The depth of an exoplanetary transit in the light curve of a distant star is commonly approximated as the squared planet-to-star radius ratio, (R_p/R_s)^2. Stellar limb darkening, however, results in significantly deeper transits. Here we derive analytical solutions to the overshoot of the mid-transit depth caused by stellar limb darkening compared to the (R_p/R_s)^2 estimate for arbitrary transit impact parameters. In turn, this allows us to compute the true planet-to-star radius ratio from the transit depth for a given parameterization of a limb darkening law and for a known transit impact parameter. We calculate the maximum emerging specific stellar intensity covered by the planet in transit and derive analytic solutions for the transit depth overshoot. Solutions are presented for the linear, quadratic, square-root, logarithmic, and non-linear stellar limb darkening with arbitrary transit impact parameters. We also derive formulae to calculate the average intensity along the transit chord, which allows us to estimate the actual transit depth (and therefore R_p/R_s) from the mean in-transit flux. The transit depth overshoot of exoplanets compared to the (R_p/R_s)^2 estimate increases from about 15% for A main-sequence stars to roughly 20% for sun-like stars and some 30% for K and M stars. The error in our analytical solutions for R_p/R_s from the small planet approximation is orders of magnitude smaller than the uncertainties arising from typical noise in real light curves and from the uncertain limb darkening. Our equations can be used to predict with high accuracy the expected transit depth of extrasolar planets. The actual planet radius can be calculated from the measured transit depth or from the mean in-transit flux if the stellar limb darkening can be properly parameterized and if the transit impact parameter is known. Light curve fitting is not required.
The detection of the first exoplanet paved the way into the era of transit photometry space missions with a revolutionary photometric precision that aim at discovering new exoplanetary systems around different types of stars. With this high precision, it is possible to derive very accurately the radii of exoplanets which is crucial for constraining their type and composition. However, it requires an accurate description of host stars, especially their center-to-limb variation of intensities (so called limb darkening) as it affects the planet-to-star radius ratio determination. We aim at improv…
Inaccurate limb-darkening models can be a significant source of error in the analysis of the light curves for transiting exoplanet and eclipsing binary star systems. To test the accuracy of published limb-darkening models, I have compared limb-darkening profiles predicted by stellar atmosphere models to the limb-darkening profiles measured from high-quality light curves of 43 FGK-type stars in transiting exoplanet systems observed by the Kepler and TESS missions. The comparison is done using the parameters $h^{\prime}_1 = I_λ(\frac{2}{3})$ and $h^{\prime}_2 = h^{\prime}_1 - I_λ(\frac{1}{3})$, …
Limb darkening is an important stellar phenomenon and must be accounted for in the study of stellar spectra, eclipsing binaries, transiting planetary systems, and microlensing events. The power-2 limb-darkening law provides a good match to the specific intensities predicted by stellar atmosphere models: it is better than other two-parameter laws and is only surpassed by the four-parameter law. Predictions of the limb-darkening coefficients for the power-2 law are not widely available. We therefore compute them, using stellar atmosphere models generated by the ATLAS (plane-parallel) code. Limb-…
Although the main goal of the Transiting Exoplanet Survey Satellite (\textit{TESS}) is to search for new transiting exoplanets, its data can also be used to study in further detail already known systems. The \textit{TESS} bandpass is particularly interesting to study the limb-darkening effect of the stellar host which is imprinted in transit lightcurves, as the widely used \textsc{phoenix} and \textsc{atlas} stellar models predict different limb-darkening profiles. Here we study this effect by fitting the transit lightcurves of 176 known exoplanetary systems observed by \textit{TESS}, which al…
Limb darkening (LD) is typically parameterised using a range of functional "laws" in models of the light curves of eclipsing binary and transiting planetary systems. The two-coefficient LD laws all suffer from a strong correlation between their coefficients, preventing a reliable determination of both coefficients from high-quality light curves. We use numerical simulations to propose re-parameterisations of the quadratic, logarithmic, square-root and cubic LD laws that show much weaker correlations, and implement them into the JKTEBOP code. We recommend that these re-parameterisations are use…
