not be relevant; the bandwidth ~ \large \gamma ~ is related to damping time, TBD

• note: perhaps I can get the relevant numbers from HITRAN, but many of the resonances (especially for tightly bound molecules and deep atomic orbitals) will be far into the ultraviolet, where HITRAN may not go. TBD

The scattering from a single resonator at frequency ~ \large \omega ~ is (from Feynman Lectures on Physics Chapter X page X) proportional to

{ \huge \int } { \Large { \omega^4 \over { ( \omega^2 ~-~ \omega_0^2 )^2 ~+~ \gamma^2 \omega^2 } } d \omega } ~=~~~~ { \Large \omega ~~ + } ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ { { \large \left( 2 \omega_0^4 ~+~ 2 \omega_0^2 \gamma \left( \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } ~-~ 2 \gamma \right) ~+~ \gamma^3 \left( \gamma - \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) \right) ~~ \tan^{-1} \left( { \Large { { \huge \omega } \over { \sqrt{ { \Large { \gamma \over 2 } } \left( \gamma ~-~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) ~-~ \omega_0^2 } } } } \right) } \over { \large \sqrt{ 2 } ~ \gamma ~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } ~ \sqrt{ \gamma ~ \left( \gamma ~-~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) ~-~ 2 \omega_0^2 } } } ~~ {\Large - } { { \large \left( 2 \omega_0^4 ~-~ 2 \omega_0^2 \gamma \left( \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } ~+~ 2 \gamma \right) ~+~ \gamma^3 \left( \gamma + \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) \right) ~~ \tan^{-1} \left( { \Large { { \huge \omega } \over { \sqrt{ { \Large { \gamma \over 2 } } \left( \gamma ~+~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) ~-~ \omega_0^2 } } } } \right) } \over { \large \sqrt{ 2 } ~ \gamma ~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } ~ \sqrt{ \gamma ~ \left( \gamma ~+~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) ~-~ 2 \omega_0^2 } } }

Using Wolfram alpha for the integration, and slightly reordered for visual symmetry.

A frightening number of radicals that could contain negative values, depending on the values of \omega_0 and \gamma ; we do know that \omega_0 > \gamma > 0 , which suggests a heap of complex numbers. I'll need to fix that, perhaps by comparing to a numerical integration around the resonance.

The actual equation will be a long series of many such terms, one per resonance. And it will really be computed with a C program, which iterates over the resonances, then over the frequency bins and values for the average vacuum solar spectrum. Much information missing, many opportunities for mistakes, and a serious lack of empirical data to compare to.