Issue 
A&A
Volume 664, August 2022



Article Number  A128  
Number of page(s)  6  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/202243827  
Published online  18 August 2022 
Power2 limbdarkening coefficients for the uvby, UBVRIJHK, SDSS ugriz, Gaia, Kepler, and TESS photometric systems
I. ATLAS 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:
20
April
2022
Accepted:
3
June
2022
Context. Limb darkening is an important stellar phenomenon and must be accounted for in the study of stellar spectra, eclipsing binaries, transiting planetary systems, and microlensing events. The power2 limbdarkening law provides a good match to the specific intensities predicted by stellar atmosphere models: it is better than other twoparameter laws and is only surpassed by the fourparameter law.
Aims. Predictions of the limbdarkening coefficients for the power2 law are not widely available. We therefore compute them, using stellar atmosphere models generated by the ATLAS (planeparallel) code.
Methods. Limbdarkening coefficients were computed for the space missions Gαiα, Kepler, and TESS as well as for the photometric systems uvby, UBVRIJHK, and SDSS ugriz. The calculations were performed by adopting the Levenberg–Marquardt leastsquares minimisation method and were computed with a resolution of 100 equally spaced viewing angles. We used 9586 model atmospheres covering 19 metallicities, effective temperatures of 3500–50 000 K, log g values from 0.0 to 5.0, and microturbulent velocities of 0, 1, 2, 4, and 8 km s^{1}.
Results. We confirm the superiority of the power2 law, in terms of the quality of the fits, over other twoparameter laws. This is particularly relevant for the quadratic law, which is widely used.
Conclusions. We recommend the use of the power2 law in cases where a twoparameter law is needed.
Key words: binaries close / binaries: eclipsing / planetary systems / stars: atmospheres
Tables 1–3 are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/cat/J/A+A/664/A128
© A. Claret and J. Southworth 2022
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
The phenomenon of limb darkening (LD) is the apparent decrease in intensity of the surface of a star from the centre of its disc to the limb. The effect is not intrinsic to the star, but arises due to the viewing geometry of the observer. Viewing angles normal to the stellar surface experience photons emitted from deeper, and thus hotter, layers of the atmosphere, whereas the atmospheric opacity causes slanted viewing angles to ‘see’ only outer, and thus cooler, layers of the atmosphere.
The Sun has for centuries been known to show the phenomenon of LD (Bouguer 1760). It was first parameterised by the linear “law” (Schwarzschild 1906), which permits only a constant gradient in the decrease in surface brightness as a function of distance from the centre of the stellar disc: (1)
where I is the specific intensity of light as a function of position on the stellar disc, u is the linear limbdarkening coefficient (LDC), and µ = cos γ, where γ is the angle between the observer’s line of sight and the surface normal.
The linear LD law is too simplistic, so many other LD parameterisations have been proposed. Some of the most widely used are the quadratic law (Kopal 1950), (2)
the squareroot law (DíazCordovés & Giménez 1992), (3)
and the logarithmic law (Klinglesmith & Sobieski 1970), (4)
where a, c, and e are the linear LDCs, b is the quadratic LDC, d is the squareroot LDC, and f is the logarithmic LDC. The twoparameter laws provide an improved representation of reality as they allow for a slow dropoff of brightness near the centre of the stellar disc and a faster decrease in the region of the limb.
In addition to these, Hestroffer (1997) introduced a twoparameter law that involves a power of µ, which has become known as the power2 law: (5)
where g and h are the corresponding LDCs. Here, it can be seen that the coefficient g gives the specific normalised intensity at the limb of the star (µ = 0) and h governs how much the specific intensity departs from the linear law. The power2 law has been found to fit specific intensity profiles very well (Morello et al. 2017; Maxted 2018) and has also been implemented in an efficient algorithm for planetary transits (Maxted & Gill 2019). Short et al. (2019) have shown that the power2 law provides a better match to the specific intensities for spherical stellar atmosphere models than other twoparameter laws. To our knowledge, the only known LD law that surpasses it in quality of fit is the fourparameter law (Claret 2000).
Limb darkening is a vital phenomenon in several areas of astrophysics where a stellar disc is resolved in some way, including highprecision spectroscopy, eclipsing binary star systems, transiting extrasolar planets, interferometry, and stellar microlensing events. The power2 law is beginning to be widely used in these areas.
Theoretical LDCs for the power2 law have been presented for a subset of situations: DA, DB, and DBA white dwarfs (Claret et al. 2020), solartype stars (Maxted 2018), and the passband of the CHEOPS space mission (Claret 2021). However, theoretical LDCs covering a wide range of stellar properties and passbands have not so far been available.