We release a new grid of stellar limb-darkening coefficients (LDCs, using the quadratic, power-2 and claret-4 laws) and intensity profiles for the Kepler, U, B, V and R passbands, based on STAGGER model atmospheres. The data can be downloaded from Zenodo (doi:10.5281/zenodo.5593162). We compare the newly-released LDCs, computed by ExoTETHyS, with previously published values, based on the same atmospheric models using a so-called "SPAM" procedure. The SPAM method relies on synthetic light curves in order to compute the LDCs that best represent the photometry of exoplanetary transits. We confirm…
We release the first grid of stellar limb-darkening coefficients (LDCs) and intensity profiles (IPs) computed by the consortium of the PLAnetary Transits and Oscillations of stars (PLATO), the next medium-class (M3) mission under development by the European Space Agency (ESA) to be launched in 2026. We have performed spectral synthesis with \texttt{TurboSpectrum} on a grid of \texttt{MARCS} model atmospheres. Finally, we adopted \texttt{ExoTETHyS} to convolve the high-resolution spectra ($R=2\times10^5$) with the state-of-the-art response functions for all the PLATO cameras, and computed the L…
This paper is part of an effort to correct the transmission spectra of a transiting planet orbiting an active star. In Paper I (Cracchiolo et al. 2020) we have demonstrated a methodology to minimize the potential bias induced by unocculted star spots on the transmission spectrum, assuming a spot model parameterized by filling factor and temperature. In this work we introduce the limb darkening effect, therefore the position of the spot in the stellar disk and the impact parameter of the transiting planet now play a key role. The method is tested on simulations of planetary transits of three re…
We provide here tables of stellar limb-darkening coefficients (LDCs) for the Ariel ESA M4 space mission. These tables include LDCs corresponding to different wavelength bins and white bands for the NIRSpec, AIRS-Ch0 and AIRS-Ch1 spectrographs, and those corresponding to the VISPhot, FGS1 and FGS2 photometers. The LDCs are calculated with the open-source software ExoTETHyS for three complete grids of stellar atmosphere models obtained with the ATLAS9 and PHOENIX codes. The three model grids are complementary, as the PHOENIX code adopts more modern input physics and spherical geometry, while the…
I present RoadRunner, a fast exoplanet transit model that can use any radially symmetric function to model stellar limb darkening while still being faster to evaluate than the analytical transit model for quadratic limb darkening by Mandel & Agol (2002). CPU and GPU implementations of the model are available in the PyTransit transit modelling package, and come with platform-independent parallelisation, supersampling, and support for modelling complex heterogeneous time series. The code is written in numba-accelerated Python (and the GPU model in OpenCL) without C or Fortran dependencies, which…
A Fast Concentric-disk Contour Integration Method For Microlensing Limb-darkening Effect
The Kavli Institute for Astronomy and Astrophysics, Peking University（北京大学科维理天文与天体物理研究所）；Department of Astronomy, School of Physics, Peking University（北京大学物理学院天文学系）
We present a novel, iterative method using an empirical Bayesian approach for modeling the limb darkened WASP-121b transit from the TESS light curve. Our method is motivated by the need to improve $R_{p}/R_{\ast}$ estimates for exoplanet atmosphere modeling, and is particularly effective with the limb darkening (LD) quadratic law requiring no prior central value from stellar atmospheric models. With the non-linear LD law, the method has all the advantages of not needing atmospheric models but does not converge. The iterative method gives a different $R_{p}/R_{\ast}$ for WASP-121b at a signific…
Analytical transit light curves for arbitrary power-law limb darkening: a unified framework
Peaceful Society Science and Innovation Foundation（和平社会科学与创新基金会）；Department of Occupational Science and Occupational Therapy, University of British Columbia（不列颠哥伦比亚大学职业科学与职业治疗系）；Department of Physics and Astronomy, Tarleton State University（塔伦州立大学物理与天文系）
Power-2 limb-darkening coefficients for the $uvby$, $UBVRIJHK$, SDSS $ugriz$, Gaia, Kepler, TESS, and CHEOPS photometric systems II. PHOENIX spherically symmetric stellar atmosphere models
