Spherically symmetric model stellar atmospheres and limb darkening
II. Limbdarkening laws, gravitydarkening coefficients and angular diameter corrections for FGK dwarf stars^{⋆}
^{1}
Department of Physics & AstronomyEast Tennessee State
University, Box
70652
Johnson City, TN
37614, USA
email:
neilsonh@etsu.edu
^{2}
Department of Chemical and Physical Sciences, University of
Toronto Mississauga, 3359
Mississauga, Ontario, L5L 156, Canada
^{3}
Department of Astronomy & Astrophysics, University of
Toronto, Toronto,
Ontario, MSS 3H4,
Canada
email:
lester@astro.utoronto.ca
Received:
13
May
2013
Accepted:
24
June
2013
Limb darkening is a fundamental ingredient for interpreting observations of planetary transits, eclipsing binaries, optical/infrared interferometry and microlensing events. However, this modeling traditionally represents limb darkening by a simple law having one or two coefficients that have been derived from planeparallel model stellar atmospheres, which has been done by many researchers. More recently, researchers have gone beyond planeparallel models and considered other geometries. We previously studied the limbdarkening coefficients from spherically symmetric and planeparallel model stellar atmospheres for cool giant and supergiant stars, and in this investigation we apply the same techniques to FGK dwarf stars. We present limbdarkening coefficients, gravitydarkening coefficients and interferometric angular diameter corrections from Atlas and SAtlas model stellar atmospheres. We find that sphericity is important even for dwarf model atmospheres, leading to significant differences in the predicted coefficients.
Key words: stars: atmospheres / binaries: eclipsing / stars: evolution / planetary systems / techniques: interferometric
Tables 2–17 and model intensity profiles are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/556/A86
© ESO, 2013
1. Introduction
One of the great astronomical advances of the past two decades has been the discovery and study of extrasolar planets via the transit method, i.e. from the minute drop of a star’s flux as a planet passes in front of it. The transit not only constrains the planet’s properties but also the star’s properties, such as limb darkening. However, interpreting planetary transits typically assumes that limb darkening can be parametrized by a simple relation (Mandel & Agol 2002) with a few free parameters that can be fit directly from the observations or assumed from model stellar atmospheres.
Limb darkening is important not only for understanding planetary transits (e.g. Croll et al. 2011), but also for interpreting optical interferometric observations (e.g. Davis et al. 2000) and microlensing observations (e.g. An et al. 2002) and eclipsing binary light curves (e.g. Bass et al. 2012). Like transit measurements, interferometric and microlensing observations are typically fit by simple limbdarkening laws with coefficients derived from model stellar atmospheres (AlNaimiy 1978; Wade & Rucinski 1985; van Hamme 1993; Claret 2000; Claret & Bloemen 2011; Claret et al. 2012). However, these simple limbdarkening laws have become less suitable as the observations have improved. For example, Fields et al. (2003) showed that fluxnormalized limbdarkening laws fit to Atlas planeparallel model atmospheres disagreed with microlensing observations. Limbdarkening coefficients derived from planetary transit observations with large impact parameters differ more from the limbdarkening coefficients from model atmospheres, but the discrepancy still exists when the impact parameter is taken into account (Barros et al. 2012).
This discrepancy might be due to a number of physical processes, including granulation, multidimensional convection and/or the presence of magnetic fields in the stellar atmosphere. However, the simplest step is to assume a more realistic geometry for the model stellar atmospheres. Limbdarkening coefficients presented in the literature are based on two forms: planeparallel model stellar atmospheres computed using the Atlas (Kurucz 1979) and Phoenix code (Hauschildt et al. 1999) and spherically symmetric model stellar atmospheres also computed from the Phoenix code (Sing 2010; Howarth 2011a; Claret & Bloemen 2011; Claret et al. 2012). In particular, Claret & Hauschildt (2003) and Claret et al. (2012, 2013) explored limb darkening using spherically symmetric Phoenix model stellar atmospheres specifically for main sequence stars. They also introduced the concept of “quasispherical” models, defined as the sphericallysymmetric intensity profile restricted to inner part of the stellar disk (μ ≥ 0.1), to compare limbdarkening coefficients with those from planeparallel models.
