Issue 
A&A
Volume 544, August 2012



Article Number  A117  
Number of page(s)  7  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/201117244  
Published online  09 August 2012 
Using limb darkening to measure fundamental parameters of stars
^{1} Argelander Institute for Astronomy, University of Bonn, 53012 Bonn, Germany
email: hneilson@astro.unibonn.de
^{2} Department of Chemical and Physical Sciences, University of Toronto Mississauga, Mississauga, ON L5L 1C6, Canada
^{3} Department of Astronomy & Astrophysics, University of Toronto, Toronto, M5S 1K7, Canada
email: lester@astro.utoronto.ca
Received: 12 May 2011
Accepted: 19 July 2012
Context. Limb darkening is an important tool for understanding stellar atmospheres, but most observations measuring limb darkening assume various parameterizations that yield no significant information about the structure of stellar atmospheres.
Aims. We use a specific limbdarkening relation to study how the bestfit coefficients relate to fundamental stellar parameters from spherically symmetric model stellar atmospheres.
Methods. Using a grid of spherically symmetric ATLAS model atmospheres, we compute limbdarkening coefficients, and develop a novel method to predict fundamental stellar parameters.
Results. We find our proposed method predicts the mass of stellar atmosphere models given only the radius and limbdarkening coefficients, suggesting that microlensing, interferometric, transit and eclipse observations can constrain stellar masses.
Conclusions. This novel method demonstrates that limbdarkening parameterizations contain important information about the structure of stellar atmospheres, with the potential to be a valuable tool for measuring stellar masses.
Key words: stars: atmospheres / stars: fundamental parameters / stars: latetype
© ESO, 2012
1. Introduction
Limb darkening, the change of surface intensity from the center to the edge of a stellar disk, is a powerful measure of the physical structure of stellar atmospheres. However, it is difficult to observe the intensity profile across a stellar disk except for the Sun. Most observations of stellar limb darkening are indirect, coming from light curves of eclipsing binaries (Claret 2008) or planetary transits (Knutson et al. 2007), interferometric visibilities (Haubois et al. 2009) and microlensing light curves (An et al. 2002), and these indirect methods have limited precision (e.g. Popper 1984; Zub et al. 2011). Because of this, limb darkening is commonly treated as a parameterized function of μ ≡ cosθ, where θ is the angle between the direction to the distant observer and the normal direction at each location on the stellar surface. An example of a linear parameterization is (Schwarzschild 1906) (1)As numerical model atmospheres became robust, the computed intensity profiles were represented by more elaborate limbdarkening parameterizations that include higher order terms of μ or are expressed in terms of powers of r = sinθ (Heyrovský 2007), as well as being normalized with respect to the stellar flux instead of the central intensity (Wade & Rucinski 1985). These more detailed limbdarkening laws were used to interpret the improving observations noted above.
However, even with these advances, current limbdarkening observations are still unable to constrain the model stellar atmospheres. Recent interferometric observations of nearby red giants are not yet precise enough to differentiate between the centertolimb intensity profiles from PHOENIX and ATLAS model atmospheres or even between planeparallel or spherically symmetric models (Wittkowski et al. 2004, 2006a,b; Neilson & Lester 2008). However, the combination of the observations and models do provide constraints for fundamental stellar parameters such as the effective temperature and gravity.
In this work, we continue our previous analysis (Neilson & Lester 2011, hereafter Paper I) to explore the connection between limb darkening and stellar parameters. We adopt the fluxconserving law (2)where ℋ_{λ} is the Eddington flux defined as (3)because this law was used by Fields et al. (2003) to analyze the microlensing event EROSBLG20005 (An et al. 2002). We fit this limbdarkening law to the intensities of spherically symmetric model stellar atmospheres computed with the SATLAS code (Lester & Neilson 2008) to explore the relation between fundamental parameters and limbdarkening coefficients hinted at in Paper I.
