Rayleigh scattering

About Rayleigh scattering

The following article gives a clear overview on Rayleigh scattering cross sections :

Bodhaine (1999) On Rayleigh Optical Depth Calculations : http://web.gps.caltech.edu/~vijay/Papers/Rayleigh_Scattering/Bodhaine-etal-99.pdf

Have a look especially on equations (2) and (9).

References

LMDZ

LMDZ uses formalism from :

Hansen (1974) Light scattering in planetary atmospheres : https://ui.adsabs.harvard.edu/link_gateway/1974SSRv...16..527H/ADS_PDF

Have a look on equations (2.29) to (2.32).

exo_k

Rayleigh routine in exo_k is avalaible here :

http://perso.astrophy.u-bordeaux.fr/~jleconte/exo_k-doc/_modules/exo_k/rayleigh.html#Rayleigh.sigma_mol

Exo_k uses formalism from :

Caldas (2019) Effects of a fully 3D atmospheric structure on exoplanet transmission spectra: retrieval biases due to day–night temperature gradients : https://hal.archives-ouvertes.fr/hal-02005332/document

Have a look on equation (12) & appendix D

Formalism

We consider a layer .

DPR(K) is the difference of pressure between the two levels that define the layer.

dm is the mass per m2 of the layer

We consider the channel NW

LMDZ formalism

In LMDZ, in optcv.F90 we have :

TRAY(K,NW) = TAURAY(NW) * DPR(K)

exo_k formalism

TRAY \( \displaystyle = \sigma_{exok} dN \) with \( \displaystyle \sigma_{exok} \) the cross section and dN in molecules/m2

which gives : TRAY \( \displaystyle = \sigma_{exok} \frac{dm}{m_{molecule}} \) with dm in kg/m2

and then : TRAY \( \displaystyle = \frac{\sigma_{exok}}{g * m_{molecule}} dP\)

Relations between LMDZ & Exo_k formalisms

LMDZ & exo_k formalism are linked as following \[ \displaystyle \text{TAURAY} = \frac{\sigma_{exok}}{g * m_{molecule}} \]

Be careful with units !!! (cm-1 for wavenumbers in exo_k, microns for wavelengths in LMDZ, not to forget the scalep factor in LMDZ)

To be noticed

TAURAY(NW) is calculated in calc_rayleigh.F90.

It is in fact TAUVAR which calculated, and then averaged by the black body function for each channel to give TAURAY \[ \text{TAURAY(NW)} = \frac{\int_{\lambda' \in \text{channel}} \text{TAUVAR} (\lambda') B_{\lambda} \, \mathrm{d}\lambda'}{\int B_{\lambda} \, \mathrm{d}\lambda'} \]

TAUVAR is cut into two parts : TAUCONSTI et TAUVARI with TAUVAR = TAUCONSTI * TAUVARI

The \( \lambda \) dependence is in the TAUVARI