In our previous study (Neilson & Lester 2013, hereafter Paper I), we presented coefficients for six typical limbdarkening laws fit to the surface intensities for grids of planeparallel and spherical model atmospheres (Lester & Neilson 2008) representing red giant and supergiant stars. The intensities were for the wavebands of the JohnsonCousins (Johnson & Morgan 1953; Bessell 2005), CoRot (Auvergne et al. 2009) and Kepler (Koch et al. 2004) filters. We also computed gravitydarkening coefficients and interferometric angular diameter corrections. We found that the predicted limbdarkening coefficients computed from spherical model atmospheres differ from those computed from planeparallel model atmospheres, which was not unexpected; the height of the atmospheres of red giants and supergiants relative to the stellar radius is many times greater than the relative heights of the atmospheres of mainsequence stars, i.e. the assumed model geometry is important. We found similar differences for the angular diameter corrections as a function of geometry but little difference between gravitydarkening coefficients as a function of geometry. While model atmosphere geometry is clearly important for understanding the extended atmospheres of red giant and supergiant stars, it is not as obvious that geometry also changes predictions for model stellar atmospheres of main sequence dwarf stars (e.g. Claret & Hauschildt 2003).
In this work, we explore the role of model atmosphere geometry in understanding limb darkening in dwarf stars and compute limbdarkening coefficients, gravitydarkening coefficients and interferometric angular diameter corrections from grids of model stellar atmospheres of dwarf stars. In Sect. 2, we briefly describe the grids of model atmospheres used. In Sect. 3, we describe various limbdarkening laws and compare predicted limbdarkening coefficients, while in Sect. 4 we compute gravitydarkening coefficients. We present interferometric angular diameter corrections as a function of geometry in Sect. 5 and summarize our results in Sect. 6.
2. Model stellar atmospheres
The Atlas/SAtlas code was used to compute model stellar atmospheres assuming either planeparallel or spherically symmetric geometry. Details of the code can be found in Lester & Neilson (2008), Neilson & Lester (2011, 2012) and Paper I. We computed model stellar atmospheres with parameters 3500 K ≤ T_{eff} ≤ 8000 K in steps of 100 K, and 4.0 ≤ log g ≤ 4.75 in steps of 0.25. For the spherically symmetric models, which require an additional parameter, such as stellar mass, to characterize the atmosphere, we chose M = 0.2 to 1.4 M_{⊙} in steps of 0.3 M_{⊙}. For each model stellar atmosphere we compute intensities at each wavelength for 1000 uniformly spaced values of μ, the cosine of the angle formed by the lineofsight point on the stellar disk and the disk center, spanning 0 ≤ μ ≤ 1. Typically, Atlas models compute intensities at only seventeen angles (Kurucz 1979), but some have employed 100 μpoints (Claret & Bloemen 2011). We compute intensity profiles for each model atmosphere for the BVRIH and Kbands as well as the CoRot and Keplerbands. As an example, Fig. 1 shows the Keplerband intensity profiles for planeparallel and spherical models with T_{eff} = 5800 K, log g = 4.5 and M = 1.1 M_{⊙}. Using the wavebands outlined above, we compute limbdarkening coefficients, gravitydarkening coefficients and interferometric angular diameter corrections.
Fig. 1 Predicted Keplerband intensity profiles for planeparallel (solid line) and spherically symmetric (dotted line) model stellar atmosphere with T_{eff} = 5800 K, log g = 4.5 and M = 1.1 M_{⊙}. 

