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 $φ=108.3^{+19.0}_{-15.4}$~$^{\circ}$. 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.
Rapid planetary rotation can cause the equilibrium shape of a planet to be oblate. While planetary oblateness has mostly been probed by examining the subtle ingress and egress features in photometric transit light curves, we investigate the effect of oblateness on the spectroscopic Rossiter-McLaughlin (RM) signals. We found that a giant planet, with planet-to-star radius ratio of 0.15 and Saturn-like oblateness of 0.098, can cause spectroscopic signatures with amplitudes up to 1.1 ms$^{-1}$ which is detectable by high-precision spectrographs such as ESPRESSO. We also found that the spectroscop…
A General, Differentiable Transit Model for Ellipsoidal Occulters: Derivation, Application, and Forecast of Planetary Oblateness and Obliquity Constraints with JWST
Knowledge of an exoplanet's oblateness and obliquity would give clues about its formation and internal structure. In principle, a light curve of a transiting planet bears information about the planet's shape, but previous work has shown that the oblateness-induced signal will be extremely difficult to detect. Here we investigate the potentially larger signals due to planetary spin precession. The most readily detectable effects are transit depth variations (T$δ$V) in a sequence of light curves. For a planet as oblate as Jupiter or Saturn, the transit depth will undergo fractional variations of…
We use Kepler short cadence light curves to constrain the oblateness of planet candidates in the Kepler sample. The transits of rapidly rotating planets that are deformed in shape will lead to distortions in the ingress and egress of their light curves. We report the first tentative detection of an oblate planet outside of the solar system, measuring an oblateness of $0.22 \pm 0.11$ for the 18 $M_J$ mass brown dwarf Kepler 39b (KOI-423.01). We also provide constraints on the oblateness of the planets (candidates) HAT-P-7b, KOI-686.01, and KOI-197.01 to be < 0.067, < 0.251, and < 0.186, respect…
When studying transiting exoplanets it is common to assume a spherical planet shape. However short rotational periods can cause a planet to bulge at its equator, as is the case with Saturn whose equatorial radius is almost 10% larger than its polar radius. As a new generation of instruments comes online, it is important to continually assess the underlying assumptions of models to ensure robust and accurate inferences. We analyze bulk samples of known transiting planets and calculate their expected signal strength if they were to be oblate. We find that for noise levels below 100ppm, as many a…
We show that the gas giant exoplanet HD 189733b is less oblate than Saturn, based on Spitzer Space Telescope photometry of seven transits. The observable manifestations of oblatenesswould have been slight anomalies during the ingress and egress phases, as well as variations in the transit depth due to spin precession. Our nondetection of these effects gives the first empirical constraints on the shape of an exoplanet. The results are consistent with the theoretical expectation that the planetary rotation period and orbital period are synchronized, in which case the oblateness would be an order…
Main-sequence stars earlier than spectral type ~F6 or so are expected to rotate rapidly due to their radiative exteriors. This rapid rotation leads to an oblate stellar figure. It also induces the photosphere to be hotter (by up to several thousand Kelvin) at the pole than at the equator as a result of a process called gravity darkening that was first predicted by von Zeipel (1924). Transits of extrasolar planets across such a non-uniform, oblate disk yield unusual and distinctive lightcurves that can be used to determine the relative alignment of the stellar rotation pole and the planet orbit…
Several processes can cause the shape of an extrasolar giant planet's shadow, as viewed in transit, to depart from circular. In addition to rotational effects, cloud formation, non-homogenous haze production and movement, and dynamical effects (winds) could also be important. When such a planet transits its host star as seen from Earth, the asphericity will introduce a deviation in the transit lightcurve relative to the transit of a perfectly spherical (or perfectly oblate) planet. We develop a theoretical framework to interpret planet shapes. We then generate predictions for transiting planet…
A couple of dozen Earth-like planets orbiting M dwarfs have been discovered so far. Some of them have attracted interest because of their potential long-term habitability; such a possibility is currently vigorously debated in the literature. I show that post-Keplerian (pK) orbit precessions may impact the habitability of a fictitious telluric planet orbiting an oblate late-type M dwarf of spectral class M9V with $M_\star=0.08\,M_\odot$ at $a=0.02\,\mathrm{au}$, corresponding to an orbital period $P_\mathrm{b}\simeq 4\,\mathrm{d}$, inducing long-term variations of the planetary obliquity $\vare…
