Issue 
A&A
Volume 674, June 2023



Article Number  A63  
Number of page(s)  7  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/202346478  
Published online  01 June 2023 
Power2 limbdarkening coefficients for the uvby, UBVRIJHK, SDSS ugriz, Gaia, Kepler, TESS, and CHEOPS photometric systems
II. PHOENIX spherically symmetric stellar atmosphere models^{★}
^{1}
Instituto de Astrofísica de Andalucía, CSIC,
Apartado 3004,
18080
Granada,
Spain
email: claret@iaa.es
^{2}
Dept. Física Teórica y del Cosmos, Universidad de Granada,
Campus de Fuentenueva s/n,
10871
Granada,
Spain
^{3}
Astrophysics Group, Keele University,
Staffordshire
ST5 5BG,
UK
Received:
22
March
2023
Accepted:
6
April
2023
Context. The phenomenon of limbdarkening is relevant to many topics in astrophysics, including the analysis of light curves of eclipsing binaries, optical interferometry, measurement of stellar diameters, line profiles of rotating stars, gravitational microlensing, and transits of extrasolar planets
Aims. Multiple parametric limbdarkening laws have been presented, and there are many available sources of theoretical limbdarkening coefficients (LDCs) calculated using stellar model atmospheres. The power2 limbdarkening law allows a very good representation of theoretically predicted intensity profiles, but few LDCs are available for this law from spherically symmetric model atmospheres. We therefore present such coefficients in this work.
Methods. We computed LDCs for the space missions Gαiα, Kepler, TESS, and CHEOPS and for the passbands uυby, UBVRIJHK, and SDSS ugriz, using the PHOENIXCOND spherical models. We adopted two methods to characterise the truncation point, which sets the limb of the star: the first (M1) uses the point where the derivative dI(r)/dr is at its maximum – where I(r) is the specific intensity as a function of the normalised radius r – corresponding to µ_{cri}, and the second (M2) uses the midpoint between the point µ_{cri} and the point located at µ_{cri–1}. The LDCs were computed adopting the Levenberg–Marquardt leastsquares minimisation method, with a resolution of 900 equally spaced µ points, and covering 823 model atmospheres for a solar metallicity, effective temperatures of 2300–12000 K, log g values from 0.0 to 6.0, and microturbulent velocities of 2 km s^{−1}. As our previous calculations of LDCs using spherical models included only 100 µ points, we also updated the calculations for the fourparameter law for the passbands listed above, and compared them with those from the power2 law.
Results. Comparisons between the quality of the fits provided by the power2 and fourparameter laws show that the latter presents a lower merit function, χ^{2}, than the former for both cases (M1 and M2). This is important when choosing the best approach for a particular science goal.
Key words: binaries: close / binaries: eclipsing / planetary systems
Tables 1–12 are only available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/vizbin/cat/J/A+A/674/A63
© The Authors 2023
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1 Introduction
Stars show a decrease in specific intensity from their centre to their limb. This is because opacity causes the sightlines of the observer to penetrate less far when they enter the photosphere at an angle. This effect, known as limbdarkening, occurs both for real stars and for theoretical stellar atmosphere models, and is important in any situation where a star is spatially resolved, such as the study of eclipsing binaries, transiting planetary systems, stellar interferometry, microlensing, and spectral line profiles.
The specific intensity profiles generated by model atmospheres with spherical symmetry differ from their planeparallel equivalents mainly in the region near the limb, where the former shows a sharp drop while the latter presents a finite specific intensity. This important difference is due to the fact that in the planeparallel case, the curvature of the atmosphere is neglected. Neglecting curvature effects is justified in stars with low atmospheric scale heights, but not for cool giants or supergiants. In planeparallel model calculations, the medium is infinite, there is no natural boundary condition, and the specific intensity is not zero at the limb but depends on the local temperature in this region. However, in the case of models with spherical symmetry, the temperature tends to zero at small optical depths (see Eq. (7.182) in Mihalas 1978). A concise explanation for this crucial difference can be found in Mihalas (1978, p. 246–247); see also Larson (1969). In the theoreticalobservational context of flux and polarisation calculations, Kostogryz et al. (2017) report the superiority of spherical models over planeparallel models in some configurations, such as systems with grazing eclipses, transits with Earthsize planets, or for hotter planet host stars (effective temperatures T_{eff} > 6000 K).
In Claret & Southworth 2022 (hereafter Paper I) we showed the superiority of the power2 law (Hestroffer 1997) in terms of the quality of fits of limbdarkening coefficients (LDCs) to theoretical predictions over the other twoparameter laws in the case of planeparallel model atmospheres. However there remain few sources of LDCs calculated for the power2 law, especially using spherical model atmospheres. The exceptions are LDCs for DA, DB, and DBA white dwarfs using planeparallel models (Claret et al. 2020), solartype stars using 3D models (Maxted 2018), extensive tables using the ATLAS planeparallel models (Paper I), and LDCs for the CHEOPS space mission, where spherical models are adopted (Claret 2021).