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3. Limbdarkening laws
We consider the same six limbdarkening laws as in Paper I:
As in Paper I, we use a general leastsquares fitting algorithm to compute the limbdarkening coefficients for each law in the BVRIH and Kbands as well as for the CoRot and Keplerbands. Using the Keplerband as an example, Fig. 2 shows the bestfit limbdarkening coefficient for the linear law (Eq. (1)), Fig. 3 shows the coefficients for the quadratic (Eq. (2)) and squareroot (Eq. (3)) laws, Fig. 4 shows the coefficients for the exponential (Eq. (5)) and logarithmic (Eq. (6)) laws and Fig. 5 shows the coefficients for the Claret (2000) fourparameter law (Eq. (4)).
Fig. 2 The limbdarkening coefficient u, used in the linear law (Eq. (1)), applied to the Kepler photometric band. Crosses are the planeparallel model stellar atmospheres, and the squares are the spherical models. 

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Fig. 3 Limbdarkening coefficients a and b used in the quadratic law (Eq. (2)) (left panel), and the coefficients c and d used in the squareroot law (Eq. (3)) (right panel), all applied to the Kepler photometric band. The symbols have the same meanings as in Fig. 2. 

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Fig. 4 Limbdarkening coefficients g and h used in the exponential law (Eq. (5)) (left panel), and the coefficients m and n used in the logarithmic law (Eq. (6)) (right panel), all applied to the Kepler photometric band. The symbols have the same meanings as in Fig. 2. 

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Fig. 5 Limbdarkening coefficients f_{1}, f_{2}, f_{3} and f_{4} used in the Claret (2000) fourparameter law, Eq. (4), applied to the Kepler photometric band. The symbols are the same as in Fig. 2. 

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The results shown in Fig. 2 demonstrate how the geometry of the model atmosphere affects the bestfit linear Keplerband limbdarkening coefficients, with squares representing fits to the spherically symmetric model atmospheres and crosses representing fits to the planeparallel models. The values of the ucoefficient for the spherical models are larger than those for the planar models, particularly for models with T_{eff} > 4500 K. At these higher effective temperatures the difference due to geometry, Δu_{Kepler}, is ~0.3. There is also a greater spread for the spherical model coefficients at a given effective temperature. This is caused by the spherical models being defined by three parameters, with mass and radius being separated, as opposed to the two parameters for planeparallel model atmospheres, where mass and radius are combined in the surface gravity. At T_{eff} < 4500 K the ucoefficients computed for both geometries shift to similar values. A likely cause of this change relative to the higher effective temperatures is the shift in dominant opacities from H^{−} to TiO.
The more complex limbdarkening laws have similar differences between coefficients from planeparallel and spherically symmetric models. For the quadratic and squareroot laws, the coefficients of the linear term (a and c, respectively) shows similar behavior as a function of effective temperature as does the ucoefficients, while the coefficients of the nonlinear terms (b and d) appear correlated to the coefficients of the linear terms, as was seen previously for other laws (Fields et al. 2003; Neilson & Lester 2011, 2012).
For the exponential and logarithmic laws, the bestfit coefficients again differ as a function of model atmosphere geometry. The limbdarkening coefficients also appear to be correlated for each law. It is notable that the bestfit mcoefficients of the logarithmic law from spherically symmetric models are approximately constant with respect to effective temperature, whereas the nonlinear term is not constant. The limbdarkening coefficients from spherically symmetric models for both exponential and logarithmic laws vary significantly for any given effective temperature, suggesting the coefficients are sensitive to the mass and gravity of a model stellar atmosphere.
The bestfit coefficients for the Claret (2000) fourparameter limbdarkening laws do not agree for spherical and planeparallel models. For effective temperatures greater than 4000 K, the limbdarkening coefficient f_{1} varies from −2 to + 4 for the spherical models but only from −0.5 to 0.5 for the planeparallel models. The dramatic difference is due to the more complex structure of spherically symmetric model intensity profiles, even when considering the smaller atmospheric extensions for models used in this work as opposed to those considered in Paper I, which indicates that even this more sophisticated limbdarkening law is not ideal for fitting spherically symmetric model intensity profiles.
Fig. 6 The error of the bestfit limbdarkening relation, defined by Eq. (7), for every model atmosphere (crosses represent planeparallel models, squares spherical models) for each of the six limbdarkening laws at Keplerband wavelengths. 