2. Limb darkening and fundamental parameters
Many previous studies have used the limb darkening predicted by model stellar atmospheres as a tool to achieve a better determination of a stellar diameter (Mérand et al. 2010) or a better characterization of a transiting planet (Lee et al. 2012). Our goal is different; we want to determine how limb darkening, represented by Eq. (2), depends on the fundamental parameters of stellar atmospheres. Because there are several steps leading to our conclusion, we describe our approach stepbystep to make it as clear as possible.
Fig. 1 Bestfit Vband limbdarkening relations using Eq. (2) for grids of 5 M_{⊙} spherically symmetric and planeparallel ATLAS model atmospheres. A fixed μpoint is clearly seen in both cases. 
2.1. Intensity fixed point
Figure 1 plots the curves produced by the limbdarkening law given by Eq. (2) for the Vband intensities computed for both planeparallel and spherical ATLAS models. The cube of spherical models is from Paper I, with luminosities and radii corresponding to the range of 3000 K ≤ T_{eff} ≤ 8000 K in steps of 100 K, − 1 ≤ log g ≤ 3 in steps of 0.25 and 2.5 ≤ M/M_{⊙} ≤ 10 in steps of 2.5 M_{⊙}, which we have supplemented with additional models having masses of M = 0.5 and 1.0 M_{⊙}. The grid of planeparallel models spans the equivalent range of effective temperature and gravity. Figure 1 shows that both sets of models have a fixed point, μ_{1}, where the curves intersect, although there is a larger spread for the spherical models because of the combination of the three fundamental stellar parameters, L_{⋆}, M_{⋆} and R_{⋆}. The immediate vicinity of the spherical fixed point is displayed in more detail in Fig. 6 of Paper I.
2.2. Relationship between A and B
In Paper I we showed that the fixed point, μ_{1}, is caused by the limbdarkening law’s A_{λ}coefficient being linearly related to the B_{λ}coefficient, A_{λ} ∝ α_{λ}B_{λ}. The proportionality term, α_{λ}, depends only on two quantities, both of which are functions of the stellar intensity: the mean intensity, J_{λ}, and the pseudomoment , defined as (4)Combining these two radiation terms defines a new quantity, (5)which is similar to the Eddington factor, f = K/J (Mihalas 1978). As we showed in Eq. (19) of Paper I, α_{λ} depends only on η_{λ}, (6)
2.3. Relationship between α and μ_{1}
Our analysis in Paper I also showed that the fixed point, μ_{1}, is a function of α_{λ}. Rearranging our result from Paper I, we obtain (7)Working back through this chain of logic, we find that because μ_{1} depends on α_{λ}, which depends on η_{λ}, we are led to the conclusion that I_{λ}(μ_{1})/2ℋ_{λ} depends solely on the two angular moments of the stellar intensity, J_{λ} and .
The analytic derivations of Eqs. (6) and (7) in Paper I assumed, for convenience, that η_{λ} is the same for all models. This assumption is true for planeparallel models because they share the same flat geometry. However, η_{λ} varies for spherical atmospheres because they have different amounts of curvature depending on the values of their particular fundamental parameters. As a result, the values of α_{λ} and μ_{1} also vary for spherical atmospheres. However, this variation does not alter the logical connections between these quantities, as we show numerically in the following section.
2.4. Dependence on surface gravity
Having used our analytic results to trace the logical connection between the key variables, we now continue our exploration using large grids of planeparallel and spherical model atmospheres to compute the key quantities numerically, which enables us to drop the assumption that η_{λ} is constant. Figure 2 shows the results for the Vband.
Fig. 2 Surface gravity dependence of η_{λ}, α_{λ}, fixed point μ_{1} and the normalized intensity at the fixed point, I_{λ}(μ_{1})/2ℋ_{λ}, for the V spectral band. Planeparallel model atmospheres are denoted by black crosses, and spherical model atmospheres are represented by other symbols. Filled red circles are T_{eff} = 3000 K, open green squares 4000 K, open blue circles 5000 K, downward pointing magenta triangles 6000 K and upward pointing pale blue triangles are 7000 K. 