The aim of the current work is to provide users with LDCs for the power2 law covering a wide range of stellar types and passbands (see Claret 2000; Claret & Bloemen 2011). The passbands are those for the Strömgren uvby, JohnsonCousins UBVRIJHK, and SDSS ugriz photometric systems, plus those for the Gaia, Kepler, and TESS space telescopes. In Sect. 2, we briefly describe the details of the computational method for deriving the LDCs adopting the ATLAS models. Sections 3 and 4 present the results of our calculations plus comparisons to the other twoparameter laws. We conclude in Sect. 5.
2 Numerical method
We adopted the leastsquares method (LSM), adapted to the nonlinear case, to compute the LDCs. Before applying the LSM to each passband, the specific intensities for the uvby, UBVRIJHK, ugriz, Gaia, Kepler, and TESS photometric systems were integrated using the following equation: (6)
where h is Planck’s constant, λ is the wavelength, I_{a}(µ) is the specific intensity in passband a, I(λ,µ) is the monochromatic specific intensity, and S (λ) is the response function. For the uvby, UBVRIJHK, and ugriz passbands, the function also takes into account the transmission of one airmass of the Earth’s atmosphere response. The passbands used were obtained from Spanish Virtual Observatory Filter Profile Service^{1}, except for the cases of Strömgren uvby, the passband of which was obtained from the Observatorio de Sierra Nevada, Granada, Spain (C. Cárdenas, priv. comm.), and JHK, obtained from the Observatorio del TeideIAC, Spain (see also Alonso 1994).
The LDC calculations were performed adopting a total of 9586 ATLAS stellar model atmospheres^{2}. These covered 19 metallicities [M/H] (from 10^{5} to 10^{+1} times the solar abundance), surface gravities log g from 0.0 to 5.0, effective temperatures T_{eff} from 3500 K to 50000 K, and five microturbulent velocities (V_{ξ} = 0, 1, 2,4, 8 km s^{1}). In each case, we used a total of 100 µ points equally spaced in µ, rather than the 17 points normally adopted in the ATLAS models.
The corresponding merit function to show the quality of the fit to the I(µ) values can be written as (7)
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.
3 Results and comparison with the quadratic law
The power2 LDCs calculated in this work are made available in three tables (available at the CDS; see Table A.1). Table 1 contains the LDCs and χ^{2} values for the Gaia, Kepler, and TESS passbands, Table 2 contains this information for the SDSS ugriz bands, and Table 3 contains this information for the uvby and UBVRIJHK bands. The LDCs for the CHEOPS mission, computed using the ATLAS and PHOENIX models, can be found in Claret (2021).
Figure 1 illustrates the behaviour of the coefficients g and h for a series of models with differing T_{eff} and [M/H] values. It can be seen that the coefficient h does not depend very strongly on metallicity, except at low T_{eff}. The change in slope, around log T_{eff} = 3.9, is due to the onset of convection. It can also be seen that coefficient h has little variation with T_{eff} or [M/H] in the interval 4.0 ≤ log T_{eff} ≤ 4.4, staying within the range 0.5 to 0.4. This confirms a previous calculation of LDCs by DíazCordovés et al. (1995), who found that the related squareroot law works well at high T_{eff} values.
In this paper, we have focused our attention on a comparison between the power2 and quadratic laws, as the quadratic one is by far the most widely used biparametric law. Figures 2 and 3 illustrate the behaviour of the normalised specific intensity near the limb of the star as a function of µ for the models with T_{eff} values of 4500 K and 23 000 K, respectively. The other input parameters are the same for both models: log g = 5.0, [M/H] = 0.0, and Vξ = 2 km s^{1}. In both cases, the superiority of the power2 law over the quadratic one is clear, mainly in the regions near the limb, where the differences in the normalised specific intensities can be of the order of 0.06. On the other hand, a comparison between Figs. 2 and 3 shows that the power2 law works better for models with high T_{eff} values, mainly at the limb. We conducted tests with models with different input parameters, and the results are similar, although they vary slightly in relation to the previously mentioned values, depending on the region of the Hertzsprung–Russell diagram analysed.
A more general way of checking such differences in the quality of the fittings can be done by inspecting Figs. 4–8. These figures show the merit functions, χ^{2}, for the space missions Gaia, Kepler, and TESS. A direct comparison shows that the power2 law (red asterisks) presents a much higher quality of fit than the corresponding quadratic one (black asterisks). However, we should recall that the twoparameter laws are accurate only for some regions of the Hertzsprung–Russell diagram. For example, for atmospheres in radiative equilibrium, the power2 law works very well. However, for models in convective equilibrium (log T_{eff} ≤ 3.85), the power2 law begins to present significant χ^{2} values, albeit still much lower than in the case of the quadratic law.
Fig. 1 Coefficients g and h for the filter Gbp of Gaia as a function of T_{eff} and [M/H]. Black lines represent [M/H] =0.0, red lines [M/H] = 1.0, and green lines [M/H] = 1.0. In all cases, log g = 4.5 and V_{ξ}= 2 km s^{1}. 
Fig. 2 Angular distribution of the specific intensity for a model with T_{eff} = 4500 K, log g = 5.0, [M/H] = 0.0, and V_{ξ} = 2km s^{1} for the TESS passband (black continuous line). Red crosses denote the fitting adopting the power2 law, and the green line represents the quadratic LD law approach. 
4 Comparison with the squareroot and logarithmic laws