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Figures 2–5 demonstrate that the bestfit coefficients from spherical models differ from those computed from planar models, but these figures do not quantify the fits for either geometry. To do this, we employ the parameter defined in Paper I, (7)to measure the difference for every model between the computed intensity distribution and the best fit to those intensities for each limbdarkening law. Unfortunately, as we showed in Paper I, the computed error depends on how the models are sampled and the number of intensity points. If one fits intensity profiles for μpoints near the center of the stellar disk then the limbdarkening coefficients and predicted errors differ from limbdarkening coefficients and errors predicted from a sample of μpoints near the edge of the stellar disk. However, we can predict the relative quality of fits as a function of geometry. We show in Fig. 6 the predicted errors for each limbdarkening law as a function of effective temperature.
As expected, Fig. 6 shows that all six limbdarkening laws fit the planeparallel model atmosphere intensity profiles better than intensity profiles from spherical models. The definition of planeparallel radiative transfer (Feautrier 1964) assumes that I(μ) ∝ e^{−τ/μ}, where τ is the monochromatic optical depth. As μ → 0, then I(μ) → 0, i.e. the intensity and the derivative of the intensity, dI/dμ, both change monotonically. These properties allow simple limbdarkening laws to fit planeparallel model intensity profiles well.
For spherically symmetric model atmospheres the radiative transfer is calculated for a set of rays along the lineofsight between the observer and points on the stellar disk. The rays nearer the center of the stellar disk come from depths that are assumed to be infinitely optically thick. The rays farther from the center of the stellar disk penetrate to depths where the optical depth is assumed never to reach infinity (Rybicki 1971; Lester & Neilson 2008), although the rays can reach extremely large optical depths. Rays located toward the limb of the star can penetrate the tenuous outer atmosphere, never reaching large optical depths. As a result, the computed intensity profiles have a point of inflection (see Fig. 1) where the intensity derivative, dI/dμ, is not changing monotonically, which prevents the simple limbdarkening laws from fitting as well.
While it is difficult to draw conclusions from the predicted errors, we can reliably state that the linear and exponential limbdarkening laws do not fit the spherical model atmospheres. The predicted errors for those limbdarkening laws range from 0.05 to 0.2 and are significantly greater than the errors for the fits to planeparallel models. The bestfitting relations are the squareroot law and the fourparameter limbdarkening law of Claret (2000), which have errors less than 0.08.
Another thing to note is that based on fits to planeparallel model atmospheres, DiazCordoves et al. (1995) suggested that the squareroot law is more adequate for fitting hotter stars (T_{eff} > 8000 K), although they were unclear which law is preferred for cooler stars. For spherical model atmospheres we find that the predicted errors for the squareroot limbdarkening law are less than the errors for the quadratic law, making the former the clear preference. Also, the quadratic limbdarkening law is of particular interest because it is the most commonly used limbdarkening law for analyzing planetary transit observations (Mandel & Agol 2002). However, numerous comparisons of quadratic limbdarkening laws fit directly to observations and those fit to model stellar atmospheres suggest disagreement for a number of cases (Howarth 2011b). The results presented here suggest it may be advantageous to consider fitting transit observations with a squareroot limbdarkening law or the more accurate fourparameter limbdarkening law.
Fig. 7 Vband central intensity derivatives and gravitydarkening coefficients as function of effective temperature (left) and gravity (right) computed from planeparallel (red crosses) and spherically symmetric (blue squares) model stellar atmospheres. 

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4. Gravitydarkening coefficients
Claret & Bloemen (2011) computed wavelengthdependent gravitydarkening coefficients from Atlas planeparallel model stellar atmospheres based on the analytic relation developed by Bloemen et al. (2011). In Paper I we used this same prescription for both planeparallel and spherically symmetric model stellar atmospheres to compute gravitydarkening coefficients for cool giant stars, and we found that model geometry played a negligible role in determining gravitydarkening coefficients except for T_{eff} < 4000 K. At the cooler effective temperatures, the spherically symmetric model gravitydarkening coefficients are predicted to be vary significantly, and are up to an orderofmagnitude greater than those predicted from planeparallel model atmospheres.
We repeat that analysis here for our higher gravity model stellar atmospheres. As described by Bloemen et al. (2011), the gravitydarkening coefficient, y(λ) for a star is (8)As described in Paper I, von Zeipel (1924) showed that T_{eff} ~ (g_{eff})^{β1/4}, where β_{1} ≡ dlnT_{eff}/dlng. As previously, we assume β_{1} = 0.2 for models with T_{eff} < 7500 K and β_{1} = 1 otherwise. Using these constant values for β_{1} provides only a limited analysis of the gravitydarkening because β_{1} is a function of effective temperature, but assuming these two values does enable us to gain some perspective on the role of model atmosphere geometry. The other terms are the partial derivatives of the wavelengthdependent intensity with respect to gravity and effective temperature, respectively.
We compute the two intensity derivatives and predicted gravitydarkening coefficients for our grids of planeparallel and spherically symmetric model atmospheres and plot the predicted values in Fig. 7 for the Kepler waveband. The predicted derivatives and gravitydarkening coefficients are similar to those computed in Paper I, for which there is little difference between spherically symmetric and planeparallel model predictions for effective temperatures greater than 4000 K. The spherical and planar predictions then diverge for cooler effective temperatures. However, the range of values for the spherical model predictions is less for the higher gravity models explored in this work relative to the lower gravity models studied in Paper I.
5. Interferometric angular diameter corrections
Interferometry provides precise measurements of stellar angular diameters. However, stellar interferometry measures the combination of angular diameter and intensity profile and the two quantities are degenerate. One route to break the degeneracy is to assume a uniform intensity profile and measure the uniformdisk angular diameter. The limbdarkened angular diameter can then be predicted from the uniformdisk angular diameter using corrections computed from stellar atmosphere models (Davis et al. 2000).
Another technique for measuring limbdarkened angular diameters is to assume a simple limbdarkening law and coefficients from model stellar atmospheres to fit the interferometric observations (e.g. Boyajian et al. 2012). However, this technique might also predict incorrect angular diameters because planeparallel model atmospheres are typically used for fitting limbdarkening coefficients. We can assess the potential error of assuming planeparallel limbdarkening coefficients to fit the angular diameter by comparing predicted angular diameter corrections from spherically symmetric model stellar atmospheres with those from planeparallel models.
Fig. 8 Interferometric angular diameter correction computed in Vband (top) and Kband (bottom) as functions of effective temperature (left) and gravity (right). Corrections computed from planeparallel model atmospheres are denoted with red x’s and spherically symmetric models blue squares. 