Beginning with the planeparallel models, the black crosses in the upper left plot of Fig. 2 shows that the η parameter is nearly constant over a very wide range of surface gravity for the models being considered. As required by Eq. (6), the near constancy of η causes the α coefficient also to be almost constant, as shown by the black crosses in lower left panel of Fig. 2. Next, as a result of Eq. (7), the fixed point, μ_{1}, must be nearly constant, as shown by the black crosses in the upper right panel of Fig. 2. Finally, when the nearly constant μ_{1} is used in Eq. (2), the result is a nearly constant value of the normalized intensity at the fixed point. This conclusion is confirmed by the black crosses in the lower right plot of Fig. 2.
The spherically symmetric models are fundamentally different. The various colored symbols in the upper left plot in Fig. 2 confirms the result of Sect. 2.3 that the parameter ηdoes vary with surface gravity. It then follows from Eq. (6) that α must also vary, as shown by the colored symbols in the lower left plot. As before, Eq. (7) requires that the variation of α leads to the variation of μ_{1} shown by the colored symbols in the upper right plot of Fig. 2. Finally, this leads to the variation of the normalized intensity at the fixed point shown by the colored symbols in the lower right plot. For spherically symmetric models, the conclusion is that the location and intensity of the fixed point depend on the fundamental stellar parameters.
The beginning step of this logical chain, the dependence of η on the atmospheric parameters, is physically reasonable. As shown in the upper left panel of Fig. 2, as log g decreases, η increases towards unity, meaning that . This is the result of the intensity profile becoming extremely centrally concentrated, or, equivalently, the atmospheric extension approaches the stellar radius. Because η depends upon the extension of the atmosphere, so do α, the fixed point μ_{1}, and the intensity at the fixed point.
3. Parameterization of atmospheric extension
To explore further the results shown in Fig. 2, we replace log g by a more explicit representation of the extension of the stellar atmosphere, for which we adopt (8)This dependence is physically plausible because the extension is larger for stars of larger radius, but greater masses pull the atmospheres into a more compact configuration.
One way of deriving Eq. (8) begins with the definition of the pressure scale height, (9)where H is the distance over which the pressure changes by a factor of e, k is the Boltzmann constant, T is the gas temperature, μ is the mean molecular weight of the gas, m_{H} is the mass of the hydrogen atom, and the surface gravity . Using H to represent the thickness of the atmosphere, the relative extension of the atmosphere is (10)both sides of which are dimensionless. Assuming that T and μ are nearly constant over the distance H, the relative extension of the atmosphere becomes (11)which is our expression.
Fig. 3 Extension parameter R_{⋆}/M_{⋆} compared to the relative atmospheric extension, ΔR/R_{⋆}, derived for more than 3000 spherical models. 
Fig. 4 (Left) Values of the pseudomoment η_{V} and variable α_{V} as a function of atmospheric extension for spherical atmospheres, R_{⋆}/M_{⋆} in solar units. (Right) The top panel shows the value of the primary fixed point, μ_{1}, for the linearplussquareroot parametrization, also plotted as a function of the extension of the spherical atmosphere. The bottom panel shows the dependence of the normalized Vband intensity of the fixed point on the atmospheric extension. The symbols have the same meaning as in Fig. 2. When comparing to Fig. 2, note that increasing extension corresponds to decreasing log g. 
To test the correlation of R_{⋆}/M_{⋆} with the actual atmospheric extension, we use the Rosseland optical depth to determine for each of our several thousand spherical models the quantity ΔR/R_{⋆}, where ΔR is the physical distance from the location where τ_{Ross} = 1 out to the location of where τ_{Ross} = 0.001, which is slightly more than one pressure scale height, and R_{⋆} is the radius where τ_{Ross} = 2/3. Figure 3 plots the extension parameter R_{⋆}/M_{⋆} for each model versus that model’s ΔR/R_{⋆}. While there is a spread because T and μ do vary over the distance H, it is obvious that R_{⋆}/M_{⋆} tracks the actual extension of the models quite well.
