Exocomet Orbit Fitting: Accelerating Coma Absorption During Transits of $\beta$ Pictoris

Comets are a remarkable feature in our night sky, visible on their passage through the inner Solar system as the Sun's energy sublimates ices and liberates surface material, generating beautiful comae, dust, and ion tails. Comets are also thought to orbit other stars, and are the most promising interpretation of sporadic absorption features (i.e. transits) seen in spectra of stars such as $\beta$ Pictoris and 49 Ceti. These `exocomets' are thought to form and evolve in the same way as in the Solar system, and as in the Solar system we may gain insight into their origins by deriving their orbits. In the case of $\beta$ Pictoris, orbits have been estimated indirectly, using the radial velocity of the absorption features coupled with a physical evaporation model to estimate the stellocentric distance at transit $d_{\rm tr}$. Here, we note that the inferred $d_{\rm tr}$ imply that some absorption signatures should accelerate over several hours, and show that this acceleration is indeed seen in HARPS spectra. This new constraint means that orbital characteristics can be obtained directly, and the pericentre distance and longitude constrained when parabolic orbits are assumed. The results from fitting orbits to 12 accelerating features, and a handful of non-accelerating ones, are in broad agreement with previous estimates based on an evaporation model, thereby providing some validation of the exocomet hypothesis. A prediction of the evaporation model, that coma absorption is deeper for more distant transits, is also seen here.