Transiting exoplanets provide unparalleled access to the fundamental
parameters of both extrasolar planets and their host stars. We present
limb-darkening coefficients (LDCs) for the exoplanet hunting CoRot and Kepler
missions. The LDCs are calculated with ATLAS stellar atmospheric model grids
and span a wide range of Teff, log g, and metallically [M/H]. Both CoRot and
Kepler contain wide, nonstandard response functions, and are producing a large
inventory of high-quality transiting lightcurves, sensitive to stellar limb
darkening. Comparing the stellar model limb darkening to results from the first
seven CoRot planets, we find better fits are found when two model intensities
at the limb are excluded in the coefficient calculations. This calculation
method can help to avoid a major deficiency present at the limbs of the 1D
stellar models.Comment: Accepted for publication in A&A. 4 pages, 2 figures, 2 tables. Full
  versions of tables 1 and 2 containing limb-darkening coefficients available
  at http://vega.lpl.arizona.edu/~sing

Auvergne

Barge

Bigot

Claret

Czesla

D. K. Sing

Deleuil

Léger

Mandel

Neckel

Rauer

Sing

Southworth

Astronomy and Astrophysics

English

arXiv

Transiting exoplanets provide unparalleled access to the
fundamental parameters of both extrasolar planets and their host
stars. We present limb-darkening coefficients (LDCs) for the
exoplanet hunting CoRot and Kepler missions. The LDCs are calculated with ATLAS stellar
atmospheric model grids and span a wide range of $T_{\rm eff}$, log g, and metallically [M/H]. Both CoRot and Kepler
use wide nonstandard photometric filters, and are producing a large
inventory of high-quality transiting lightcurves, sensitive to stellar
limb darkening.
Comparing the stellar model limb darkening to results from the first seven CoRot planets, we find better fits 
are found when two model intensities at the limb are excluded in the
coefficient calculations. This calculation method can help to 
avoid a major deficiency present at the limbs of the 1D stellar models

EDP Sciences OAI-PMH repository (1.2.0)

Stellar limb-darkening coefficients for CoRot and Kepler

International audienceTransiting exoplanets provide unparalleled access to the fundamental parameters of both extrasolar planets and their host stars. We present limb-darkening coefficients (LDCs) for the exoplanet hunting CoRot and Kepler missions. The LDCs are calculated with ATLAS stellar atmospheric model grids and span a wide range of Teff, log g, and metallically [M/H]. Both CoRot and Kepler use wide nonstandard photometric filters, and are producing a large inventory of high-quality transiting lightcurves, sensitive to stellar limb darkening. Comparing the stellar model limb darkening to results from the first seven CoRot planets, we find better fits are found when two model intensities at the limb are excluded in the coefficient calculations. This calculation method can help to avoid a major deficiency present at the limbs of the 1D stellar models. Full Tables 1 and 2 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/510/A21</A

Sing, D. K.

HAL: Hyper Article en Ligne

Copyright © ESO, 2010Transiting exoplanets provide unparalleled access to the fundamental parameters of both extrasolar planets and their host stars. We present limb-darkening coefficients (LDCs) for the exoplanet hunting CoRot and Kepler missions. The LDCs are calculated with ATLAS stellar atmospheric model grids and span a wide range of , log g, and metallically [M/H]. Both CoRot and Kepler use wide nonstandard photometric filters, and are producing a large inventory of high-quality transiting lightcurves, sensitive to stellar limb darkening. Comparing the stellar model limb darkening to results from the first seven CoRot planets, we find better fits are found when two model intensities at the limb are excluded in the coefficient calculations. This calculation method can help to avoid a major deficiency present at the limbs of the 1D stellar models.CNE

Sing, David K.

