Research Note
New limbdarkening coefficients for Phoenix/1d model atmospheres
II. Calculations for 5000 K ≤ T_{eff} ≤ 10 000 K Kepler, CoRot, Spitzer, uvby, UBVRIJHK, Sloan, and 2MASS photometric systems^{⋆}
^{1} Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004, 18080 Granada, Spain
email: claret@iaa.es
^{2} Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany
Received: 18 December 2012
Accepted: 13 February 2013
Aims. We present an extension of our investigations on limbdarkening coefficients computed with spherical symmetrical Phoenix models. The models investigated in this paper cover the range 5000 K ≤ T_{eff} ≤ 10 000 K and complete our previous studies of low effective temperatures computed with the same code.
Methods. The limbdarkening coefficients are computed for the transmission curves of the Kepler, CoRoT, and Spitzer space missions and the Strömgren, JohnsonCousins, Sloan, and 2MASS passbands. These computations were performed by adopting the leastsquares method.
Results. We have used six laws to describe the specific intensity distribution: linear, quadratic, square root, logarithmic, exponential, and a general law with four terms. The computations are presented for the solar chemical composition and cover the range 3.0 ≤ log g ≤ 5.5. The adopted microturbulent velocity and the mixinglength parameter are 2.0 km s^{1} and 2.0.
Key words: stars: atmospheres / binaries: eclipsing / planetary systems
Tables 2−25 are available in electronic form 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/552/A16
© ESO, 2013
1. Introduction
Some years ago, Claret & Hauschildt (2003) presented calculations of limbdarkening coefficients (LDC) for spherical symmetrical Phoenix models with 5000 K ≤ T_{eff} ≤ 10 000 K. At that occasion, the LDC were computed only for the classical photometric systems. Here we expand those calculations to the Kepler, CoRoT, and Spitzer space missions as well as to the Sloan and 2MASS passbands. For completeness, we also provide the calculations for the Strömgren and JohnsonCousins photometric systems. This short paper completes our previous paper on LCD calculated with PHOENIX models (Claret et al. 2012), where we analysed models for latetype stars (1500 K ≤ T_{eff} ≤ 4800 K).
Fig. 1 Specific intensity distribution for a [6000, 4.5] model. Continuous line represents the actual intensities while crosses denote the fits by adopting Eq. (6). Sloan and 2MASS photometric systems. 

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2. Limbdarkening coefficients for Phoenix models
For a description of the Phoenix code that was used to compute the specific intensities, we refer to Claret & Hauschildt (2003). The laws of limbdarkening used in the present paper are the linear law (1)the quadratic law (2)the square root law (3)the logarithmic law (4)the exponential law (5)and a more general law with four terms (6)The symbols u,a,b,c,d,e,f,g,h, and a_{k} denote the corresponding LDC and μ = cos(γ), γ being the angle between the line of sight and the emerging intensity. The specific intensities were convolved with a response function containing the filter transmission curves for Kepler, CoRoT, Spitzer (IRAC), uvby (Strömgren), UBVRIJHK (JohnsonCousins), Sloan, and 2MASS, double reflection from an aluminiumcoated mirror, and detector sensitivity. All calculations were performed by adopting the leastsquares method (LSM). To provide users with a tool for evaluating the theoretical error bars in the LDC we also computed the coefficients with the flux conservation method (FCM) for the biparametric and linear approximations. For a more detailed description of the adopted numerical methods to derive the LDC, see Claret & Hauschildt (2003) and Claret et al. (2012).
In Fig. 1 we show the resulting LDC for the Sloan and 2Mass photometric systems. The actual integrated intensities are compared with the fits provided by Eq. (6) for a model with T_{eff} = 6000 K and log g = 4.5. The interagreement is good, except for longer effective wavelengths, although they can be considered acceptable. This is due to the steeper profiles of the integrated specific intensities at dropoff zones for larger wavelengths. A similar pattern was also found for models with 1500 K ≤ T_{eff} ≤ 4800 K (Claret et al. 2012).
Fig. 2 Linear LDC for quasispherical PHOENIX calculations (crosses) and ATLAS planeparallel models (continuous lines). Log g = 4.0, solar composition, Kepler, CoRot, and Spitzer passbands. 

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Limbdarkening coefficients for the Kepler, CoRoT, Spitzer, UBVRIJHK, uvby, Sloan, and 2MASS photometric systems.
A quasispherical model is defined as the one computed with spherical symmetry but without considering the dropoff region. (for a more detailed definition of these models, see Claret et al. 2012). These models can be compared with previous calculations of linear LDC, such as those based on the planeparallel Atlas code (Kurucz, priv. comm.) This comparison is shown in Fig. 2. For the Kepler and Corot passbands we detect a systematic difference: the Phoenix linear LDC are slightly larger than the corresponding to the Atlas models. On the other hand, for Spitzer passbands, we notice an opposite effect: Phoenix LDC are slightly smaller than those derived with the Atlas code. These differences are approximately of the same order as the semiempirical errors of the LDC. It would be useful if observers could compare the LDC provided by Phoenix and Atlas models with the objective of providing some clues to the stellar atmosphere modellers so that they in turn can improve the theoretical models. Despite these systematic effects, the interagreement is good. The discontinuities in the linear LDC around log T_{eff} = 3.9 for both models are also notorious. These small jumps are characteristic of the atmosphere models and are connected to the onset of convection (Phoenix and Atlas adopted 2.0 and 1.25 for the mixinglength parameter, respectively). These discontinuities in the linear LDC are dependent on the passband, which are smaller for the larger effective wavelengths.
On the other hand, it is known that it is very difficult to observationally detect the low intensities near the extreme limb predicted by spherical models. As explained in Claret et al. (2012), only biparametric semiempirical LDC can be, at best, inferred by using highquality light curves of extrasolar planet transits or doublelined eclipsing binaries, for example. However, the intensity profiles of the spherical models are complicated and the biparametric laws are not able to reproduce these profiles well, except for the exponencial law (Eq. (5)). Therefore, the concept of quasispherical models is very useful in these cases since their theoretical biparametric (or linear) LDC can be directly compared with the semiempirical ones.
In Table 1 we summarise the results for the LDC calculations (Tables 2−25). These tables can be downloaded directly from CDS. Additional calculations for other photometric systems not covered in this paper can be provided on request.
Acknowledgments
The Spanish MEC (AYA200606375, AYA200914000C0301) is gratefully acknowledged for its support during the development of this work. This research has made use of the SIMBAD database, operated at the CDS, Strasbourg, France, and of NASA’s Astrophysics Data System Abstract Service.
References
 Claret, A., & Hauschildt, P. H. 2003, A&A, 412, 241 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Claret, A., Hauschildt, P. H., & Witte, S. 2012, A&A, 546, A14 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
All Tables
Limbdarkening coefficients for the Kepler, CoRoT, Spitzer, UBVRIJHK, uvby, Sloan, and 2MASS photometric systems.
All Figures
Fig. 1 Specific intensity distribution for a [6000, 4.5] model. Continuous line represents the actual intensities while crosses denote the fits by adopting Eq. (6). Sloan and 2MASS photometric systems. 

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In the text 
Fig. 2 Linear LDC for quasispherical PHOENIX calculations (crosses) and ATLAS planeparallel models (continuous lines). Log g = 4.0, solar composition, Kepler, CoRot, and Spitzer passbands. 

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In the text 