For completeness, we also compared the quality of fit for the power2 law with those computed by adopting the squareroot and logarithmic LD laws. These are rarely used in the analysis of transiting planets but are often used in the study of eclipsing binary systems (e.g. Van Hamme 1993).
Figure 9 shows a comparison of the merit functions for the G passband using the squareroot law (black asterisks) and the power2 (red asterisks). We have kept the same scale to facilitate a comparison with the results shown in Fig. 5. It can be seen that the superiority of the fits provided by the power2 law (Fig. 9) is not as great as for the quadratic law (Fig. 5) because the squareroot law is a better representation than the quadratic law. In fact, DíazCordovés & Giménez (1992) have shown that the values of σ for the squareroot law are of the order of ten times smaller than those provided by the quadratic one. To assess the quality of a fit, DíazCordovés & Giménez (1992) used a definition of the merit function slightly different from the one given by Eq. (7):
where N, y_{i}, and Yi have the same meaning as in Eq. (7).
Figure 10 shows the same comparison but for the logarithmic law. It can be seen that the logarithmic law provides a better fit to the model atmosphere intensities than does the quadratic law (Fig. 5), but it is clearly inferior to the squareroot (Fig. 9) and power2 laws. This conclusion holds particularly at higher T_{eff} values, as the logarithmic law performs relatively well for 5500 < T_{eff} < 9000 K. In summary, in addition to the fact that Eq. (5) is a nonlinear law, as noted by Hestroffer (1997), it exhibits steeper gradients at the limb (see, for example, Figs. 2 and 3). This makes it the biparametric law that best fits the distribution of intensities from atmosphere models, as we have seen by inspecting Figs. 4–10.
Fig. 3 Same as Fig. 2 but for the model with T_{eff} = 23 000 K, log g = 5.0, [M/H] = 0.0, and V_{ξ} = 2km s^{1}. 
Fig. 4 Merit function for the ATLAS models as a function of log T_{eff} for the Gaia G_{BP} passband. The power2 law is shown using red asterisks and the quadratic law using black asterisks. The results for all models are shown. 
Fig. 9 Same as Fig. 5 but for the squareroot law (black asterisks) versus the power2 law (red asterisks). 
Fig. 10 Same as Fig. 5 but for the logarithmic law (black asterisks) versus the power2 law (red asterisks). 
5 Summary
The power2 LD law is seeing increasing use in several areas of stellar physics. In order to help this work, we have calculated LDCs for the power2 law using ATLAS model atmospheres covering a wide range of stellar T_{eff} values, gravities, and metallicities. This has been done for a large number of passbands used by both groundbased and spacebased telescopes.
In addition to the LDCs, we have calculated the quality of fit to the specific intensities from the model atmospheres. A comparison between these and other twoparameter LD laws shows that the power2 law produces the best results in all cases, followed by the squareroot and logarithmic laws. The quadratic law is by far the worst of the four options. The differences between the χ^{2} provided by the power2 law and the quadratic one are particularly important for regions near the limb and for low T_{eff} values, and can reach more than one order of magnitude. We find similar results for comparisons in all of the passbands considered. Additional calculations for other photometric systems and/or the transmissions of the filters and the response of the Earth’s atmosphere are available upon request.
Therefore, we recommend the power2 law to users who prefer to use a twoparameter LD law. We note that the fourparameter law provides a better fit, particularly in the cases of eclipsing binaries and extrasolar planet transits with stars that have low T_{eff} values. For the specific case of shortperiod eclipsing binaries, it would also be advisable to take the mutual irradiation into account because it can significantly alter both the corresponding LDCs and the bolometric albedo. In fact, irradiated stellar atmospheres show different distributions of brightness if we compare them with standard models without the action of an external radiation field. More details on this matter are available in Claret (2001, 2004) and Claret & Giménez (1992).
Acknowledgements
We thank the anonymous referee for his/her helpful comments that have improved the manuscript. The Spanish MEC (AYA201571718R and ESP201787676C52R) is gratefully acknowledged for its support during the Spanish MEC (ESP201787676C52R, PID2019107061GBC64, and PID2019109522GBC52) is gratefully acknowledged for its support during the development of this work. A.C. acknowledges financial support from the State Agency for Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa” award for the Instituto de Astrofísica de Andalucía (SEV20170709). 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 1–3 (available at the CDS)
Power2 LDCs for the Gaia, Kepler, TESS, SDSS Sloan ugriz and uvby UBVRIJHK photometric systems.
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All Tables
Power2 LDCs for the Gaia, Kepler, TESS, SDSS Sloan ugriz and uvby UBVRIJHK photometric systems.
All Figures
Fig. 1 Coefficients g and h for the filter Gbp of Gaia as a function of T_{eff} and [M/H]. Black lines represent [M/H] =0.0, red lines [M/H] = 1.0, and green lines [M/H] = 1.0. In all cases, log g = 4.5 and V_{ξ}= 2 km s^{1}. 

