Issue 
A&A
Volume 554, June 2013



Article Number  A98  
Number of page(s)  10  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/201321502  
Published online  11 June 2013 
Sphericallysymmetric model stellar atmospheres and limb darkening
I. Limbdarkening laws, gravitydarkening coefficients and angular diameter corrections for red giant 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, ON L5L 1C6, Canada
^{3} Department of Astronomy & Astrophysics, University of Toronto, ON M55, 3H4, Canada
email: lester@astro.utoronto.ca
Received: 18 March 2013
Accepted: 30 April 2013
Model stellar atmospheres are fundamental tools for understanding stellar observations from interferometry, microlensing, eclipsing binaries and planetary transits. However, the calculations also include assumptions, such as the geometry of the model. We use intensity profiles computed for both planeparallel and spherically symmetric model atmospheres to determine fitting coefficients in the BVRIHK, CoRot and Kepler wavebands for limb darkening using several different fitting laws, for gravitydarkening and for interferometric angular diameter corrections. Comparing predicted variables for each geometry, we find that the spherically symmetric model geometry leads to different predictions for surface gravities log g < 3. In particular, the most commonly used limbdarkening laws produce poor fits to the intensity profiles of spherically symmetric model atmospheres, which indicates the need for more sophisticated laws. Angular diameter corrections for spherically symmetric models range from 0.67 to 1, compared to the much smaller range from 0.95 to 1 for planeparallel models.
Key words: stars: atmospheres / stars: latetype / binaries: eclipsing / stars: evolution / techniques: interferometric
Tables 2–17, and the 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/554/A98
© ESO, 2013
1. Introduction
Stellar limb darkening is an important tool for interpreting interferometric, microlensing and eclipsing binary observations of red giant and supergiant stars. It also provides critical information about the temperature structure of a stellar atmosphere (Schwarzschild 1906) as well as a measure of the radial extension of an atmosphere (Neilson & Lester 2012).
Interferometric observations measure the angular diameter of a star as well as the intensity variation across the stellar surface. Some of the first interferometric observations measured only uniformdisk angular diameters, that is the angular diameter for a star assumed to have a constant surface brightness (Hanbury Brown et al. 1974). Wittkowski et al. (2004) presented Kband interferometric observations of the M3 giant ψ Phoenicis with measurements of the first and second lobes of the visibility curve, which constrain limb darkening. Unfortunately, these observations were not precise enough to distinguish between different model stellar atmospheres. Advances in interferometric observations have allowed for observations of convective cells in Betelgeuse (Haubois et al. 2009) and measurements of gravity darkening in Altair (van Belle et al. 2001). In terms of model stellar atmospheres, Aufdenberg et al. (2005) constrained threedimensional models using observations of Procyon.
Microlensing observations, like interferometry, also probe stellar limb darkening, but unlike interferometry, which targets specific nearby stars, microlensing observations are random. An et al. (2002) and Fields et al. (2003) constrained nonlinear limbdarkening relations from microlensing observations of a K3 giant and compared them to model stellar atmospheres. They found significant disagreement between the observed and predicted limbdarkening relation. More recently, however, microlensing observations have only constrained linear limbdarkening relations for red giant stars (Fouqué et al. 2010; Zub et al. 2011).
Eclipsing binaries and planetary transits provide yet another avenue for measuring stellar limb darkening. In terms of red giant stars, there are a number of known eclipsing binary systems, specifically the ζ Aurigae systems that have a K45 red giant primary and a mainsequence Btype companion. Eaton et al. (2008) fit the orbits for several of these systems assuming a simple linear limbdarkening law. There is also the potential of observing planets transiting red giant stars, which would provide powerful constraints of theories of planetary evolution. Currently, extrasolar planets have been observed orbiting dwarf and subgiant stars (Howell et al. 2012), but not giant stars; future missions such as PLATO may remedy this (Catala et al. 2010).
