Radial profiles of surface density in debris discs

Resolved observations of debris discs can be used to derive radial profiles of azimuthally averaged surface density (ASD), which carries important information about the disc structure even in the presence of non-axisymmetric features and has improved signal-to-noise characteristics relative to the two-dimensional surface brightness measurements. We develop a (semi-)analytical formalism allowing one to relate ASD to the underlying semimajor axis and eccentricity distributions of the debris particles in a straightforward manner. This approach does not involve the distribution of particle apsidal angles, thus simplifying calculations. It is a much faster, more flexible and effective way of calculating ASD than the Monte Carlo sampling of orbital parameters of debris particles. We present explicit analytical results based on this technique for a number of particle eccentricity distributions, including two cases of particular practical importance: a prescribed radial profile of eccentricity, and the Rayleigh distribution of eccentricities. We then show how our framework can be applied to observations of debris discs and rings for retrieving either the semimajor axis distribution or (in some cases) the eccentricity distribution of debris, thus providing direct information about the architecture and dynamical processes operating in debris discs. Our approach also provides a fast and efficient way of forward modelling observations. Applications of this technique to other astrophysical systems, e.g. the nuclear stellar disc in M31 or tenuous planetary rings, are also discussed.