Using R_{⋆}/M_{⋆} to represent the atmospheric extension transforms Fig. 2 into 4, where the extension parameter for the planeparallel models is determined by assuming a mass of 5 M_{⊙} and computing the radius for each model from its surface gravity. Figure 4 shows, as expected, that η_{λ} is a tight function of log (R_{⋆}/M_{⋆}), except for the models with T_{eff} = 3000 K. It follows, because of the connections established in Sect. 2, that all other quantities also depend on R_{⋆}/M_{⋆}. For planeparallel atmospheres the quantities are nearly constant, as expected, and as they were found to be in Fig. 2. Therefore, the values of μ_{1} and I_{λ}(μ_{1})/2ℋ_{λ} depend on the fundamental stellar parameters R_{⋆} and M_{⋆}. The deviation of the models with T_{eff} = 3000 K may be due to changes in the atmospheres at the lowest temperature.
Excluding the models with T_{eff} = 3000 K, we fit for each waveband the dependence of μ_{1} and I_{λ}(μ_{1})/2ℋ_{λ} on log (R_{⋆}/M_{⋆}) shown in Fig. 4 using the functional forms (12)and (13)The values of C_{μ}, D_{μ}, C_{I}, and D_{I} for these fits to the spherical models are given in Table 1 for the five wavebands B,V,R,I, and H. We include fits for models with effective temperatures T_{eff} = 5000 K and models within the effective temperature range of 4100 − 4300 K (T_{eff} = 4200 ± 100 K). Equations (12) and (13) will be employed in later sections.
4. Iterative method
The right panels of Fig. 4 show that μ_{1} and I_{λ}(μ_{1})/2ℋ_{λ} depend on R_{⋆}/M_{⋆} for the spherical model atmospheres. It follows from the limbdarkening law, Eq. (2), that the coefficients A_{λ} and B_{λ} must also be functions of the atmospheric extension, which is confirmed in Fig. 5. Because the limbdarkening coefficients A_{λ} and B_{λ} in Eq. (2) can be derived from observations, this establishes an observational method for determining the atmosphere’s extension.
In Fig. 5 the open colored symbols show that the limbdarkening coefficients of stars with T_{eff} > 3000 K share a common dependence on the atmospheric extension for log (R_{⋆}/M_{⋆}) ≳ 1.5 in solar units. This suggests it is possible to derive a unique value of R_{⋆}/M_{⋆} for stars with atmospheric extensions R_{⋆}/M_{⋆} ≳ 30 R_{⊙}/M_{⊙}. Stars with log (R_{⋆}/M_{⋆}) < 1.5 show a clear dependence on T_{eff}.
The tight dependence of the μ_{1} and I_{λ}(μ_{1})/2ℋ_{λ} on R_{⋆}/M_{⋆} displayed in Fig. 4 follows directly from Sect. 2.3, where we showed that μ_{1} is a function of α_{λ}, which depends only on . The more scattered dependence of the coefficients A_{λ} and B_{λ} on the extension shown in Fig. 5 is due to these coefficients having a less direct functional dependence on the extension through the separate terms J_{λ}/2ℋ_{λ} and , as we showed in Paper I. Because of this, as we will show in the following sections, the values of A_{λ} and B_{λ} do not measure atmospheric extension as cleanly as using μ_{1} and I_{λ}(μ_{1})/2ℋ_{λ} as constraints.