These three types of observations are ideal tools for probing stellar atmospheres and constraining the physics employed in numerical models. Likewise, predictions from model stellar atmospheres help constrain these types of observations. Recently, Sing (2010), Howarth (2011a) and Claret & Bloemen (2011) presented limbdarkening laws fit to planeparallel model stellar atmosphere intensity profiles. Even more recently, Claret et al. (2012, 2013) fit limbdarkening laws to sphericallysymmetric PHOENIX model stellar atmospheres of cool brown dwarf stars. In this work, we study how the assumed geometry of the model stellar atmosphere, plane parallel versus spherically symmetric, affects predictions of stellar limb darkening, gravity darkening and interferometric angular diameter corrections. We examine model atmospheres spanning the effective temperature and gravity range consistent with yellow and red giant and supergiant stars. Tables of limbdarkening and gravitydarkening coefficients, as well as new angular diameter corrections are presented as more physically based tools for understanding these bright stars.
In Sect. 2 we describe the stellar atmosphere code used in this work, as well as the model atmosphere grids computed for both planeparallel and spherically symmetric geometries. In Sect. 3 limbdarkening coefficients are presented for several commonly used limbdarkening relations. We compute gravitydarkening coefficients in Sect. 4 and angular diameter corrections in Sect. 5. Computations in these three sections provide insight into how intensity profiles depend on the assumed model geometry that can be directly compared to observations.
2. Model stellar atmospheres
Model stellar atmospheres form a key foundation of our understanding of stars, arguably a great success of computational astrophysics. However, the early success of model atmosphere codes transformed them into standard tools, and only in the past decade have these codes moved beyond simple planeparallel, localthermodynamicequilibrium (LTE) models to full threedimensional, statisticalequilibrium codes that can model nonLTE physics as well as stellar convection. Unfortunately, computing power is still limited for calculating largescale model atmosphere grids varying stellar gravity, effective temperature, stellar mass and composition.
A step toward more realistic geometry is achieved by shifting from onedimensional planeparallel model stellar atmosphere codes to onedimensional spherically symmetric codes, which can be used to compute large grids of models atmospheres that include physics that is more appropriate to stars where the depth of the stellar photosphere is a significant fraction of the stellar radius, such as evolved giant and supergiant stars and premain sequence stars. One such code for modeling atmospheres assuming spherically symmetric geometry is the SAtlas code (Lester & Neilson 2008). This code is based on the ATLAS code developed by Kurucz (1979), and continues its assumption of local thermodynamic and hydrostatic equilibrium. However, the radiative transfer is computed assuming spherical geometry using the Rybicki (1971) version of the Feautrier (1964) raytracing method, while radiative and convecting equilibrium is enforced using an updated version of the Avrett & Krook (1963) temperature correction method. Models computed using this code have been compared to sphericallysymmetric Phoenix and MARCS models (Hauschildt et al. 1999; Gustafsson et al. 2008) and shown to produce similar results (Lester & Neilson 2008; Neilson & Lester 2008).
In this work we use the grid of spherical model atmospheres from Neilson & Lester (2011), extended in mass up to M = 20 M_{⊙}. The grid assumes solar composition and spans the gravities from log g = − 1 to log g = 3 in steps of 0.25, effective temperatures from T_{eff} = 3000 to 8000 K and masses from M = 2.5 to 20 M_{⊙} in steps of 2.5 M_{⊙} and includes models with masses M = 0.5 and 1 M_{⊙}. Surface intensities are computed for each model at 1000 equally spaced values of μ = cosθ, where θ is the angle between the vertical direction and the direction toward a distant observer. Limbdarkening profiles are computed for JohnsonCousins BVRIHKwavebands (Johnson & Morgan 1953; Bessell 2005) along with the CoRot (Auvergne et al. 2009) and Kepler (Koch et al. 2004) wavebands. Angular diameter corrections for interferometric observations, gravitydarkening coefficients and various limbdarkening relations are computed using these wavelengthintegrated intensity profiles.
