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We derive solutions to transit light curves of exoplanets orbitingrapidly-rotating stars. These stars exhibit significant oblateness and gravitydarkening, a phenomenon where the poles of the star have a higher temperatureand luminosity than the equator. Light curves for exoplanets transiting thesestars can exhibit deviations from those of slowly-rotating stars, evendisplaying significantly asymmetric transits depending on the system'sspin-orbit angle. As such, these phenomena can be used as a protractor tomeasure the spin-orbit alignment of the system. In this paper, we introduce anovel semi-analytic method for generating model light curves forgravity-darkened and oblate stars with transiting exoplanets. We implement themodel within the code package starry and demonstrate several orders ofmagnitude improvement in speed and precision over existing methods. We test themodel on a TESS light curve of WASP-33, whose host star displays rapid rotation($v \sin i_* = 86.4$ km/s). We subtract the host's $\delta$-Scuti pulsationsfrom the light curve, finding an asymmetric transit characteristic of gravitydarkening. We find the projected spin orbit angle is consistent with Dopplertomography and constrain the true spin-orbit angle of the system as$\varphi=108.3^{+19.0}_{-15.4}$~$^{\circ}$. We demonstrate the method's uses inconstraining spin-orbit inclinations of such systems photometrically withposterior inference. Lastly, we note the use of such a method for inferring thedynamical history of thousands of such systems discovered by TESS.

Efficient and precise transit light curves for rapidly-rotating, oblate stars