Although A_{λ} and B_{λ} are less direct measures of the atmospheric extension, they are determined by the modeling of the lensing observations. Therefore, we propose an iterative method to use A_{λ} and B_{λ} to determine the atmospheric extension. The method begins by assuming an initial value for log (R_{⋆}/M_{⋆}), which is used to compute μ_{1} with Eq. (12). This value of μ_{1} and the coefficients of the fit given in Table 1 enable us to compute the normalized intensity at the fixed point using Eq. (2). Finally, we compute a new value of log (R_{⋆}/M_{⋆}) using Eq. (13). This new value of log (R_{⋆}/M_{⋆}) replaces our initial assumption, starting a second iteration, which is continued until the value of log (R_{⋆}/M_{⋆}) converges. If the stellar radius is determined by other methods, such as from interferometric observations and distance estimates, the converged value of log (R_{⋆}/M_{⋆}) can be solved for the star’s mass.
Fig. 5 Vband values of the A limbdarkening coefficient (top panel) and the B limbdarkening coefficient (bottom panel) for the linearplussquareroot parametrization given by Eq. (2), plotted as a function of the extension parameter, R_{⋆}/M_{⋆} in solar units. The symbols have the same meaning as in Fig. 2. 
5. Test using a model atmosphere
The previous section describes a potential method for determining the fundamental parameters of a star using stellar limbdarkening laws and the underlying physics behind these laws. To test this method, we arbitrarily selected a model with T_{eff} = 5000 K, log g = 2, M_{⋆} = 5 M_{⊙}, corresponding to R_{⋆}/M_{⋆} = 7.42 R_{⊙}/M_{⊙} or log (R_{⋆}/M_{⋆}) = 0.87. Note that this value of the atmospheric extension falls in the region of Fig. 5 where the limbdarkening coefficients also depend on the T_{eff}, whereas the fixed point and intensity in Fig. 4 are less dependent on the T_{eff}. By selecting a trial model for which the A_{λ} and B_{λ}coefficients depend on T_{eff} as well as the atmospheric extension, we can determine how the iterative method is affected by this additional dependence. Table 2 lists the test model’s A and B limbdarkening coefficients for each of the five wavebands. Using the iterative method outlined in Sect. 4, we computed the R_{⋆}/M_{⋆} for each waveband, with the results given in Table 2. By assuming a stellar radius of R_{⋆} = 37.1 R_{⊙}, we determined the surface gravity from R_{⋆}/M_{⋆}, which is also given in Table 2.
Even though the values of A_{λ} and B_{λ} for our test model have no observational uncertainty, the results in Table 2 differ slightly from the test model’s values. One reason for this difference is that the coefficients given in Table 1 have uncertainties because they are fits to a grid of thousands of models with the range of fundamental parameters given earlier; this leads to the spread of points shown in Fig. 4.
The variation with wavelength also contributes to the uncertainty. Averaging over all wavelengths, the mean difference of the predicted R_{⋆}/M_{⋆} from the model’s value is only 5%, with the largest deviation being about 13% for the Vband. The wavelength variation of R_{⋆}/M_{⋆} is related to the definition of the stellar radius, which we define as R_{⋆} = R(τ_{Ross} = 2/3). However, in Table 2 the different wavelength bands produce values of R_{⋆} = R(τ_{B,V,R,I,H} = 2/3), which will differ slightly from the radius where τ_{Ross} = 2/3.
Using the five wavebands in Table 2, we find (R_{⋆}/M_{⋆})_{avg} = 7.01 ± 0.45 R_{⊙}/M_{⊙}, and log g_{avg} = 2.05 ± 0.05, both of which are consistent with the actual values. Therefore, our proposed method for using limbdarkening coefficients recovers the actual stellar parameters for the theoretical test case with a statistical uncertainty of ≈ 8% in R_{⋆}/M_{⋆}.
We improved the test by fitting simultaneously the limbdarkening coefficients for all the wavelength bands using the fixpoint relations for all models with T_{eff} > 3000 K. This resulted in the bestfit values R_{⋆}/M_{⋆} = 6.83 ± 0.38 R_{⊙}/M_{⊙}, and , where the error bars come from the formal uncertainty of the fits to the coefficients in Table 1, which are used in Eqs. (12) and (13). When the fit was repeated using only the relations for models with T_{eff} = 5000 K, the result was and . We conclude from this test using a model stellar atmosphere that our proposed method for measuring stellar properties yields consistent results, although, as expected, the uncertainty of R_{⋆}/M_{⋆} increases for stars with smaller atmospheric extension.