In the text 
Fig. 2 Angular distribution of the specific intensity for a model with T_{eff} = 4500 K, log g = 5.0, [M/H] = 0.0, and V_{ξ} = 2km s^{1} for the TESS passband (black continuous line). Red crosses denote the fitting adopting the power2 law, and the green line represents the quadratic LD law approach. 

In the text 
Fig. 3 Same as Fig. 2 but for the model with T_{eff} = 23 000 K, log g = 5.0, [M/H] = 0.0, and V_{ξ} = 2km s^{1}. 

In the text 
Fig. 4 Merit function for the ATLAS models as a function of log T_{eff} for the Gaia G_{BP} passband. The power2 law is shown using red asterisks and the quadratic law using black asterisks. The results for all models are shown. 

In the text 
Fig. 5 Same as Fig. 4 but for the Gaia G passband. 

In the text 
Fig. 6 Same as Fig. 4 but for the Gaia G_{RP} passband. 

In the text 
Fig. 7 Same as Fig. 4 but for the Kepler passband. 

In the text 
Fig. 8 Same as Fig. 4 but for the TESS passband. 

In the text 
Fig. 9 Same as Fig. 5 but for the squareroot law (black asterisks) versus the power2 law (red asterisks). 

In the text 
Fig. 10 Same as Fig. 5 but for the logarithmic law (black asterisks) versus the power2 law (red asterisks). 

In the text 
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