Inversion of occultation observation of a dusty atmosphere using hypergeometric functions.

Occultation of solar radiation by a planetary atmosphere is a very accurate method to obtain high signal vs noise spectral measurement of the properties of the atmospheric gas, not only owing to the overwhelmingly large photon flux from our host star, but also because the method is nearly not dependent on instrument calibration. On the other hand, the method can only be applied near the terminator. Using occultation techniques in other regions of the atmosphere can nevertheless be done using stars as a radiation source, but the instrument then has to be more sensitive to cope with the severely reduced photon flux. The method nevertheless remains independent on the absolute calibration of the instrument.
Occultation observation directly provides the optical thickness (or the extinction coefficient) of the absorbing and scattering constituents when multiple scattering can be safely neglected. Under those conditions, the measurement gives the line-of-sight integrated density of the absorbing and scattering constituents, and simultaneous measurements at several wavelength are then needed to discriminate between the effects of the several species. Retrieval of the vertical density profile of the different constituents requires an inversion method, basically an inverse Abel transform when a spherical (or cylindrical) symmetry assumption can be made.
Efficient inverse Abel transform methods rely on least squares fit techniques taking advantage of easy-to-compute analytical indefinite integrals constructed from the Abel transform integral operator. In the case of a dusty atmosphere, the contribution of dusts to the extinction depends on the properties of the dust grains controlling their scattering cross section, which is generally represented using the so-called alpha parameter appearing as an exponent of the wavelength in the expression of the cross section. As the properties of the dusts vary with altitude, so does the alpha parameter, which severely complicates the computation of the indefinite integrals needed for the inverse Abel transform fitting. We propose a method that allows to express those indefinite integrals using Gauss’s hypergeometric 2F1 function, which can be applied to the observation of the Earth as well as of planet Mars, as it is done by the ESA EXOMARS-NOMAD instrument.