The aim of this work is to provide users with LDCs for the power2 law based on spherical model atmospheres, covering a wide range of T_{eff} values, surface gravities, and passbands. The passbands adopted here are Strömgren uυby, Johnson–Cousins UBVRIJHK, and SDSS ugriz photometric systems, plus those for the space missions Gaia, Kepler, TESS, and CHEOPS. The results of the present study supersede those of Claret (2021) for the CHEOPS mission.
The structure of the paper is as follows. Section 2 is dedicated to the description of the spherical models (PHOENIXCOND) and the numerical details of the calculations of the LDCs. A comparison between the power2 and fourparameter LDCs is presented in Sect. 3. Section 4 gives a comparison between the planeparallel and spherical LDCs, and Sect. 4 summarises our findings.
2 The spherical atmosphere models and numerical methods
In this paper, we use specific intensity profiles from the PHOENIXCOND spherical model atmospheres. The main characteristics of these models are described in Husser et al. (2013); see also Claret et al. (2012). We adopt a LevenbergMarquardt leastsquares minimisation method to compute the LDCs. The specific intensities for the uυby, UBVRIJHK, ugriz, Gaia, Kepler, TESS, and CHEOPS photometric systems were integrated using the following equation:
where h is Planck’s constant, c is the speed of light in vacuum, λ is the wavelength in Å, and µ is given by cos(γ), where γ is the angle between the line of sight and the outward surface normal. I_{a}(µ) is the specific intensity for a given passband a, I(λ,µ) is the monochromatic specific intensity, and S_{a}(λ) is the response function. For the uυby, UBVRIJHK, and ugriz photometric systems, the response function includes the transmission of one airmass of the Earth’s atmosphere. The passbands used were obtained from Spanish Virtual Observatory Filter Profile Service^{1}. For the specific case of Strömgren uυby, the response function was obtained from the Observatorio de Sierra Nevada, Granada, Spain (C. Cárdenas, priv. comm.). For JHK, the corresponding S(λ) were obtained from the Observatorio del TeideIAC, Spain (Alonso et al. 1994).
The specific intensities generated by the spherical model atmospheres show a much more pronounced curvature near the limb than the planeparallel models, making it much more difficult to obtain a good fit to them. For example, the fourparameter law is able to fit the full profiles well but only for some filters (see e.g. Fig. 1 in Claret et al. 2012). Some alternative methods have been proposed to better describe the specific intensity profiles for models with spherical symmetry, such as Claret & Hauschildt (2003), where the concept of quasispherical models was introduced.
Later, Wittkowski et al. (2004) introduced a more elaborate method: instead of truncating the models at a certain value of µ, the truncation was defined by searching for the maximum of the derivative of the specific intensity with respect to r, where . This point corresponds to τ_{R} ≈ 1.0, where τ_{R} is the Rosseland mean optical depth. Fortunately, the PHOENIX models were computed with sufficient points in the dropoff region to enable us to accurately determine the corresponding derivatives. Wittkowski et al. (2004) gave two ways to determine this critical point. The first method (M1) uses the point where the derivative is maximum, which is known as the point µ_{cri}, while the second method (M2) uses the midpoint between the point µ_{cri} and the point located at µ_{cri–1} (G. Morello, priv. comm.). The specific intensity using M2 at this average point is larger than that obtained by the M1 method. Therefore, it gives rescaled profiles that are more similar to the planeparallel models than those provided by M1. However, M2 does not accurately represent the specific intensity at the critical point as defined by Wittkowski et al. (2004). The corresponding critical points for both methods are characterised by µ_{cri,1} and µ_{cri,2}. In the current work, we present calculations for both M1 and M2 for completeness.
The LDC calculations were performed adopting 823 PHOENIX stellar model atmospheres for solar metallicity, surface gravities log g from 0.0 to 6.0, T_{eff} values from 2300 K to 12 000 K, and microturbulent velocities V_{ξ} = 2.0 km s^{−1}. We adopted 900 equally spacedpoints in µ, rather than the 78 original ones adopted in the PHOENIX models. In our previous papers on LDCs from spherical models, we used only 100 µ points, and so we have updated our calculations using 900 points for the photometric systems studied in this work.
The power2 LD law introduced by Hestroffer (1997) is
where g and h are the corresponding LDCs. Given that the profiles generated by PHOENIX are much more complicated and difficult to adjust than in the planeparallel approximation, we also calculated LDCs using the fourparameter law (Claret 2000) which is
where a_{k} are the four LDCs. The greater flexibility of the fourparameter law allows it to reproduce the specific intensity profiles to a higher precision than limbdarkening laws using only two LDCs.
We define a merit function, which measures the quality of the fit to a given set of I(µ) values as
where y_{i} is the model intensity at point i, Y_{i} is the fitted function at the same point, and N is the number of µ points.