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In Fig. 8 we plot the V and Kband angular diameter corrections as a function of effective temperature and gravity for both spherical and planar model atmospheres. The Vband corrections vary from 0.93 to 0.97 for the planeparallel model atmospheres and from 0.92 to 0.95 for spherical models. The difference is more apparent if one considers stellar atmospheres with T_{eff} > 4500 K, where the difference between planeparallel and spherical model corrections is about 0.01 to 0.02. This suggests that employing planeparallel model corrections for measuring stellar angular diameters from interferometric observations will lead to a 1 to 2% underestimate of the angular diameter.
Similarly, the Kband corrections also vary as a function of model atmosphere geometry; planeparallel models suggest values of θ_{UD}/θ_{LD} = 0.98 to 0.99 while spherical models suggest θ_{UD}/θ_{LD} = 0.97 to 0.985. Again, using planeparallel model corrections to fit Kband interferometric observations will underestimate the actual angular diameter by about 1%. Thus, for precision measurements of angular diameters, hence fundamental stellar parameters from optical interferometry, one should employ more physically representative spherical model atmospheres. This appears to be the case even for main sequence stars with large gravities and small atmospheric extensions.
6. Summary
In this work, we followed up on the study of Paper I to measure how model stellar atmosphere geometry affects predicted limbdarkening coefficients, gravitydarkening coefficients and interferometric angular diameter corrections for main sequence FGK dwarf stars. As in Paper I, we find significant differences between predictions from planeparallel and spherically symmetric model atmospheres computed with the Atlas/SAtlas codes. The results in this article are surprising because geometry is believed to be not important for stars with smaller atmospheric extension, i.e. main sequence stars with log g ≥ 4. As atmospheric extension gets smaller, defined as the ratio of the atmospheric depth to stellar radius, then it is expected that a spherical model atmosphere should appear more and more like a planeparallel model atmosphere. However, even for small atmospheric extension models, we find differences in predicted intensity profiles, hence differences in limbdarkening and angular diameter corrections.
As in Paper I, there is negligible difference between gravitydarkening coefficients predicted from planar and spherical model atmospheres. This is because gravitydarkening coefficients depend heavily on the central intensity of the star, not the entire intensity profile. The central intensity is approximately a function of the effective temperature at τ_{Ross} = 1 according to the EddingtonBarbier relation (Mihalas 1978). Lester & Neilson (2008) showed that the atmospheric temperature structure computed from planeparallel and spherically symmetric model atmospheres for the same effective temperature and gravity primarily differs closer to the surface, τ_{Ross} < 2/3, and converges as τ → ∞. Because the computed temperatures at depth are very similar for the two geometries, the central intensity is also similar for both model geometries, making the gravitydarkening coefficients insensitive to model geometry. However, geometry is important for stars with T_{eff} < 4000 K, which is due to differences in the opacity structure and convection, which lead to changes in the temperature structure.
Angular diameter corrections do vary as a function of geometry. The corrections account for the degeneracy between the intensity profile and limbdarkened angular diameter in modeling interferometric observations. Therefore, differences between the intensity profiles of planeparallel and spherically symmetric model stellar atmospheres lead directly to differences between predicted angular diameter corrections. We find that spherically symmetric model corrections are about 1 to 2% smaller than planar model corrections for the main sequence stars analyzed here.
Similarly, we computed limbdarkening coefficients for six different limbdarkening laws. As in Paper I, we find that the linear law is least consistent with predicted intensity profiles and that the fourparameter law is best. We also find that the commonly used quadratic limbdarkening law does not fit spherically symmetric model atmosphere intensity profiles as precisely as the similar squareroot or fourparameter limbdarkening laws. This suggests that as planetarytransit observations become increasingly precise, the fourparameter law combined with the more physically representative spherically symmetric model stellar atmospheres will be more appropriate for fitting observations or, better still, using intensity profiles directly.
Summary of limbdarkening coefficient, gravitydarkening coefficient and interferometric angular diameter correction tables found online.
The angulardiameter corrections, limbdarkening and gravitydarkening coefficients are publicly available as online tables. Each table has the format T_{eff} (K), log g and M (M_{⊙}) and then the appropriate variables for each waveband, such as linear
limbdarkening coefficients. Tables for planeparallel model fits do not include mass. Tables of gravitydarkening coefficients also contain values of the intensity derivatives with respect to gravity and effective temperature. For planeparallel models, values of mass, radius and luminosity are presented as zero in the tables. We list the properties of these tables in Table 1 that are available from CDS. Tabulated grids of the model atmosphere intensity profiles used in this work are also available.
Acknowledgments
This work has been supported by a research grant from the Natural Sciences and Engineering Research Council of Canada, the Alexander von Humboldt Foundation and NSF grant (AST0807664).
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All Tables
Summary of limbdarkening coefficient, gravitydarkening coefficient and interferometric angular diameter correction tables found online.
All Figures
Fig. 1 Predicted Keplerband intensity profiles for planeparallel (solid line) and spherically symmetric (dotted line) model stellar atmosphere with T_{eff} = 5800 K, log g = 4.5 and M = 1.1 M_{⊙}. 