3. Limbdarkening laws
An understanding of stellar limb darkening is required to model the properties of interferometric, eclipsing binarystar, microlensing, and planetarytransit observations. As these observations become more precise and more accurate, models of stellar limb darkening must also improve. Limb darkening is typically treated as a simple parametrization as a function of θ (e.g. Fouqué et al. 2010; Croll et al. 2011), which makes fitting the stellar intensity profile much simpler and reduces the number of free parameters. The most common parametrizations are linear and quadratic relations (AlNaimiy 1978; van Hamme 1993; DiazCordoves et al. 1995), but other suggested relations include a fourparameter relation (Claret 2000a), a squareroot relation (Wade & Rucinski 1985) as well as exponential and logarithmic relations (Claret 2000a; Claret & Hauschildt 2003).
Fig. 1
Keplerband model intensity profiles (blacksolid) predicted for both planeparallel (left) and spherically symmetric (right) model stellar atmospheres with T_{eff} = 5000 K, log g = 2 and M = 10 M_{⊙}. Along with the intensity profiles, bestfit linear (greendashed), quadratic (orangeshortdashed), squareroot (bluedotted), fourparameter (violetlongdashdotted), logarithmic (brownshortdashdotted), and exponential (greydoubledash) limbdarkening relations are plotted. Bottom panels show the difference, Δ ≡ I_{model} − I_{law}, between model intensities and bestfit limbdarkening laws. 
3.1. Bestfit limbdarkening laws
We fit the following limbdarkening relations to the grids of planeparallel and spherically symmetric model stellar atmospheres: We derive the bestfit coefficients for each of the limbdarkening laws using a general leastsquares algorithm. This was done using the computed surface intensities for the BVRIHK and CoRot and Keplerwavebands. Figure 1 shows the Keplerband intensity profile and corresponding bestfit limbdarkening laws for both spherical and planeparallel model atmospheres with the properties T_{eff} = 5000 K, log g = 2 and M = 10 M_{⊙} (mass is defined for the spherical model only). The chosen limbdarkening laws all fit the planeparallel model intensity profiles well. This is not surprising because planeparallel model atmosphere intensity profiles do not deviate significantly from being linear, and a linear term is included in all of the chosen limbdarkening laws. However, sphericallysymmetric model stellar atmospheres have intensity profiles that are significantly nonlinear, and the bestfit limbdarkening relations for these intensity profiles match less well than for the planeparallel models because of this nonlinearity. For the model shown in Fig. 1, limbdarkening laws predict intensities that vary by Δ ≡ I_{model} − I_{law} = 0.15 for the spherical model while Δ < 0.04 for the planeparallel model. Although limbdarkening laws fit the intensities of planeparallel model atmospheres better than spherically symmetric models, the spherical models are more physically realistic, making them the more appropriate choice to use in modeling observations. We explore the uncertainty of the limbdarkening fits later.
Fig. 2
Limbdarkening coefficient u, used in Eq. (0), applied to the Kepler photometric band. Red crosses are the planeparallel model stellar atmospheres, and the blue squares are the spherical models. 
We present in Figs. 2–5 the coefficients derived by leastsquares fitting for the limbdarkening laws given by Eqs. (1)–(6) respectively for the Kepler photometric band as a function of effective temperature for both planeparallel and spherically symmetric model stellar atmospheres. It is clear that more realistic spherically symmetric model stellar atmospheres predict limbdarkening coefficients that vary much more as a function of effective temperature than those for planeparallel models. For the simplest case of the linear limbdarkening law, the ucoefficient determined from planeparallel models in the Keplerband vary from u = 0.2 to 0.5, whereas spherical models with the same effective temperatures and gravities vary from u = 0.6 to 1.4. The coefficients predicted for almost all limbdarkening laws examined here show the same behavior as the limbdarkening coefficients predicted for a fluxconserving linear+squareroot law (Neilson & Lester 2011, 2012). This uniform dependence of the coefficients on T_{eff} is surprising and suggests all of these laws carry essentially the same information regarding the moments of the intensity and the atmospheric extension about the stellar atmosphere in question. The one exception is the Claret (2000a) fourparameter limbdarkening law, for which the coefficients appear to vary much more as a function of effective temperature.
Fig. 3
Limbdarkening coefficients a and b used in Eq. (1) (left panel), and the coefficients c and d used in Eq. (2) (right panel), all applied to the Kepler photometric band. The symbols have the same meanings as in Fig. 2. 