We present a reanalysis of visible/near-infrared (480-930 nm) observations of Neptune, made in 2018 with the MUSE instrument at the Very Large Telescope (VLT) in Narrow Field Adaptive Optics mode, reported by Irwin et al., Icarus, 311, 2019. We find that the inferred variation of methane abundance with latitude in our previous analysis, which was based on central meridian observations only, underestimated the retrieval errors when compared with a more complete assessment of Neptune's limb darkening. In addition, our previous analysis introduced spurious latitudinal variability of both the abun…
The CARMENES search for exoplanets around M dwarfs. A homogeneous catalogue of projected rotational velocities accounting for limb-darkening
Instituto de Astrofísica de Andalucía, CSIC（安达卢西亚天体物理研究所，西班牙高等科学研究委员会）；INAF- Palermo Astronomical Observatory（意大利国家天体物理研究所巴勒莫天文台）；Institut für Astrophysik und Geophysik, Georg-August-Universität Göttingen（哥廷根大学天体物理学与地球物理学研究所）；Centro de Astrobiología, CSIC-INTA（天体生物学中心，西班牙高等科学研究委员会-国家航空航天技术研究所）；Universidad de Granada（格拉纳达大学）；Landessternwarte, Zentrum für Astronomie der Universität Heidelberg（海德堡大学天文学中心州立天文台）
提出一种结合真实临边昏暗模型的过采样卷积方法，从 CARMENES 光谱精确测量 M 矮星的投影自转速度，构建了目前最大的均匀星表，测量精度显著优于文献。
Limb darkening laws are convenient parameterizations of the stellar intensity center-to-limb variation, and their use is ubiquitous in eclipse and transit modeling. But they are not "laws" in any sense -- they are simple approximations of the real intensity variations, and their limitations are becoming more and more apparent as stellar atmosphere models improve and higher precision data become available. When fitting eclipses and transit light curves, one would ideally like to use model intensities that are based on fundamental stellar parameters such as the mass, radius, and effective temper…
Where does the simplified Stellar Contamination Model fail in Exoplanet Transmission Spectroscopy?
Jet Propulsion Laboratory, California Institute of Technology（加州理工学院喷气推进实验室）；Center for Radio Astronomy and Astrophysics Mackenzie (CRAAM), Mackenzie Presbyterian University（麦肯锡长老会大学射电天体物理学中心）
We have added to the Chroma+ suite of stellar atmosphere and spectrum modelling codes the ability to synthesize the exo-planet transit lightcurve for planets of arbitrary size up to 10% of the host stellar radius, and arbitrary planetary and stellar mass and orbital radius (thus determining orbital velocity) and arbitrary orbital inclination. The lightcurves are computed in situ, integrated with the radiative transfer solution for the radiation field emerging from the stellar surface, and there is no limb-darkening parameterization. The lightcurves are computed for the Johnson-Bessel photometr…
We present a solution for the light curve of two bodies mutually transiting a star with polynomial limb darkening. The term "mutual transit" in this work refers to a transit of the star during which overlap occurs between the two transiting bodies. These could be an exoplanet with an exomoon companion, two exoplanets, an eclipsing binary and a planet, or two stars eclipsing a third in a triple star system. We include analytic derivatives of the light curve with respect to the positions and radii of both bodies. We provide code that implements a photodynamical model for a mutual transit. We inc…
WASP-33 is one of the few $δ$ Sct stars with a known planetary companion. By analyzing the stellar oscillations, we search for possible star-planet interactions in the pattern of the pulsation. We made use of the Transit and Light Curve Modeller (TLCM) to solve the light curve from the Transiting Exoplanet Survey Satellite (TESS). We include gravity darkening into our analysis. The stellar oscillation pattern of WASP-33 clearly shows signs of several tidally perturbed modes. We find that there are peaks in the frequency spectrum that are at or near the $3$rd, $12$th and $25$th orbital harmonic…
High-resolution transmission spectroscopy is a powerful method for probing the extended atmospheres of short-period exoplanets. With the advancement of ultra-stable echelle spectrographs and the advent of 30-meter class telescopes on the horizon, even minor observational and physical effects will become important when modeling atmospheric absorption of atomic species. In this work we demonstrate how the non-uniform temperature across the surface of a fast rotating star, i.e., gravity darkening, can affect the observed transmission spectrum in a handful of atomic transitions commonly observed i…
