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Comets are a remarkable feature in our night sky, visible on their passagethrough the inner Solar system as the Sun's energy sublimates ices andliberates surface material, generating beautiful comae, dust, and ion tails.Comets are also thought to orbit other stars, and are the most promisinginterpretation of sporadic absorption features (i.e. transits) seen in spectraof stars such as $\beta$ Pictoris and 49 Ceti. These `exocomets' are thought toform and evolve in the same way as in the Solar system, and as in the Solarsystem we may gain insight into their origins by deriving their orbits. In thecase of $\beta$ Pictoris, orbits have been estimated indirectly, using theradial velocity of the absorption features coupled with a physical evaporationmodel to estimate the stellocentric distance at transit $d_{\rm tr}$. Here, wenote that the inferred $d_{\rm tr}$ imply that some absorption signaturesshould accelerate over several hours, and show that this acceleration is indeedseen in HARPS spectra. This new constraint means that orbital characteristicscan be obtained directly, and the pericentre distance and longitude constrainedwhen parabolic orbits are assumed. The results from fitting orbits to 12accelerating features, and a handful of non-accelerating ones, are in broadagreement with previous estimates based on an evaporation model, therebyproviding some validation of the exocomet hypothesis. A prediction of theevaporation model, that coma absorption is deeper for more distant transits, isalso seen here.

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