Fig. 4
Limbdarkening coefficients g and h used in Eq. (4) (left panel), and the coefficients m and n used in Eq. (5) (right panel), all applied to the Kepler photometric band. The symbols have the same meanings as in Fig. 2. 
Fig. 5
Limbdarkening coefficients f_{1}, f_{2}, f_{3} and f_{4} used in the Claret (2000a) fourparameter law, Eq. (3), applied to the Kepler photometric band. The symbols are the same as in Fig. 2. 
To explore the interdependence of the coefficients, we plot in Fig. 6 the Keplerband bcoefficient from the quadratic law as a function of the acoefficient. This plot is typical of all the twoparameter limbdarkening laws considered in this work as well as the limbdarkening law employed by Neilson & Lester (2011, 2012), including the apparent hook in the correlation between coefficients. Figure 6 also plots the values of f_{2} + f_{4} as a function of f_{1} + f_{3} for the fourparameter law, again for the Kepler photometric band. The correlation for both planeparallel and spherical models is readily apparent. A bestfit linear relation to the coefficients for spherical models is (7)The correlation is different for planeparallel models for which the slope is − 0.978 and the intercept is 0.493.
Fig. 6
Correlation between the Keplerband limbdarkening coefficients for the quadratic law (left) and for the fourparameter law, f_{2} + f_{4} as a function of f_{1} + f_{3} (right). Red crosses represent coefficients computed from planeparallel models and blue squares spherical models. 
These correlations are caused by the limbdarkening coefficients being linear combinations of various angular moments of the intensity. For instance, in planeparallel model atmospheres the moments J ≡ ^{∫}I(μ)dμ and K ≡ ^{∫}I(μ)μ^{2}dμ are related such that J = 3K (Mihalas 1978). In spherical symmetry, this ratio varies, causing the moments of the intensity to differ in spherical symmetry from those predicted moments for planeparallel model stellar atmospheres. This difference in geometry is reflected in the difference between the zeropoints of the relation Eq. (7) for spherical models and that for planeparallel models. One can potentially use this difference to test observations and test model geometry.
3.2. Error analysis
Various limbdarkening laws, such as those given in Eqs. (0)–(5), are fit to the surface intensities computed with model stellar atmospheres, and it is important to understand how well these laws represent the actual intensities. For instance, DiazCordoves et al. (1995) argued that a squareroot law fit intensity profiles for hotter stars (T_{eff} > 9000 K) better than a quadratic law, whereas no limbdarkening law is preferred for cooler stars. We compute the relative error of the limbdarkening fit, Δ, using the relation (8)which quantifies the deviation of the bestfit limbdarkening law from the surface intensities of the model atmosphere. We compute the relative error for each bandpass as a function of the fundamental stellar parameters for both planeparallel and spherical geometries, and show in Fig. 7 the relative errors as a function of effective temperature for fits in the Keplerband. The relative error of the fits for spherical models is greater than the error for planeparallel fits for all the limbdarkening laws. The errors are similar only for T_{eff} ~ 3500 K, where the spherical model atmospheres predict intensity profiles that are closest to being linear, with the error of the linear limbdarkening law approaching a minimum value. This result appears to suggest that these limbdarkening laws are inappropriate for fitting light curves and interferometric observations, but this is not true.
Fig. 7
Error of the bestfit limbdarkening relation, defined by Eq. (8), for every model atmosphere (red crosses represent planeparallel models, blue squares spherical models) for each of the six limbdarkening laws at Keplerband wavelengths. 
There are a number of issues with how the relative error is computed and what the error tells us, such as how the limbdarkening laws are defined, how they are fit to the surface intensities and the effect of sampling.

Defining limbdarkening laws: The intensity profiles computed using the planeparallel and spherical model atmospheres employed in this work are normalized with respect to the central intensity so thatI(μ = 1) ≡ 1. Furthermore, all limbdarkening laws, except the exponential law, are defined so that the I(μ = 1) ≡ 1, regardless of the values of the bestfit coefficients. As a result, every fit to an intensity profile is anchored to the center of the stellar disk before representing the remainder of the intensity profile. This definition alone results in a perfect fit to the center of the stellar disk and a deteriorating fit as μ → 0 as the intensity profile deviates from the assumed structure of a particular limbdarkening law.