Open Research Exeter

A&A 510, A21 (2010)DOI: 10.1051/0004-6361/200913675c© ESO 2010Astronomy&AstrophysicsStellar limb-darkening coefficients for CoRot and Kepler(Research Note)D. K. Sing1,21 Institut d’Astrophysique de Paris, CNRS/UPMC, 98bis boulevard Arago, 75014 Paris, France2 Astrophysics Group, School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UKe-mail: sing@astro.ex.ac.ukReceived 16 November 2009 / Accepted 11 December 2009ABSTRACTTransiting exoplanets provide unparalleled access to the fundamental parameters of both extrasolar planets and their host stars. Wepresent limb-darkening coeﬃcients (LDCs) for the exoplanet hunting CoRot and Kepler missions. The LDCs are calculated withATLAS stellar atmospheric model grids and span a wide range of Teﬀ , log g, and metallically [M/H]. Both CoRot and Kepler usewide nonstandard photometric filters, and are producing a large inventory of high-quality transiting lightcurves, sensitive to stellarlimb darkening. Comparing the stellar model limb darkening to results from the first seven CoRot planets, we find better fits are foundwhen two model intensities at the limb are excluded in the coeﬃcient calculations. This calculation method can help to avoid a majordeficiency present at the limbs of the 1D stellar models.Key words. stars: atmospheres – planetary systems – binaries: eclipsing1. IntroductionTransiting exoplanets have provided an unprecedented opportu-nity to directly measure physical parameters of the planet andhost star, such as mass and radius. The lightcurve shape of a tran-sit (where the planet passes in front of the star as viewed from theEarth) is primarily determined by the planet-to-star radius ratio,impact parameter, and stellar limb darkening. Thus, an accuratetreatment of limb darkening is critical when deriving planetaryradii and transmission spectra from transit data.With high signal-to-noise (S/N) transit light curves, the limbdarkening can be fit and compared to theoretical stellar atmo-spheric models, providing a method to scrutinize and test dif-ferent models (e.g. Southworth 2008). In a comparison with thesun, Sing et al. (2008) found that for the widely used Kurucz 1DATLAS stellar models1, the largest diﬀerences between existinglimb-darkening data (Neckel & Labs 1994) and the 1D stellarmodels was at the limb, where ATLAS models predict a dra-matic increase in the strength of limb darkening. For the sun,the ATLAS models over-predict the strength of limb darkeningby >20% at μ = cos(θ) values below 0.05, though seemed toperform well otherwise, over-predicting the strength by only afew percent. While there remain few other observational con-straints, theoretical atmospheric models have also been unableto provide a satisfactory fits to the observations of eclipsing bi-nary stars (Claret 2008) and the transiting data of HD 209458(Claret 2009). However, diﬀerent LDC calculation methods mayhelp improve the situation, with (Sing et al. 2009) recentlyfitting near-infrared LDCs to a level of a few percent with Full Tables 1 and 2 are only available in electronic form at the CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) orviahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/510/A211 http://kurucz.harvard.edu/HST/NICMOS transit data of HD 189733, finding good agree-ment with the theoretical stellar models.The exoplanet hunting missions CoRot and Kepler providean excellent opportunity to further test these stellar models, ashigh S/N transits for a variety of spectral types are being dis-covered. Both CoRot and Kepler operate at optical wavelengths,where the eﬀects of limb-darkening are strong, using wide bandfilters with wavelengths between ∼4000 and ∼9000 Å. Here wepresent stellar limb-darkening coeﬃcients for both the CoRotand Kepler satellites, along with some initial results compar-ing diﬀerent calculation methods to the observed stellar LD. Wepresent our calculation methods in Sect. 2, compare our resultsto CoRot transit data in Sect. 3, and make concluding remarks inSect. 