New limbdarkening coefficients and synthetic photometry for modelatmosphere grids at Galactic, LMC, and SMC abundances.
Abstract
New grids of Atlas9 models have been calculated using revised convection parameters and updated opacitydistribution functions, for chemical compositions intended to be representative of solar, [M/H] = +0.3, +0.5, Large Magellanic Cloud (LMC), and Small Magellanic Cloud (SMC) abundances. The grids cover kK, from to the effective Eddington limit. Limbdarkening coefficients and synthetic photometry are presented in the UBVRIJHKLM, uvby, ugriz, WFCAM, Hipparcos/Tycho, and Kepler passbands for these models, and for Castelli’s comparable ‘newODF’ grids. Flux distributions are given for the new models. The sensitivity of limbdarkening coefficients to the adopted physics is illustrated.
1 Introduction
The significance of limb darkening as a probe of atmospheric structure was recognized from the earliest modelling of stellar atmospheres (e.g., Schwarzschild, 1906; Jeans, 1917; Milne, 1921), and its importance in the quantitative photometric analysis of eclipsing binaries similarly noted (Russell, 1912; Russell & Shapley, 1912). Reliable empirical determinations of limb darkening are rarely possible in eclipsing binaries (in part because of degeneracies with other parameters), and so values derived from modelatmosphere calculations retain their importance through to the present day, where new applications include modelling of exoplanetary transits, microlensing events, and spatially resolved stellar surfaces (e.g., Southworth, 2008; Witt, 1995; Hestroffer, 1997; Aufdenberg, Ludwig & Kervella, 2005).
Wholesale calculation of limbdarkening coefficients became feasible only after production of the first large grids of model atmospheres (cf. Grygar, Cooper & Jurkevich, 1972, who review earlier studies). In particular, the extensive modelatmosphere grids generated by Kurucz (1979, 1993) form the basis of work by Wade & Rucinski (1985), DíazCordovés & Giménez (1992), van Hamme (1993), Claret (2000), and others.
Although several programs are now available in the public domain for generating lineblanketed model atmospheres which relax the approximation of Local Thermodynamic Equilibrium (LTE), Kurucz’s Atlas codes remain the benchmark for LTE calculations, and have two major attractions for the present purposes. First, while the models may not accurately reproduce individual spectral lines for which nonLTE effects are significant, they nonetheless appear to be quite successful in generating accurate atmospheric structures and broadband fluxes.
Secondly, it is straightforward, and computationally cheap, to generate selfconsistent Atlas grids covering large ranges in parameters of interest. The aspect of internal consistency is of particular importance when modelling the spectra of systems whose temperatures and gravities may show considerable variations over their surfaces (such as the components of close binary systems, rapidly rotating stars, and nonradial pulsators), and when attempting uniform analyses of samples encompassing a range of stellar parameters (for example, in synoptic studies of lightcurves of exoplanetary transits).
Almost all extensive calculations of grids of Atlasbased limbdarkening coefficients published to date have used the formulations of opacitydistribution functions (ODFs) and atmospheric convection inherent to the models published by Kurucz (1993).^{1}^{1}1An exception is work reported by Barban et al. (2003) However, improved treatements in both areas have emerged subsequently, leading in some cases to quite significant changes in the models (Castelli & Kurucz, 2004). The principal purpose of the present paper is to present calculations of limbdarkening coefficients made using new ODFs and improved parameterizations of convection. Because of growing interest in modelling eclipsing binaries in the Magellanic Clouds (e.g., Harries, Hilditch & Howarth, 2003; Hilditch, Howarth & Harries, 2005; Bonanos, 2000; North et al., 2010), new models with tailored MC abundances are included.
2 Models
Atlas has undergone more or less continuous development over its 40year lifetime, and currently exists in two distinct forms, Atlas9 (which uses opacitydistribution functions) and Atlas12 (which uses opacity sampling). The models discussed here were all computed using the Trieste port of Atlas9 to gnulinux systems (Sbordone, Bonifacio & Castelli, 2007).
2.1 ODFs and Abundances
Opacitydistribution functions are required as a basic input to Atlas9 models. Castelli & Kurucz (2004) describe ODFs which incorporate a variety of improvements over those used in the Kurucz (1993) grids, of which perhaps the most important is the inclusion of improved and additional molecular opacities. The models described here use new ODFs, generated using dfsynthe (Kurucz, 2005; Castelli, 2005), based on the Castelli & Kurucz (2004) line lists.
The baseline models use solar abundances reported by Asplund, Grevesse & Sauval (2005). Limb darkening plays an important role in the analysis of lightcurves of stars with transiting exoplanets, and the sample of such stars currently known shows a propensity for enhanced metallicity (e.g., Gonzalez, 1997; Fischer & Valenti, 2005; Sousa et al., 2008). With this in mind, additional models (and ODFs) were calculated with metallicities enhanced by +0.3 and +0.5 dex over solar values. For test purposes, ODFs (but not model grids) were computed at several other metallicities, including [M/H] = for the Vega models discussed in Section 5.
The routine study of eclipsing binaries in external galaxies, and especially in the Magellanic Clouds (MCs), has become feasible in recent years. While no one set of MC abundances is likely to be generally satisfactory, most interest in eclipsing binaries in the Clouds has concentrated on earlytype stars (in part because of their intrinsic brightness). In order to have an appropriate (Pop. I) reference set of representative Large and Small Magellanic Cloud abundances that are at least well defined, and apparently reasonable, results have been compiled from a number of sources. These abundances, listed in Appendix A (Table 4), have been used to compute new LMC and SMC ODFs.
All the ODFs newly calculated here are available online.
2.2 Convection
All modelatmosphere codes incorporate approximations and parameterizations of physics that may be difficult or expensive to model, or is simply poorly understood (cf., e.g., Kurucz, 1996). Convection is a prime example, and is characterized in Atlas9 by a mixinglength approximation with optional overshooting.
The widely distributed Kurucz (1993) grids were calculated with a ratio of mixing length to pressure scale height of and overshooting included, but Castelli, Gratton & Kurucz (1997) argue that moreconsistent results are obtained with Atlas9 when overshooting is switched off. Furthermore, Smalley (2005, and personal communication) advocates smaller values of the mixinglength parameter over at least the lateA spectral range.
All the models presented here were calculated with no overshooting. The baseline grids use , but supplementary grids at were also computed.
Structure source  Grid  Grid  