Fitting limbdarkening laws: Limbdarkening laws are typically fit to intensity profiles using a leastsquare algorithm. Neilson & Lester (2011) showed that the bestfit coefficients for a given law are functions of weighted integrals of the intensity profile. For example, the linear limbdarkening coefficient from Eq. (1) is a function of the mean intensity, J, and the stellar flux, ℋ ≡ ^{∫}I(μ)μdμ, and both of these quantities are more sensitive to the central intensity than to the much smaller intensity near the limb. As with the definition of the limbdarkening laws, using a least square fitting algorithm fits the central part of the intensity structure better. Similarly, one might fit limbdarkening coefficients by enforcing flux conservation, but because the flux is the μweighted integral of the intensity, any fluxconserving fit is constrained weakly by the intensity at the stellar limb relative to the intensity near the center of the stellar disk.
Fig. 8 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.
Fig. 9 Interferometric angular diameter correction computed in Vband (top) and Kband 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.

Sampling issues: Sampling is the most important of the three issues affecting the computed error of the fit of the intensity profile. For instance, Wade & Rucinski (1985) and Heyrovský (2007) noted that fitting an intensity profile that is uniformly sampled in μ has a larger error than fitting the same profile that is uniformly sampled in r = sinθ = sin(cos^{1}μ). Uniform rspacing emphasizes the intensity profile near the center of the disk while a uniform μspacing emphasizes the limb. Adopting any of the limbdarkening laws presented here, that law will fit the central part of the stellar surface more precisely than the limb because of the normalization at the center of the disk. If, in addition, the surface intensity is sampled uniformly in r, that will give added weight to the central region. These two factor combine to make the computed error of the fit smaller. Similarly, Howarth (2011b) found that limbdarkening coefficients derived from planetary transits with large impact factors do not agree with model stellar atmosphere predictions. This is because the planet passes across only the limb of the star and not the center, therefore probing only part of the intensity profiles. Claret (2008, 2009) also found disagreement between theoretical limbdarkening coefficients and empirical coefficients measured from eclipsing binary light curves and comparisons to the planetary system HD 209458. Limbdarkening coefficients from stellar atmosphere models fit the whole profile yielding different results.
The combination of these three factors lead to calculated errors that are relative and not an absolute measure of the quality of the fit. In this work, differences in the error between fits to planeparallel and spherically symmetric model stellar atmosphere intensity profiles computed with the same properties are due solely to differences in the intensity profile near the limb where the spherical models provide more realistic predictions. Therefore, the error analysis suggests that the various limbdarkening laws lack the necessary complexity to precisely fit intensity profiles from spherical models. The only exception is the Claret (2000a) fourparameter law, which fits the laws best, but appears to have unique properties.
4. Gravity darkening coefficients
Rapid rotation distorts the shape of a star, making it aspheric, with flattened poles and a bulged equator. As shown first by von Zeipel (1924), the gravity and effective temperature vary in a coordinated way across the stellar surface such that at any point the effective temperature is proportional to the effective gravity, , where β_{1} = 1 for radiative stars. However, this value of β_{1} is valid only for bolometric radiation, and Kopal (1959) later derived monochromatic gravitydarkening corrections, y(λ). Claret (2000a), Claret & Hauschildt (2003) and Claret & Bloemen (2011) have computed wavebanddependent gravitydarkening corrections as a a function of the central intensity of the star, as well as the gravity, effective temperature and the variable, β_{1} from planeparallel models. Bloemen et al. (2011) derived (9)and noted that (dlnT_{eff}/dlng) = β_{1}/4. The variable β_{1} is a function of effective temperature, but for the purpose of this analysis we assume β_{1} = 0.2 for T_{eff} < 7500 K and β_{1} = 1 for hotter stars. However, the value of β_{1} based on von Zeipel’s theorem is not strictly valid for radiative or convective stellar envelopes (Claret 2000b, 2012; Espinosa Lara & Rieutord 2011).