Fig. 1 Angular distribution of the specific intensity for a model with T_{eff} = 4500 K, log g = 4.5, [M/H] = 0.0, and V_{ξ} = 2 km s^{¯1} for the TESS passband. The red line represents the specific intensity distribution and the red cross indicates µ_{cri,1}. The green line denotes the fitting adopting the power2 law, while the green cross indicates the fitting at µ_{cri,1}. The blue line denotes the fourparameter law approach and the blue cross indicates the fitting at µ_{cri,1}. This plot is for case M1, and the specific intensity profiles have not been rescaled. 
3 Results
An extensive comparison of the different limbdarkening laws was carried out in Paper I. It was found that the power2 law produced better fits to the specific intensities from planeparallel model atmospheres than the other twoparameter laws (quadratic, logarithmic, and squareroot). Given this result and the increased complexity of the specific intensities from spherical model atmospheres, it was not thought necessary to repeat that analysis here. However, we find it useful to investigate the relative success of the power2 and fourparameter laws, as this information is helpful to users of the LDCs.
The resulting χ^{2} values are generally small and are dominated by the difficulty of fitting the µ_{cri,1} and µ_{cri,2} points, which is most conspicuous for the power2 laws in the M1 case. Figures 1–4 show the specific intensity profiles for the M1 and M2 cases using both the power2 (Eq. (2)) and fourparameter (Eq. (3)) law. For the sake of clarity, only the regions near the limb are shown, because the fits for µ > µ_{cri} are almost perfect. For mainsequence stars (Figs. 1 and 2), and in both the M1 and M2 cases, the fourparameter law (Eq. (3)) provides better fits than those provided by the power2 law (Eq. (2)). The situation is the same but even clearer for the giant stars (Figs. 3 and 4). It is important to emphasise that one of the main differences between the planeparallel models and the spherical ones is in the dropoff region, and so it is necessary to characterise this region well through the LDCs. The best test of the spherical models would be achieved using Eq. (3) and case M1.
The relative quality of the fits can be computed using the following expression:
Figures 5–10 show the quantity log α for the M1 case, a range of T_{eff} and log g values, and for the Gaia G_{BP}, G, G_{RP}, Kepler, TESS, and CHEOPS passbands. It can be seen that the χ^{2} provided by Eq. (3) is typically a factor of 8 ± 2 smaller than the χ^{2} given by Eq. (2), depending on the effective wavelength of the passband. We also see that there is a common maximum in all passbands centred on log T_{eff} ≈ 3.65 (≈4500K), where the fourparameter law performs much better than the power2 law. A comparison of the effects for the SDSS u and z passbands (not shown) indicates that the ratio α is larger at shorter wavelengths, which is in agreement with the results for the Gaia G_{BP} passband (Fig. 5). It is also apparent that the power2 law has a better relative performance at higher T_{eff} values, but this is still poorer than the fourparameter law.
In Figs. 5–10, we trace the relative performance for two surface gravities: log g = 6.0 for the most compact stars and log g = 3.5 to represent subgiants. This shows how the relative quality of the fits depends on the surface gravity of a star. In general, the power2 law is better at higher surface gravities, but is still not as good as the fourparameter law. This is the same in all passbands except for the interval 3.4 ⪆ log T ⪆ 3.55 in the G_{BP} band. For models in radiative equilibrium, there is also a tendency for the relative quality of the fits to not depend strongly on the surface gravity.
Fig. 5 Comparison of the quality of fits provided for the Gaia G_{BP} passband. The quantity α is the ratio between the χ^{2} for the power2 law and the χ^{2} for the fourparameter law (both for case M1), for all 823 models. The red line traces the results for the most compact models (log g = 6.0) whilst the green line indicates the results for subgiant ones (log g = 3.5). 
Fig. 11 Comparison between the power2 g (upper panel) and h (lower panel) LDCs in the case of planeparallel model atmospheres (solid lines) and spherical model atmospheres for cases M1 (dotted lines) and M2 (dashed lines). 
4 Comparison between planeparallel and spherical LDCs
A comparison between the power2 LDCs predicted by planeparallel and spherical model atmospheres is useful. For the planeparallel model atmospheres, we adopted the results from Paper I, which were calculated using the ATLAS code. For the spherical model atmospheres, we used the results from the current work for both cases M1 and M2.
Figure 11 shows the variation of the g and h LDCs as a function of T_{eff}, for log g = 4.0, solar metallicity, and V_{ξ} = 2.0 km s^{−1}. Cases M1 and M2 show much closer agreement than the ATLAS LDCs, in particular at lower T_{eff} values where spherical model atmospheres are more reliable. The anticorrelation between g and h is also clear, particularly at low T_{eff} values for M1 and M2, and at 6000 < T_{eff} < 8000 K for the ATLAS LDCs.
