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

Abstract

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 semi-analytic 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). 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 ϕ = 108.3 +19.0 −15.4• . 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.

Gravity darkening and tidally perturbed stellar pulsation in the misaligned exoplanet system WASP-33

“…After calculating the Bayesian and Akaike Information Criteria (BIC and AIC), we can infer that the β = 0.25 case is the most probable out of the three, as it is ∆BIC = 75.8, ∆AIC = 98.4, ∆BIC = 10.1, and ∆AIC = 10.1 lower than the β = 0 and λ = −111.59 • cases, respectively. Thus we confirm the presence of gravity darkening in the WASP-33b system first noted by Dholakia et al (2021), and adopt this model.…”

“…There is a 3.1σ difference between our fitted value of u + compared to the same parameter from von Essen et al ( 2020), but we note that the authors of that paper used theoretical limb darkening coefficients. Of more interest is the ratio of the planetary and stellar radii as our value of 0.1098±0.0012 is in good agreement with Dholakia et al (2021) Computed Fourier spectrum (top panel, black), the orbital frequency and the first 47 orbital harmonics (top panel, red dashed lines) that cover the range of frequencies < 40 days −1 , above which no significant peaks are found. The bottom row shows the 3rd, 12th and 25th orbital harmonics (red dashed lines) and the zoomed-in parts of the spectrum that correspond to these frequencies.…”

“…from the 0.1071 ± 0.0002 derived by Herrero et al (2011). Comparing the results of i , λ and ϕ to those from Dholakia et al (2021) (who also fixed the value of the gravity darkening exponent), we find good agreements for the projected and true spin orbit angles, while the stellar inclinations are inconsistent with each other.…”

“…Using the TESS light curve, von Essen et al (2020) improved the precision of the known transit parameters. Dholakia et al (2021) argued that after substracting the pulsation from the TESS light curve, an asymmetry emerges during the transits that is characteristic of a gravity darkened stellar disk.…”

Mapping stellar surfaces III: An Efficient, Scalable, and Open-Source Doppler Imaging Model

“…Many of the Doppler imaging targets are massive stars, which tend to be fast-rotating and therefore oblate. We recently expanded the starry framework to model stellar oblateness Dholakia et al (2021), and will similarly adapt our Doppler imaging framework in an upcoming paper. In the absence of differential rotation and convective blueshift, oblateness can be accounted for by simply weighting the integration limits in Equation (A1) by the semi-minor axis of the elliptical boundary of the star in projection.…”