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In the text 
Fig. 2 The limbdarkening coefficient u, used in the linear law (Eq. (1)), applied to the Kepler photometric band. Crosses are the planeparallel model stellar atmospheres, and the squares are the spherical models. 

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In the text 
Fig. 3 Limbdarkening coefficients a and b used in the quadratic law (Eq. (2)) (left panel), and the coefficients c and d used in the squareroot law (Eq. (3)) (right panel), all applied to the Kepler photometric band. The symbols have the same meanings as in Fig. 2. 

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In the text 
Fig. 4 Limbdarkening coefficients g and h used in the exponential law (Eq. (5)) (left panel), and the coefficients m and n used in the logarithmic law (Eq. (6)) (right panel), all applied to the Kepler photometric band. The symbols have the same meanings as in Fig. 2. 

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In the text 
Fig. 5 Limbdarkening coefficients f_{1}, f_{2}, f_{3} and f_{4} used in the Claret (2000) fourparameter law, Eq. (4), applied to the Kepler photometric band. The symbols are the same as in Fig. 2. 

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In the text 
Fig. 6 The error of the bestfit limbdarkening relation, defined by Eq. (7), for every model atmosphere (crosses represent planeparallel models, squares spherical models) for each of the six limbdarkening laws at Keplerband wavelengths. 

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In the text 
Fig. 7 Vband central intensity derivatives and gravitydarkening coefficients as function of effective temperature (left) and gravity (right) computed from planeparallel (red crosses) and spherically symmetric (blue squares) model stellar atmospheres. 

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In the text 
Fig. 8 Interferometric angular diameter correction computed in Vband (top) and Kband (bottom) as functions of effective temperature (left) and gravity (right). Corrections computed from planeparallel model atmospheres are denoted with red x’s and spherically symmetric models blue squares. 

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In the text 