A coordinate-independent characterization of a black hole shadow

A large international effort is under way to assess the presence of a shadow in the radio emission from the compact source at the centre of our Galaxy, Sagittarius A$^*$ (Sgr A$^*$). If detected, this shadow would provide the first direct evidence of the existence of black holes and that Sgr A$^*$ is a supermassive black hole. In addition, the shape of the shadow could be used to learn about extreme gravity near the event horizon and to determine which theory of gravity better describes the observations. The mathematical description of the shadow has so far used a number of simplifying assumptions that are unlikely to be met by the real observational data. We here provide a general formalism to describe the shadow as an arbitrary polar curve expressed in terms of a Legendre expansion. Our formalism does not presume any knowledge of the properties of the shadow, e.g. the location of its centre, and offers a number of routes to characterize the distortions of the curve with respect to reference circles. These distortions can be implemented in a coordinate independent manner by different teams analysing the same data. We show that the new formalism provides an accurate and robust description of noisy observational data, with smaller error variances when compared to previous approaches for the measurement of the distortion.