Efficient transit light curves for oblate and rapidly rotating stars

Exoplanets that transit rapidly-rotating stars can be a unique tool to learn about the dynamical history of exoplanet systems photometrically. Two rotational effects on such stars are oblateness and gravity darkening, where the poles are hotter and more luminous than the equator due to latitude-dependent stellar surface gravity. Both of these effects break spherical symmetry and can allow a photometric measurement of the true spin-orbit angle of the system. We create a fast, semi-analytic transit model in the starry package that incorporate these effects. We model gravity darkening using a spherical harmonic expansion of the Von Zeipel Law to arbitrary order. We take into account stellar oblateness by modeling the star as an ellipse in projection. Using Green’s theorem, we write the 2D flux integral as an equivalent 1D integral around the projected boundary of the star and planet. The implementation in starry is 4 orders of magnitude faster and more precise than equivalent numerical methods and well suited to posterior inference. We note its use as a complement to spectroscopic measurements that provide the projected spin-orbit angle and also as a standalone method to constrain spin-orbit angles photometrically for hundreds of such systems in TESS.