Measurement of the Spin-Orbit Misalignment of KOI-13.01 from Its Gravity-darkened Kepler Transit Lightcurve

We model the asymmetry of the KOI-13.01 transit lightcurve assuming a gravity-darkened rapidly rotating host star in order to constrain the system's spin-orbit alignment and transit parameters. We find that our model can reproduce the Kepler lightcurve for KOI-13.01 with a sky-projected alignment of λ = 23° ± 4° and with the star's north pole tilted away from the observer by 48° ± 4° (assuming M_star = 2.05 M_sun). With both these determinations, we calculate that the net misalignment between this planet's orbit normal and its star's rotational pole is 56° ± 4°. Degeneracies in our geometric interpretation also allow a retrograde spin-orbit angle of 124° ± 4°. This is the first spin-orbit measurement to come from gravity darkening and is one of only a few measurements of the full (not just the sky-projected) spin-orbit misalignment of an extrasolar planet. We also measure accurate transit parameters incorporating stellar oblateness and gravity darkening: R_star = 1.756 ± 0.014 R_sun, R_p = 1.445 ± 0.016 R_Jup, and i = 85.9 ± 0.4 deg. The new lower planetary radius falls within the planetary mass regime for plausible interior models for the transiting body. A simple initial calculation shows that KOI-13.01's circular orbit is apparently inconsistent with the Kozai mechanism having driven its spin-orbit misalignment; planet-planet scattering and stellar spin migration remain viable mechanisms. Future Kepler data will improve the precision of the KOI-13.01 transit lightcurve, allowing more precise determination of transit parameters and the opportunity to use the Photometric Rossiter-McLaughlin effect to resolve the prograde/retrograde orbit determination degeneracy.