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

Line-of-sight integration of emissions from planetary and cometary atmospheres is the Abel transform of the emission rate, under the spherical symmetry assumption. Indefinite integrals constructed from the Abel transform integral are useful for implementing remote sensing data analysis methods, such as the numerical inverse Abel transform. We propose analytical expressions obtained by a suitable, non-alternating, series development to compute those indefinite integrals. We establish expressions allowing absolute accuracy control of the convergence of these series and illustrate how this accuracy depends on the number of terms involved in the series computation. We compare the analytical method with numerical computation techniques, which are found to be sufficiently accurate as well. Inverse Abel transform fitting is then tested in order to establish that the expected emission rate profiles can be retrieved from the observation of both planetary and cometary atmospheres. We show that the method is robust, i.e. that it can be applied even when the properties of the observed atmosphere depart from the assumed ones, especially when Tikhonov regularization is included. A first application is conducted over observation of comet 46P/Wirtanen, showing some variability, possibly attributable to an evolution of the contamination by dust and icy grains.