Kepler-13Ab (KOI-13) is an exoplanet orbiting a rapidly rotating A-type star. The system shows a significant spin-orbit misalignment and a changing transit duration most probably caused by the precession of the orbit. Here we present a self-consistent analysis of the system combining {\it Kepler} and TESS observations. We model the light curves asssuming a planet transits a rotating oblate star which has a strong surface temperature gradient due to rotation-induced gravity darkening. The transit chord moves slowly as an emergent feature of orbital precession excited by the oblate star with a d…
While scattering of light by atoms and molecules yields large amount of polarization at the B-band of both T- and L-dwarfs, scattering by dust grains in cloudy atmosphere of L-dwarfs gives rise to significant polarization at the far-optical and infra-red wavelengths where these objects are much brighter. However, the observable disk averaged polarization should be zero if the clouds are uniformly distributed and the object is spherically symmetric. Therefore, in order to explain the observed large polarization of several L-dwarfs, rotation-induced oblateness or horizontally inhomogeneous cloud…
We measure the bulk system parameters of the seismically active, rapidly-rotating $δ$-Scuti KOI-976 and constrain the orbit geometry of its transiting binary companion using a combined approach of asteroseismology and gravity-darkening light curve analysis. KOI-976 is a $1.62\pm0.2~\mathrm{M_\odot}$ star with a measured $v\sin(i)$ of $120\pm2$ km/s and seismically-induced variable signal that varies by $\sim$ 0.6\% of the star's total photometric brightness. We take advantage of the star's oblate shape and seismic activity to perform three measurements of its obliquity angle relative to the pl…
We investigate how rapid stellar rotation commonly seen in A/F stars can influence planet habitability. Specifically, we model how rapid rotation influences a planet's irradiation and determine the location of the habitable zone for stars in the mass range $1.3M_\odot\leq M_\star\leq 2.2M_\odot$. Rapid stellar rotation can dramatically change a star's luminosity and spectral energy distribution and, therefore, can impact the habitability of any surrounding planets. Stars of mass $M_\star\gtrsim1.3M_\odot$ commonly rotate near their breakup speeds, which causes two effects relevant to planet ha…
KELT-9 b is an ultra hot Jupiter transiting a rapidly rotating, oblate early-A-type star in a polar orbit. We model the effect of rapid stellar rotation on KELT-9 b's transit light curve using photometry from the Transiting Exoplanet Survey Satellite (\tess) to constrain the planet's true spin-orbit angle and to explore how KELT-9 b may be influenced by stellar gravity darkening. We constrain the host star's equatorial radius to be $1.089\pm0.017$ times as large as its polar radius and its local surface brightness to vary by $\sim38$\% between its hot poles and cooler equator. We model the ste…
Rapidly rotating early-type main-sequence stars with transiting planets are interesting in many aspects. Unfortunately, several astrophysical effects in such systems are not well understood yet. Therefore, we performed a photometric mini-survey of three rapidly rotating stars with transiting planets, namely KELT-17b, KELT-19Ab, and KELT-21b, using the Characterising Exoplanets Satellite (CHEOPS), complemented with Transiting Exoplanet Survey Satellite (TESS) data, and spectroscopic data. We aimed at investigating the spin-orbit misalignment and its photometrical signs, therefore the high-quali…
The presence of silicate material in known rings in the Solar System raises the possibility of ring systems existing even within the snow line -- where most transiting exoplanets are found. Previous studies have shown that the detection of exoplanetary rings in transit light curves is possible, albeit challenging. To aid such future detection of exoplanetary rings, we present the Polygon+Segments model for modelling the light curve of an exoplanet with rings. This high-precision model includes full ring geometry as well as possible ring transparency and the host star's limb darkening. It is al…
The CoRoT exoplanet science team announces the discovery of CoRoT-11b, a fairly massive hot-Jupiter transiting a V=12.9 mag F6 dwarf star (M*=1.27 +/- 0.05 Msun, R*=1.37 +/- 0.03 Rsun, Teff=6440 +/- 120 K), with an orbital period of P=2.994329 +/- 0.000011 days and semi-major axis a=0.0436 +/- 0.005 AU. The detection of part of the radial velocity anomaly caused by the Rossiter-McLaughlin effect shows that the transit-like events detected by CoRoT are caused by a planet-sized transiting object in a prograde orbit. The relatively high projected rotational velocity of the star (vsini=40+/-5 km/s…
We report the discovery of a multi-planet system orbiting HD 106315, a rapidly rotating mid F-type star, using data from the K2 mission. HD 106315 hosts a $2.51\pm0.12\,R_\oplus$ sub-Neptune in a 9.5 day orbit, and a $4.31_{-0.27}^{+0.24}\,R_\oplus$ super-Neptune in a 21 day orbit. The projected rotational velocity of HD 106315 (12.9 km s$^{-1}$) likely precludes precise measurements of the planets' masses, but could enable a measurement of the sky-projected spin-orbit obliquity for the outer planet via Doppler tomography. The eccentricities of both planets were constrained to be consistent wi…
