Analytic methods for the Abel transform of exponential functions describing planetary and cometary atmospheres.

Remote sensing of planetary and cometary atmosphere is one of the most important source of data and knowledge of the gas layers surrounding the celestial objects of our solar system, including our own planet. Most of the instruments used up to now and that will be used in a near future study the emission of radiations directly produced by the atmosphere. Under optically thin conditions, this observation method provides the local volume emission rate (VER) originating from the atmosphere, integrated along the full line of sight (l.o.s.) of the instrument. Under a spherical or cylindrical symmetry assumption, the l.o.s. integration of the VER takes the form of the Abel transform of the vertical VER profile. The simplest analytical functions representing VER profiles in real planetary and cometary atmosphere include an exponential function of the altitude (or radial distance), giving the isothermal profile for a planet and the Haser model for a coma. The Abel transform of these functions can be computed analytically using combinations of special functions. Retrieving the vertical (radial) profile of the VER does however require to invert the observed Abel transform to account for possible departures from these idealized analytical expressions, so that indefinite integrals defined from the Abel integral (which we will call indefinite Abel transforms) are needed (or numerical integrations need to be performed).

In this study, we present a new method to produce a workable series development allowing accurate computation of the indefinite Abel transforms that appear in the study of optically thin emissions of planetary and cometary atmospheres. Indeed, taking the Taylor series development of the exponential function to reduce the problem to a series of indefinite Abel transforms of polynomial functions (which can be carried analytically) does not work. It leads to the computation of the difference of large, nearly equal numbers, which cannot be done accurately. Our method rather relies on an appropriate series development of the Jacobian of the Abel transform. We show that the computation can be done reliably up to near machine precision, and that accuracy control can be enforced for tailored applications. Possible applications are considered, that include the study of comas and of the upper atmosphere of Mars and the Earth