In Fig. 8, we plot the Vband values of each intensity derivative for each model stellar atmosphere in Eq. (9), as well as y(λ) computed for the assumed values of β_{1}. We find that planeparallel and spherically symmetric model stellar atmospheres predict similar gravitydarkening coefficients for T_{eff} > 4000 K, but there are significant differences for cooler stars. We interpret these differences for the cooler stars as consequences of both surface convection and the shift from the negative hydrogen ion to titanium oxide as the dominant opacity source. Both planeparallel and spherical model intensities show greater variation at these cool effective temperatures, but the intensity profiles of spherically symmetric model atmospheres vary more than that of planeparallel model atmospheres.
While the most significant differences between spherical and planar model predictions of gravitydarkening coefficients are at lower temperatures, the gravitydarkening coefficients computed from spherically symmetric models are greater than those of planeparallel models for every effective temperature. For example, a spherically symmetric model with T_{eff} = 8000 K has a Vband gravitydarkening coefficient of y_{V} ≃ 0.165 while the planeparallel model with the same effective temperature has y_{V} ≃ 0.14. The difference is small but systematic.
5. Angular diameter corrections
Interferometric observations measure the angular diameter of a star along with its limbdarkening profile, but, unfortunately, the measured angular diameter and limbdarkening profiles are not independent quantities. This is especially true when the measured visibilities do not probe the second lobe. Davis et al. (2000) measured stellar angular diameters from interferometric observations by assuming that the stellar intensity profile is uniform, i.e. the intensity at any point on a stellar disk is equal to the central intensity. In that case, the uniformdisk angular diameter can be directly fit to the observed visibilities and then converted to a limbdarkened angular diameter using model stellar atmospheres. Davis et al. (2000) computed corrections using planeparallel ATLAS models (Kurucz 1993) and found k ≡ θ_{UD}/θ_{LD} = 0.91 to 0.98 in the wavelength range λ = 400–800 nm. These limbdarkening corrections have been applied to observations of Cepheids (Gallenne et al. 2012) and Sirius (Davis et al. 2011) for example.
We compute angular diameter corrections using the recipe described by Marengo et al. (2004), where we assume a limbdarkened angular diameter of θ_{LD} = 1 mas to compute interferometric visibilities from a model atmosphere intensity profile. That synthetic visibility is then fit by a uniformdisk angular diameter. The bestfit uniformdisk angular diameter is then equivalent to the theoretical angular diameter correction. We compute angular diameter corrections for the JohnsonCousins BVRIHK wavebands and show the corrections for the V and Kbands in Fig. 9 as a function of effective temperature for planeparallel and spherically symmetric models. Corrections from spherical models clearly differ from corrections from planeparallel model atmospheres. Intensity profiles from planeparallel model stellar atmospheres predict corrections in the narrow range from k = 0.97–0.99 in Vband and approaches unity for longer wavelengths. We show in Fig. 9 the V and Kband angular diameter corrections as function of effective temperature and gravity for planeparallel and spherically symmetric model stellar atmospheres.
Fits to sphericallysymmetric model atmospheres suggest significantly different angular diameter corrections as functions of both effective temperature and gravity. The Vband corrections from spherical models, denoted k_{s}, range from k_{s} = 0.67 to 0.95, with no overlap with the planeparallel model predictions. The Kband corrections show similar behaviors except that spherical and planar corrections overlap somewhat. These results suggest that using planeparallel model atmosphere corrections systematically underestimates the stellar angular diameter. For instance, Mozurkewich et al. (2003) presented uniformdisk angular diameters for a sample of 85 stars, along with limbdarkened angular diameters corrected using limbdarkening coefficients from Claret et al. (1995) and DiazCordoves et al. (1995). Their angular diameter corrections vary from k = 0.89 to ≈1, consistent with the values found here for planeparallel model atmospheres.