4.2. The numerical methodsWe calculate LDCs for the laws most commonly used in exo-planetary transit work:The Linear lawI(μ)I(1) = 1 − u(1 − μ). (1)The Quadratic lawI(μ)I(1) = 1 − a(1 − μ) − b(1 − μ)2. (2)The Non-Linear lawI(μ)I(1) = 1−c1(1 − μ1/2)−c2(1−μ)−c3(1 − μ3/2)−c4(1 − μ2), (3)where I(1) is the intensity at the center of the stellar disk,μ = cos(θ) (which is the angle between the line of sight and theemergent intensity), while u, a, b, and cn are the LDCs. TheseArticle published by EDP Sciences Page 1 of 4A&A 510, A21 (2010)Table 1. CoRot stellar limb-darkening coeﬃcients.Teﬀ Logg [M/H] Linear Quadratic 3 parameter non-linear 4 parameter non-linear(K) u a b c2 c3 c4 c1 c2 c3 c44000 4.50 0.00 0.6690 0.4880 0.2241 1.6776 –1.5366 0.6067 0.5651 0.1233 0.2108 –0.08174250 4.50 0.00 0.7091 0.6332 0.0939 1.2813 –0.9538 0.4217 0.6895 –0.6083 1.1653 –0.41184500 4.50 0.00 0.7104 0.6528 0.0714 1.1266 –0.6859 0.2995 0.7336 –0.8830 1.5672 –0.58664750 4.50 0.00 0.6975 0.6184 0.0980 1.1159 –0.6250 0.2399 0.7099 –0.8275 1.5532 –0.61645000 4.50 0.00 0.6831 0.5767 0.1317 1.1434 –0.6285 0.2081 0.6995 –0.7701 1.5147 –0.63415250 4.50 0.00 0.6656 0.5301 0.1677 1.1917 –0.6760 0.1978 0.6904 –0.6941 1.4338 –0.63065500 4.50 0.00 0.6464 0.4830 0.2023 1.2571 –0.7658 0.2118 0.6560 –0.5308 1.2309 –0.57125750 4.50 0.00 0.6280 0.4412 0.2312 1.3297 –0.8831 0.2462 0.6060 –0.3196 0.9569 –0.47456000 4.50 0.00 0.6096 0.4008 0.2584 1.4350 –1.0719 0.3211 0.5409 –0.0366 0.5688 –0.32136250 4.50 0.00 0.5937 0.3695 0.2774 1.5321 –1.2593 0.4034 0.4857 0.2088 0.2177 –0.17536500 4.50 0.00 0.5797 0.3423 0.2939 1.6369 –1.4647 0.4980 0.4256 0.4731 –0.1624 –0.01326750 4.50 0.00 0.5687 0.3210 0.3065 1.7227 –1.6343 0.5770 0.3898 0.6510 –0.4303 0.10307000 4.50 0.00 0.5602 0.3033 0.3180 1.7944 –1.7717 0.6398 0.3682 0.7769 –0.6243 0.18687250 4.50 0.00 0.5531 0.2866 0.3299 1.8515 –1.8757 0.6842 0.3622 0.8469 –0.7397 0.23487500 4.50 0.00 0.5479 0.2708 0.3429 1.8972 –1.9489 0.7108 0.3552 0.9092 –0.8296 0.2673Notes. The full version of this table can be found at the CDS and at http://vega.lpl.arizona.edu/~singd/ which also includes an IDLprogram to read and interpolate the table. The full table covers: Teﬀ from 3500 to 50 000 K, log g from 0.0 to 5.0, and log [M/H] from +1 to –0.5with a turbulent velocity of 2 km s−1. The linear, quadratic, and 3 parameter non-linear laws are calculated with the improved calculation methodusing 15-ATLAS angles with the limb intensities excluded, while the 4 parameter non-linear law was calculated with all 17 angles including thoseat the limb (see text).laws can all be used along with the analytical transit light mod-els of Mandel & Agol (2002), and the definitions of each lawhave been set to conform with the work of Claret (2000).In addition to the above laws, we also calculate a variant ofEq. (3), which is a three parameter non-linear law,I(μ)I(1) = 1 − c2(1 − μ) − c3(1 − μ3/2)− c4(1 − μ2), (4)introduced by Sing et al. (2009) to improve the performance ofthe calculated LD at the limb, while still providing enough flex-ibility to capture the inherently non-linear nature of stellar LD.The μ1/2 term from the four parameter non-linear law mainly af-fects the intensity distribution at small μ values and is not neededwhen the intensity at the limb varies approximately linearly atsmall μ values. Dropping the c1 term is also further motivatedby both solar data (Neckel & Labs 1994) and 3D stellar mod-els (Bigot et al. 2006), which show the intensity distribution atthe limb to vary smoothly to μ = 0, with no dramatic or suddenincreases in limb-darkening