An important use of the LDCs presented in the current work is in calculating the light curves of planetary transits (see Fig. 12). We explored this by using the JKTEBOP code (Southworth 2013) to calculate the transit light curve of a system similar to HATP7 (Pál et al. 2008), using the transit parameters from Southworth (2011). This system was chosen because the small ratio of the radii of the planet and star gives a good spatial sampling of the limbdarkening over the disc of the star. To avoid interpolating LDCs to specific atmospheric parameters, we adopted fixed values of T_{eff} = 6000 K, log g = 4.0, solar metallicity, and V_{ξ} = 2.0 km s^{−1}. The differences between the ATLAS and PHOENIX LDCs cause a change in the transit shape that reaches approximately 70 ppm around second and third contact. In principle, this change is large enough to be measured from spacebased missions such as Kepler and TESS, especially when many transits can be averaged together, but correlations between the measured LDCs and other parameters of the system will reduce the significance of the signal. The M1 and M2 LDCs give much more similar transit light curves – the greatest deviation is only 4 ppm –, which suggests that the choice of which to use is unimportant.
Fig. 12 Top panel: comparison between the transit shapes using the power2 LDCs for the planeparallel (ATLAS) and spherical (cases M1 and M2) model atmospheres (labelled). Lower panels: differences between the transit shapes in each case (labelled). 
5 Summary and final remarks
We computed LDCs for the power2 law using specific intensities from spherical model atmospheres, as such results were only previously available for a small range of stellar parameters. To this end, we used the PHOENIXCOND models for T_{eff} values from 2300K to 12000K, logy values from 0.0 to 6.0, solar metallicity, V_{ξ} = 2.0 km s^{−1}, and with 900 µ points. We also computed LDCs for the fourparameter law, which supersede our previous calculations based on only 100 µ points. For both laws, we computed coefficients for the uυby, UBVRIJHK, and SDSS ugriz passbands, and for the Gaia, Kepler, TESS, and CHEOPS photometric systems. Two approaches were used to define the limb of the star based on the dropoff of intensity in the spherical model atmosphere predictions. We emphasise that the µ values have been rescaled to remove those beyond the limb of the star, and so this need not be done by users of the LDCs.
We performed comparisons between the merit function χ^{2} for the power2 and the fourparameter laws, finding that the fourparameter law is superior in all cases to the power2 law. The performance of the power2 law is better at higher temperatures and at redder wavelengths. Additional calculations of LDCs for other photometric systems not covered in this paper are available upon request to the first author.
We recommend that the fourparameter law be used whenever high precision is needed for specific intensity profiles. If a less complex law is preferred, we recommend the power2 law as it reproduces the theoretical intensity profiles to a higher precision than the other twoparameter laws (see Paper I); we also recommend not using the linear and quadratic laws because they provide the least good fit to theoretical specific intensity profiles. For modelling eclipsing binary star and transiting planetary systems, the choice of which limbdarkening law to use has a relatively small effect on the bestfitting parameter values (Southworth 2023).
The current work, together with Paper I, provides three sets of LDCs for any given set of stellar parameters and passbands, within those covered in this work. These are LDCs from planeparallel ATLAS model atmospheres (Paper I) and LDCs from spherical model atmospheres (this work) calculated using methods M1 and M2. Those from methods M1 and M2 are very similar, whereas the LDCs from ATLAS are moderately different. We show that these LDCs could be tested using highprecision photometry of transits or eclipses. In our experience, planetary transits are the best approach for this because the small size of the planet relative to the star provides a finer spatial sampling of the star’s specific intensity profile, but the size of the differences between the transit light curves from different LDCs is not large.
Acknowledgements
We thank the anonymous referee for his/her helpful comments that have improved the manuscript. The Spanish MEC (AYA201571718R, ESP201787676C52R, PID2019107061GBC64, and PID2019109522GBC52) is gratefully acknowledged for its support during the development of this work. A.C. acknowledges financial support from the grant CEX2021001131S funded by MCIN/AEI/ 10.13039/501100011033. This research has made use of the SIMBAD database, operated at the CDS, Strasbourg, France, of NASA’s Astrophysics Data System Abstract Service and of SVO Filter Profile supported from the Spanish MINECO through grant AYA201784089.
Appendix A Description of Tables 1–12 (available at the CDS)
Power2 and fourparameter LDCs for the Gaia, Kepler, TESS, CHEOPS, ugriz, uυby, and UBVRIJHK photometric systems.
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All Tables
Power2 and fourparameter LDCs for the Gaia, Kepler, TESS, CHEOPS, ugriz, uυby, and UBVRIJHK photometric systems.
All Figures
Fig. 1 Angular distribution of the specific intensity for a model with T_{eff} = 4500 K, log g = 4.5, [M/H] = 0.0, and V_{ξ} = 2 km s^{¯1} for the TESS passband. The red line represents the specific intensity distribution and the red cross indicates µ_{cri,1}. The green line denotes the fitting adopting the power2 law, while the green cross indicates the fitting at µ_{cri,1}. The blue line denotes the fourparameter law approach and the blue cross indicates the fitting at µ_{cri,1}. This plot is for case M1, and the specific intensity profiles have not been rescaled. 