Of particular interest are the results of Mozurkewich et al. (2003) for α Persei (F5 Ib), for which they measured T_{eff} = 6750 K, and for ϵ Geminorum (G8 Ib), which was measured to have T_{eff} = 4485 K. Mozurkewich et al. (2003) measured the uniformdisk angular diameters at 550 nm to be 2.986 ± 0.042 for α Per and 4.467 ± 0.115 mas for ϵ Gem. Using these they computed limbdarkened angular diameters of 3.188 ± 0.035 and 4.703 ± 0.047 mas, respectively. Our sphericallysymmetric models with log g = 1.5 and M = 10 M_{⊙} yield Vband angulardiameter corrections of 0.929 for α Per and 0.916 for ϵ Gem. Applying these to the uniform disk measurements gives larger limbdarkened angular diameters: θ_{LD} = 3.214 mas for α Per and θ_{LD} = 4.877 mas for ϵ Gem. The spherical correction for α Per yields a value for θ_{LD} that is marginally consistent with the angular diameter found using planeparallel correction, whereas the limbdarkened angular diameter of ϵ Gem measured by Mozurkewich et al. (2003) is almost 4% smaller than what would be predicted by applying spherical model corrections. This difference may appear to be small but this underestimate is systematic.
As a test, we check how the angular diameter corrections vary as function of stellar mass. Because models with low effective temperature but relatively high gravity appear to predict the smallest corrections, we hold T_{eff} = 3500 K and log g = 2. The angular diameter corrections are shown in Fig. 10 as a function of stellar mass for the six JohnsonCousins wavebands considered in this work. The figure suggests that the corrections are insensitive to the mass of the stellar model except for lowmass (M ≤ 1 M_{⊙}) models. This is reassuring and suggests that when applying these corrections, one can ignore the stellar mass. The difference between limbdarkening profiles and angular diameter corrections is small and consistent with previous results by Lester et al. (2013).
Fig. 10
Interferometric angular diameter corrections as a function of waveband and stellar mass for spherically symmetric model stellar atmospheres with log g = 2 and T_{eff} = 3500 K. 
6. Summary
In this work, we present model atmosphere intensity profiles for the BVRIHK, CoRot and Kepler passbands from both planeparallel and spherically symmetric geometries based on models computed by Neilson & Lester (2011, 2012). We fit a number of limbdarkening laws to these intensity profiles, as well as compute gravitydarkening coefficients and angular diameter corrections for interferometry. We test how these fits vary as a function of model atmosphere geometry and compile tables of limbdarkening coefficients, gravitydarkening coefficients and angular diameter corrections that can be applied to observations.
We consider six limbdarkening laws in this work: linear, quadratic, squareroot, fourparameter, exponential and logarithmic. These laws fit the intensity profiles from planeparallel model atmospheres well, but not the intensity profiles of the spherical models based on computed relative errors. The one exception is the Claret (2000a) fourparameter law, for which the difference between the spherical model intensity profiles and the predictions of the fitting law is small enough to still be applicable to observations, although the law still fits the spherical profiles more poorly than the planeparallel intensities.
While those predicted errors are useful for comparing fits to planar and spherical model intensity profiles, they are not ideal for studies of actual limb darkening. Bestfit limbdarkening coefficients depend on the definition of the laws, all of which anchor the fit to I(μ = 1) = 1, making the fit sensitive to the sampling of the intensity profile as well as to the method for fitting the data. Because intensity profiles for spherical models are more complex, the fitting error is greater than the error for simpler planeparallel model intensity profiles. However, spherically symmetric model atmospheres are a more realistic representation of actual stellar atmospheres, meaning they are better suited for limb darkening studies.
Fits to the fourparameter limbdarkening law also show correlations between the limbdarkening coefficients; we find that the linear combination of the four coefficients are approximately constant, with that constant being a function of the atmosphere’s geometry. This result suggests that the linear combination of the observed coefficients for the fourparameter law provides a simple test of whether the observations are probing the edge of the stellar disk, i.e. sphericity.
