Efficient and Precise Transit Light Curves for Rapidly Rotating, Oblate Stars

We derive solutions to transit light curves of exoplanets orbiting rapidly rotating stars. These stars exhibit significant oblateness and gravity darkening, a phenomenon where the poles of the star have a higher temperature and luminosity than the equator. Light curves for exoplanets transiting these stars can exhibit deviations from those of slowly rotating stars, even displaying significantly asymmetric transits depending on the system's spin-orbit angle. As such, these phenomena can be used as a protractor to measure the spin-orbit alignment of the system. In this paper, we introduce a novel semianalytic method for generating model light curves for gravity-darkened and oblate stars with transiting exoplanets. We implement the model within the code package starry and demonstrate several orders of magnitude improvement in speed and precision over existing methods. We test the model on a TESS light curve of WASP-33, whose host star displays rapid rotation ( $v\sin {i}_{* }=86.4$ km s^-1). We subtract the host's δ-Scuti pulsations from the light curve, finding an asymmetric transit characteristic of gravity darkening. We find the projected spin-orbit angle is consistent with Doppler tomography and constrain the true spin-orbit angle of the system as $\varphi ={108.3}_{-15.4}^{+19.0}$ °. We demonstrate the method's uses in constraining spin-orbit inclinations of such systems photometrically with posterior inference. Lastly, we note the use of such a method for inferring the dynamical history of thousands of such systems discovered by TESS (https://github.com/rodluger/starry, https://rodluger.github.io/starry, https://github.com/shashankdholakia/gravity-dark).