Fig. 6 a) Surfacebrightness distribution as a function of fractional radius. The solid line is the full set of intensities from a spherical model atmosphere with T_{eff} = 5000 K, log g = 2.0 and M_{⋆} = 5 M_{⊙}. The dashed line has removed intensity values for r/R_{⋆} ≥ 0.995 and then renormalized the fractional radius to the interval 0 − 1. The dotted line has removed intensity values for r/R_{⋆} ≥ 0.98 and then renormalized the fractional radius to the interval 0 − 1. b) Surfacebrightness distributions from panel a) plotted as a function of μ, with the lines having the same meaning as in the a) panel. c) Fits to the surfacebrightness distributions in panel b) using Eq. (2). The lines have the same meaning as the other two panels. 
6. Conclusions
The purpose of this study has been to investigate how limb darkening probes fundamental stellar properties. The limb darkening was parameterized by a fluxconserving linearplussquareroot law that has two free parameters that are functions of two angular moments of the intensity. The ratio of these two moments provides a measure of the amount of stellar atmospheric extension, represented by R_{⋆}/M_{⋆}. We tested our method for determining the extension from limbdarkening parameters using a spherically symmetric model atmosphere and found agreement of the mean value of the five spectral bands analyzed to within 8%.
We also find significant variation of R_{⋆}/M_{⋆} due to the definition of the stellar radius. From the context of the model of the stellar atmosphere, we find it natural to define the stellar radius where τ_{Ross} = 2/3, but other options are possible, such as τ_{Ross} = 1 or using a specific wavelength, such as 500 nm, to define the radius where τ_{500} = 1. Observations, of course, are done in specific wavebands, such as V, and the stellar radius is assumed to refer to some particular optical depth, such as τ_{V} = 2/3. Switching to another waveband will lead to another value of the radius. As an example, in the nearinfrared the predicted radius is larger than at optical wavelengths. Therefore, the definition of the stellar radius is ambiguous in general for studies of angular diameters (e.g. Wittkowski et al. 2004, 2006b,a), which makes the definition of the stellar radius challenging for stars with significant atmospheric extension. A recent example is Ohnaka et al. (2011), who measured the Kband angular diameter of Betelgeuse to be about 2 mas ( ≈ 5%) smaller than measured by Haubois et al. (2009) in the Hband. However, simultaneously fitting R_{⋆}/M_{⋆} to multiband limbdarkening observations provides a bolometric value for R_{⋆}/M_{⋆} analogous to how spectrophotometry can be used to measure a star’s effective temperature.
Limbdarkening parameters and predicted R_{ ∗ }/M_{ ∗ } for a model stellar atmosphere with T_{eff} = 5000 K, log g = 2 and M = 5 M_{⊙}.
The connection between atmospheric extension and stellar limbdarkening is a consequence of the assumption of spherical symmetry for the stellar atmospheres, not a unique feature of our SATLAS code. The results presented here should also be found using intensity profiles from sphericallysymmetric MARCS models and the PHOENIX model atmospheres used by Claret & Hauschildt (2003) and Fields et al. (2003). Unfortunately, in both of these works the authors truncated the intensity profiles to remove the low intensity limb and then redistributed the spacing of μpoints. Truncating the intensity profile eliminates critical information about the limb darkening, and makes the star’s intensity distribution resemble a planeparallel atmosphere, as we demonstrate in Fig. 6.
Our method for determining fundamental stellar parameters is based on the general atmospheric properties of fluxconserving limbdarkening laws, not the particular law used here. For instance, Heyrovský (2003) and Zub et al. (2011) found a fixed point in fluxconserving linear limbdarkening laws, which indicates that the linear limbdarkening coefficient is a measure of the mean intensity and flux of the atmosphere. Another example is a quadratic limbdarkening law where the limbdarkening coefficients would provide a measure of the mean intensity and second moment of the intensity, K. Because these laws measure the correlation between various moments of the intensity in an atmosphere, they would also measure the atmospheric extension of that star.