strength as observed in the ATLASmodels.We choose a least squares method to fit for the limb-darkening coeﬃcients from ATLAS models. The model specificintensities were first integrated at each angle using the CoRot PFwhite-light (Auvergne et al. 2009) and Kepler2 response func-tions. We used only values of μ ≥ 0.05 for the linear, quadratic,and three parameter non-linear laws (corresponding to 15 anglesin the ATLAS model grids) while retaining all 17 angles, includ-ing the limb intensities, for the four-parameter non-linear law.The four-parameter law best describes the tabulated ATLASmodel grid intensities and should be analogous to the widelyused LDCs of Claret (2000). Thus in our study, the four-parameter non-linear law is the best representation of the orig-inal stellar atmospheric models themselves, while the linear,quadratic and three-parameter non-linear laws are calculatedwith the intent of improving the limb intensities. The results of2 http://keplergo.arc.nasa.gov/CalibrationResponse.shtmlthe calculations are given in Tables 1 and 2, with the full versionsof these tables available3. In Fig. 1, we illustrate typical CoRotand Kepler model limb darkening profiles for F, G, and K mainsequence stars.3. Comparisons with CoRot exoplanet transitresultsWe compared the calculated LDCs with results from six of thefirst seven CoRot exoplanets, four of which have LDCs deter-mined from the transit light curve fit: CoRot-1 (Barge et al.2008), CoRot-2 (Alonso et al. 2008), CoRot-3 (Deleuil et al.2008), and CoRot-4 (Aigrain et al. 2008). The LDCs of CoRot-5and CoRot-7 were not fit by the authors (Léger et al. 2009; Raueret al. 2009), so no comparison was possible. With the limb in-tensities a potential source of disagreement between the observa-tions and models, we compared the theoretical values of I(μ = 0)with the CoRot results (see Fig. 2). This comparison is aided bythe fact that for each law considered here, I(0) = 1 − Σ(Cn),making a direct comparison between diﬀerent laws possible. Inaddition, the authors of CoRot 1, 2, and 3 choose to fit the LDCu+ = a + b, for the quadratic law, which is directly related tothe limb intensity, I(0) = 1 − u+. CoRot-4 was fit with the linearlaw. We also calculated the model uncertainty for the theoreticalLDCs, incorporating the quoted uncertainty in stellar parameters(Teﬀ, log g, [M/H]) using the partial derivative of the coeﬃcientsas a function of the stellar parameters. We find that typical errorsin Teﬀ, log g, and [M/H] lead to only a small change in modelLDCs for the linear and quadratic laws, with the model limbintensities uncertain by only a few percent or less. This errorestimation breaks down for the higher-order laws, as fitting de-generacies between the coeﬃcients lead to unrealistically largepartial derivatives and large errors. In those cases, we used theuncertainty from the linear law as a reasonable estimate of themodel I(0) error. However, the model error is small and muchless important than the observational errors.3 at the CDS or at http://vega.lpl.arizona.edu/~singd/Page 2 of 4D. K. Sing: Stellar limb-darkening coeﬃcients for CoRot & Kepler (RN)Table 2. Kepler stellar limb-darkening coeﬃcients.Teﬀ Logg [M/H] Linear Quadratic 3 parameter non-linear 4 parameter non-linear(K) u a b c2 c3 c4 c1 c2 c3 c44000 4.50 0.00 0.6888 0.5079 0.2239 1.6669 –1.4738 0.5729 0.5478 0.1602 0.2201 –0.09444250 4.50 0.00 0.7215 0.6408 0.0999 1.2877 –0.9263 0.4009 0.6928 –0.6109 1.2029 –0.43664500 4.50 0.00 0.7163 0.6483 0.0842 1.1412 –0.6724 0.2793 0.7393 –0.8838 1.5980 –0.61364750 4.50 0.00 0.6977 0.6036 0.1164 1.1431 –0.6355 0.2272 0.7156 –0.8156 1.5595 –0.63575000 4.50 0.00 0.6779 0.5528 0.1548 1.1854 –0.6693 0.2070 0.7072 –0.7489 1.4971 –0.64425250 4.50 0.00 0.6550 0.4984 0.1939 1.2500 –0.7540 0.2138 0.7011 –0.6649 1.3885 –0.62755500 4.50 0.00 0.6307 0.4451 0.2297 1.3274 –0.8765 0.2442 0.6716 –0.5029 1.1678 –0.55745750 4.50 0.00 0.6074 0.3985 0.2586 1.4038 –1.0114 0.2882 0.6283 –0.3063 0.8965 –0.45936000 4.50 0.00 0.5842 0.3539 0.2851 1.5101 –1.2143 0.3714 0.5685 –0.0364 0.5100 –0.30386250 4.50 0.00 0.5640 0.3198 0.3023 1.6062 –1.4130 0.4616 0.5170 0.1979 0.1587 –0.15416500 4.50 0.00 0.5459 0.2901 0.3167 1.7070 –1.6236 0.5611 0.4576 0.4564 –0.2249 0.01226750 4.50 0.00 0.5312 0.2672 0.3267 1.7862 –1.7936 0.6430 0.4219 0.6279 –0.4938 0.13177000 4.50 0.00 0.5191 0.2478 0.3358 1.8587 –1.9464 0.7165 0.3968 0.7640 –0.7134 0.23027250 4.50 0.00 0.5085 0.2308 0.3437 1.9144 –2.0641 0.7722 0.3905 0.8337 –0.8442 0.29027500 4.50 0.00 0.5003 0.2165 0.3512 1.9533 –2.1433 0.8075 0.3875 0.8777 –0.9264 0.3259Notes. Same as note to Table 1.Fig. 1. Stellar limb darkening for CoRot (solid lines) and Kepler (dashedlines) calculated from Atlas models appropriate for a solar metallicallyF5V, G5V, and K5V star (top to bottom: Teﬀ = 6500, 5500, 4500 K;log g= 4.5, [M/H]= 0.0). The Atlas models show earlier type stars ex-hibiting increasingly non-linear intensity distributions for the CoRotand Kepler bandpasses.From Fig. 2, the most obvious disagreement between themodels and observations is with the active star CoRot-2 (Alonsoet al. 2008). A determination of the planet-to-star radius ratio isaﬀected by stellar activity, as shown for CoRot-2 by Czesla et al.(2009) who re-determined the planetary radius, taking into ac-count stellar activity, finding a larger radius than either Alonsoet al. (2008) or Gillon et al. (2009). However, as the radii andlimb darkening are linked in a transit fit, the limb darkening co-eﬃcients are also aﬀected by stellar activity, though these pa-rameters were not re-determined. With these complications dueto stellar activity, a proper comparison of limb darkening willlikely have to wait until the limb darkening is also re-determinedin conjunction with the planetary radii.The three CoRot targets (1, 3 and 4) are suﬃcient to see asignificant improvement in model limb intensities, when calcu-lating LDCs with 15-angles and using lower order laws. For ev-ery CoRot target thus far, the calculated limb intensities of theATLAS models are significantly lower than the observed transitfit values, further proof of the model limb deficiencies previouslymentioned.There is a potential source of disagreement between com-paring I(0) when diﬀerent laws are used in the stellar model LDand transit-fit LD. In our model calculations, the general trendis to decrease I(0) when adopting a higher order law to describethe intensity. Higher order laws are increasingly sensitive to theshape of the falling intensity at the limb, thus reducing the fitvalue of I(0). However, we believe this to be a second order ef-fect which should not seriously aﬀect these results, especially asseveral of the CoRot LDCs are poorly constrained and are notsensitive to higher order terms.Comparing the I(0) calculated from the diﬀerent laws to thefit CoRot values, the 17-angle four parameter non-linear law hasa χ2ν of 5.15, the 15-angle 3-parameter law has a χ2ν of 2.85, thequadratic law a χ2ν of 1.67 and the linear law a χ2ν of 0.63. Forthe lower order laws, a better fit to I(0) is obtained by using15-angles to calculate the LDCs, disregarding the model limbvalues. Though the linear law represents the best fit for I(0) here,the expected deviations of the model I(μ) from a linear trend inF and G type stars (Fig. 1) will likely make higher order lawsnecessary to describe the full intensity distribution.Determining which limb darkening law to use in a transitlight curve fit is very dependent on the S/N of the data and thestellar type of the star being studied. From our model calcula-tions, we expect