We also predict wavelengthdependent gravitydarkening coefficients based on the Claret & Bloemen (2011) prescription. Unlike the limbdarkening coefficients, the gravitydarkening coefficients are less dependent on model atmosphere geometry. This is because the gravitydarkening coefficients depend on the change of the central intensity with respect to effective temperature and gravity, hence the difference between atmospheres for the same geometry. Gravitydarkening is also a function of the central intensity, which is insensitive to model geometry. The spherically symmetric gravitydarkening coefficients are similar to planeparallel coefficients for T_{eff} > 5000 K and begin to diverge for cooler stellar atmosphere models. Only at the coolest effective temperatures, 3000 K ≤ T_{eff} ≤ 4000 K, is the geometry of the model atmosphere important, with the spherically symmetric coefficients being approximately an orderofmagnitude greater than those predicted from planeparallel model atmospheres.
Unlike the gravity darkening coefficients, the interferometric angulardiameter corrections do depend on geometry. For planeparallel model atmospheres the angulardiameter corrections vary from about 0.95–1, whereas the corrections for spherically symmetric model atmospheres vary from 0.67–1. Previous analyses had assumed that corrections from planeparallel models are applicable to all stars, but this is not true. At low gravity, log g < 3, spherically symmetric corrections deviate significantly from planeparallel model predictions. The difference between spherical and planeparallel models is a function of both gravity and effective temperature and also appears to vary as a function of stellar mass.
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 of gravitydarkening coefficients also contain values of the intensity derivatives with respect to gravity and effective temperature. For planeparallel models, only T_{eff} and log g are given in the tables. We list the properties of these tables in Table 1, that are archived in electronic form at the CDS. Model atmosphere intensity profiles are also archived at the CDS
Summary of limbdarkening coefficient, gravitydarkening coefficient and interferometric angular diameter correction tables found online.
Techniques such as optical interferometry, microlensing observations, planetary transit and eclipsing binary observations are continuously improving the measurements of stellar limb darkening needed to test model stellar atmospheres and the physics assumed in their calculation. At lower gravities, these observations require the more physically realistic spherically symmetric models to constrain stellar properties. The predicted limbdarkening coefficients, gravitydarkening coefficients and angular diameter corrections from spherically symmetric SAtlas models are new tools that for aiding analyses of these observations.
Acknowledgments
The authors acknowledge support from a research grant from the Natural Sciences and Engineering Research Council of Canada, the Alexander von Humboldt Foundation and National Science Foundation (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
Keplerband model intensity profiles (blacksolid) predicted for both planeparallel (left) and spherically symmetric (right) model stellar atmospheres with T_{eff} = 5000 K, log g = 2 and M = 10 M_{⊙}. Along with the intensity profiles, bestfit linear (greendashed), quadratic (orangeshortdashed), squareroot (bluedotted), fourparameter (violetlongdashdotted), logarithmic (brownshortdashdotted), and exponential (greydoubledash) limbdarkening relations are plotted. Bottom panels show the difference, Δ ≡ I_{model} − I_{law}, between model intensities and bestfit limbdarkening laws. 

In the text 
Fig. 2
Limbdarkening coefficient u, used in Eq. (0), applied to the Kepler photometric band. Red crosses are the planeparallel model stellar atmospheres, and the blue squares are the spherical models. 

In the text 
Fig. 3
Limbdarkening coefficients a and b used in Eq. (1) (left panel), and the coefficients c and d used in Eq. (2) (right panel), all applied to the Kepler photometric band. The symbols have the same meanings as in Fig. 2. 

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

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

In the text 
Fig. 6
Correlation between the Keplerband limbdarkening coefficients for the quadratic law (left) and for the fourparameter law, f_{2} + f_{4} as a function of f_{1} + f_{3} (right). Red crosses represent coefficients computed from planeparallel models and blue squares spherical models. 

In the text 
Fig. 7
Error of the bestfit limbdarkening relation, defined by Eq. (8), for every model atmosphere (red crosses represent planeparallel models, blue squares spherical models) for each of the six limbdarkening laws at Keplerband wavelengths. 

In the text 
Fig. 8
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. 

In the text 
Fig. 9
Interferometric angular diameter correction computed in Vband (top) and Kband 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. 

In the text 
Fig. 10
Interferometric angular diameter corrections as a function of waveband and stellar mass for spherically symmetric model stellar atmospheres with log g = 2 and T_{eff} = 3500 K. 

In the text 
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