We conclude that the method outlined in this work is a viable way to use a limbdarkening law, such as the linearplussquareroot law employed here, to determine the fundamental physical parameters of stars. The method requires knowledge of one of the degenerate parameters effective temperature or luminosity. However, the method is able to constrain stellar parameters using limbdarkening observations even in the case when those limbdarkening observations from microlensing or eclipsing binaries cannot test the model stellar atmosphere directly. It seems as though we still have things to learn from fairly simple representations of limb darkening, and that, as observations continue
to improve, this will become an even more powerful tool in the study of stellar astrophysics.
Acknowledgments
We thank the referee for his/her many comments that led us to improve this work. This work has been supported by a research grant from the Natural Sciences and Engineering Research Council of Canada, and H.N. acknowledges funding from the Alexander von Humboldt Foundation.
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All Tables
Limbdarkening parameters and predicted R_{ ∗ }/M_{ ∗ } for a model stellar atmosphere with T_{eff} = 5000 K, log g = 2 and M = 5 M_{⊙}.
All Figures
Fig. 1 Bestfit Vband limbdarkening relations using Eq. (2) for grids of 5 M_{⊙} spherically symmetric and planeparallel ATLAS model atmospheres. A fixed μpoint is clearly seen in both cases. 

In the text 
Fig. 2 Surface gravity dependence of η_{λ}, α_{λ}, fixed point μ_{1} and the normalized intensity at the fixed point, I_{λ}(μ_{1})/2ℋ_{λ}, for the V spectral band. Planeparallel model atmospheres are denoted by black crosses, and spherical model atmospheres are represented by other symbols. Filled red circles are T_{eff} = 3000 K, open green squares 4000 K, open blue circles 5000 K, downward pointing magenta triangles 6000 K and upward pointing pale blue triangles are 7000 K. 

In the text 
Fig. 3 Extension parameter R_{⋆}/M_{⋆} compared to the relative atmospheric extension, ΔR/R_{⋆}, derived for more than 3000 spherical models. 

In the text 
Fig. 4 (Left) Values of the pseudomoment η_{V} and variable α_{V} as a function of atmospheric extension for spherical atmospheres, R_{⋆}/M_{⋆} in solar units. (Right) The top panel shows the value of the primary fixed point, μ_{1}, for the linearplussquareroot parametrization, also plotted as a function of the extension of the spherical atmosphere. The bottom panel shows the dependence of the normalized Vband intensity of the fixed point on the atmospheric extension. The symbols have the same meaning as in Fig. 2. When comparing to Fig. 2, note that increasing extension corresponds to decreasing log g. 

In the text 
Fig. 5 Vband values of the A limbdarkening coefficient (top panel) and the B limbdarkening coefficient (bottom panel) for the linearplussquareroot parametrization given by Eq. (2), plotted as a function of the extension parameter, R_{⋆}/M_{⋆} in solar units. The symbols have the same meaning as in Fig. 2. 

In the text 
Fig. 6 a) Surfacebrightness distribution as a function of fractional radius. The solid line is the full set of intensities from a spherical model atmosphere with T_{eff} = 5000 K, log g = 2.0 and M_{⋆} = 5 M_{⊙}. The dashed line has removed intensity values for r/R_{⋆} ≥ 0.995 and then renormalized the fractional radius to the interval 0 − 1. The dotted line has removed intensity values for r/R_{⋆} ≥ 0.98 and then renormalized the fractional radius to the interval 0 − 1. b) Surfacebrightness distributions from panel a) plotted as a function of μ, with the lines having the same meaning as in the a) panel. c) Fits to the surfacebrightness distributions in panel b) using Eq. (2). The lines have the same meaning as the other two panels. 

In the text 
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