that linear laws will likely fit transit light curvesquite well for both CoRot and Kepler in cooler K type stars, orfor earlier types at low S/N. Higher order terms will likely beneeded in F and G type stars at high S/N, where multiple LDCscan be fit at good precision.4. ConclusionsFor use within the community, we present ATLAS model gridLDCs calculated for the CoRot and Kepler transiting exoplanetmissions. We find better agreement between the existing CoRotobservations and model LDC when two limb intensities are notused in the calculations, and incorporate this method in the pre-sented LDCs.The future catalog of transiting planets discovered by CoRotand Kepler oﬀers the prospect of substantially improving thetheoretical models of stellar limb darkening. The very highPage 3 of 4A&A 510, A21 (2010)CoRot Transit Limb-Darkening Coeff. & Kurucz Atlas model Coeff.5000 5500 6000 6500 7000Teff (K)0.00.10.20.30.40.50.60.70.80.91.0Limb Intensity  I(μ=0) = 1-Σ(C n)Angles  Law           χ2ν   15       Linear       0.63   15       Quad        1.67   15       3-Param   2.85   17       4-Param   5.15CoRot-3CoRot-2 CoRot-1 CoRot-4CoRot-5CoRot-7Fig. 2. Limb intensity for the CoRot transitingplanets (black circles) vs. eﬀective stellar tem-perature. CoRot-5 and CoRot-7 did not havelightcurve-fit LDCs, and the adopted valueswithout y-error bars are shown. The ATLASmodel limb intensities for the linear (red),quadratic (orange), three-parameter non-linear(purple), and four-parameter non-linear (bluestars) laws are also shown, along with the errorderived from the uncertainty in the stellar pa-rameters. The model errors are small and typi-cally less than the size of the plotted point. Ourmethod of calculating for LDCs, with lowerorder laws and excluding the model limb in-tensities, shows a better agreement with theobservations.photometric precision of Kepler should allow for multiple LDCsto be fit at the percent level, which should open up many de-tailed comparisons with stellar atmospheric models. For in-stance, Kepler (at high temporal resolution) will be quite sen-sitive to non-linear LD terms, as there will be suﬃcient S/N toaccurately fit for coeﬃcients beyond just the linear term. TheLDCs presented here are intended to aid in these studies and beof general use to the community.Acknowledgements. D.K.S. is supported by CNES. We thank Robert Kuruczfor making his grid of stellar models publicly available and the referee for theirhelpful comments.ReferencesAigrain, S., Collier Cameron, A., Ollivier, M., et al. 2008, A&A, 488, L43Alonso, R., Auvergne, M., Baglin, A., et al. 2008, A&A, 482, L21Auvergne, M., Bodin, P., Boisnard, L., et al. 2009, A&A, 506, 411Barge, P., Baglin, A., Auvergne, M., et al. 2008, A&A, 482, L17Bigot, L., Kervella, P., Thévenin, F., & Ségransan, D. 2006, A&A, 446,635Claret, A. 2000, A&A, 363, 1081Claret, A. 2008, A&A, 482, 259Claret, A. 2009, A&A, 506, 1335Czesla, S., Huber, K. F., Wolter, U., Schröter, S., & Schmitt, J. H. M. M. 2009,A&A, 505, 1277Deleuil, M., Deeg, H. J., Alonso, R., et al. 2008, A&A, 491, 889Gillon, M., Lanotte, A. A., Barman, T., et al. 2009, ArXiv e-printsLéger, A., Rouan, D., Schneider, J., et al. 2009, A&A, 506, 287Mandel, K., & Agol, E. 2002, ApJ, 580, L171Neckel, H., & Labs, D. 1994, Sol. Phys., 153, 91Rauer, H., Queloz, D., Csizmadia, S., et al. 2009, A&A, 506, 281Sing, D. K., Vidal-Madjar, A., Désert, J.-M., Lecavelier des Etangs, A., &Ballester, G. 2008, ApJ, 686, 658Sing, D. K., Désert, J., Lecavelier Des Etangs, A., et al. 2009, A&A, 505, 891Southworth, J. 2008, MNRAS, 386, 1644Page 4 of 4

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Stellar Limb-Darkening Coefficients for CoRot and Kepler

Stellar Limb-Darkening Coefficients for CoRot and Kepler

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