In the text 
Fig. 2 Same as in Fig. 1, except for case M2 and µ_{cri,2} instead of case M1 and µ_{cri,1}. 

In the text 
Fig. 3 Same as in Fig. 1, except for log g = 2.5 instead of log g = 4.5. 

In the text 
Fig. 4 Same as in Fig. 2, except for log g = 2.5 instead of log g = 4.5. 

In the text 
Fig. 5 Comparison of the quality of fits provided for the Gaia G_{BP} passband. The quantity α is the ratio between the χ^{2} for the power2 law and the χ^{2} for the fourparameter law (both for case M1), for all 823 models. The red line traces the results for the most compact models (log g = 6.0) whilst the green line indicates the results for subgiant ones (log g = 3.5). 

In the text 
Fig. 6 Same as in Fig. 5, but for the Gaia G passband. 

In the text 
Fig. 7 Same as in Fig. 5, but for the Gaia G_{RP} passband. 

In the text 
Fig. 8 Same as in Fig. 5, but for the Kepler passband. 

In the text 
Fig. 9 Same as in Fig. 5, but for the TESS passband. 

In the text 
Fig. 10 Same as in Fig. 5, but for the CHEOPS passband. 

In the text 
Fig. 11 Comparison between the power2 g (upper panel) and h (lower panel) LDCs in the case of planeparallel model atmospheres (solid lines) and spherical model atmospheres for cases M1 (dotted lines) and M2 (dashed lines). 

In the text 
Fig. 12 Top panel: comparison between the transit shapes using the power2 LDCs for the planeparallel (ATLAS) and spherical (cases M1 and M2) model atmospheres (labelled). Lower panels: differences between the transit shapes in each case (labelled). 

In the text 
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