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.

dP is the difference of pressure between the two levels that define the layer.

dN is the number of molecules per m2 & dm is the mass per m2 of the layer

dTau is the optical depth for a given wavelength (or wavenumber)

\(m_{molecule}\) is the mass of one molecule of the considered gas

g is the gravity

sigma_mol is the Rayleigh scattering cross section of the molecule

LMDZ formalism

dTau = scalep * (tauconsti * tauvari) * dP

tauvari = tauvari(wavelength in microns)

scalep = 100, because P is in mBar in optcv.F90

exo_k formalism

dTau = sigma_mol * dN

sigma_mol = sigma_mol(wavenumber in cm-1)

which gives : dTau = sigma_mol \( \displaystyle \frac{dm}{m_{molecule}} \)

and then : dTau \( \displaystyle = \frac{\text{sigma_mol}}{g * m_{molecule}} dP\)

Relations between LMDZ & Exo_k formalisms

LMDZ & exo_k formalism are linked as following \[ \displaystyle \text{(tauconsti * tauvari)} = \frac{\text{sigma_mol}}{g * m_{molecule}} * \text{scalep}\]

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 for a given channel NW.

First 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 (previously mentionned) with TAUVAR = TAUCONSTI * TAUVARI

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