Atlas9.C04  m25v02  1.25  476  m40av02  1.25  476  
Atlas9.C04  m20v02  1.25  476  m25av02  1.25  476  
Atlas9.C04  m15v02  1.25  476  m20av02  1.25  476  
Atlas9.C04  m10v02  1.25  476  m15av02  1.25  476  
Atlas9.C04  m05v02  1.25  476  m10av02  1.25  476  
Atlas9.C04  p00v00  1.25  476  m05av02  1.25  476  
Atlas9.C04  p00v02  1.25  476  p00av02  1.25  476  
Atlas9.C04  p02v02  1.25  476  p02av02  1.25  476  
Atlas9.C04  p05v02  1.25  476  p05av02  1.25  476  
Atlas9.A10  P00v00  1.25  554  LMCv02  1.25  554  
Atlas9.A10  P00v02  1.25  554  LMCv02r  0.50  552  
Atlas9.A10  P00v02r  0.50  552  SMCv02  1.25  554  
Atlas9.A10  P00v04  1.25  548  SMCv02r  0.50  552  
Atlas9.A10  P03v02  1.25  543  
Atlas9.A10  P03v02r  0.50  544  
Atlas9.A10  P05v02  1.25  531  
Atlas9.A10  P05v02r  0.50  533 
2.3 Parameter space
The grid sampling adopted here matches the basic sampling of the Castelli & Kurucz (2004) models; that is, from 3.5–13kK at 0.25kK intervals, and 12.0–50.0kK at 1kK intervals, from dex (cgs) to lower values at 0.5dex intervals. This grid density is sufficient for interpolation of output products (e.g., linearly in ) to be satisfactory for most applications. Some effort has been put into generating models to as low a gravity as possible at a given temperature (the effective Eddington limit), and so the present grids extend beyond Castelli & Kurucz’s in this regard. The sampling is illustrated in Fig. 1.
Previous work suggests that the Atlas9 turbulentvelocity parameter has little consequence for limbdarkening in the optical regime, and its value (as well as its physical significance) is difficult to establish observationally in any individual case. Most of the model intensities presented here have been computed with a canonical kms. Solarabundance grids at and 4 kms confirm the insensitivity of limb darkening to .
Results of grids of 6571 new models are presented here. Castelli & Kurucz (2004) have calculated models using solar abundances listed by Grevesse & Sauval (1998), which differ only slightly from those given by Asplund, Grevesse & Sauval (2005). Their grids extend over a greater range in scaled solar abundances than those newly calculated here, and include additional modifications to abundances of alphaprocess elements. For completeness, intensities have also been calculated from the 8568 atmospheric structures they provide, along with limbdarkening coefficients and other products to match the new grids. Table LABEL:tab:Grids summarizes the grids, where ‘Atlas9.C04’ and ‘Atlas9.A10’ identify the Castelli & Kurucz (2004) and current grids, respectively.
3 Data Products
The principal motivation for the work reported here was to provide uptodate limbdarkening coefficients (particularly at MC abundances). Thus the main data products are specific intensities (monochromatic radiances) , as a function of , the cosine of the angle between the line of sight and the surface normal. These are provided in condensed form as broadband limbdarkening coefficients.
Physical fluxes,
(1) 
(where is the Eddington flux), are required as a check on the accuracy of intensity integrations, and in any case represent only a minor additional computational overhead. Model fluxes are therefore also provided, both at the 1221 wavelengths used in the structure calculations (91Å–160m), and in broadband (‘synthetic photometry’) form. Other data products (such as model structures and higherresolution intensities) will be made freely available on request.
4 Limb Darkening
4.1 Passbands
Modern detectors working in the optical are essentially photoncounting in nature. For such detectors broadband model intensities (and, mutatis mutandis, fluxes) may be calculated as
(2) 
(e.g., Bessell 2005) where is the response function of passband in a given photometric system. Historically, the equivalent formalism for energyintegrating detectors has been used:
(3) 
Results are provided for photoncounting detectors for all photometric systems considered here, and additionally for energyintegrating detectors for the Johnson and Strömgren passbands. The adopted sources for response functions are:

UBVRI (Johnson–Cousins system): Bessell (1990).

JHKLLM (Johnson–Glass system): Bessell & Brett (1988).

uvby (Strömgren system): Crawford & Barnes (1970; filters), Stubbs et al. (2007; atmospheric transmission), Hamamatsu (1999; S4 response), Allen (1976, p. 108; aluminium reflectivity), Praezisions Glas & Optik (BK7 glass transmission).^{2}^{2}2The Strömgren system is close to being filter defined, but atmospheric transmission, 1P21S4 photomultiplier sensitivity, and BK7 glass transmission all make roughly equal inroads into the shortwavelength edge of the passband.

ZYJHK (WFCAM system): Hewett et al. (2006)

H, B, V (Hipparcos/Tycho system): Bessell (2000)

ugriz (Sloan system): online,
http://www.sdss3.org/instruments/camera.php 
Kepler: online,
http://keplergo.arc.nasa.gov/kepler_response_hires1.txt
Simple geometric treatements of mutual irradiation (the ‘reflection effect’) in binarystar systems require the angular dependence of the bolometric intensity (Wilson, 1990), so bolometric limbdarkening coefficients have also been calculated ().
4.2 Analytical characterization
Specific intensities are traditionally, and conveniently, characterized by simple functional forms, of which the most venerable is the linear limbdarkening law
(4) 
the analytical solution for a source function that is linear in optical depth (Schwarzschild 1906, Milne 1921; for a grey atmosphere).
Morerealistic models do not have analytical solutions, and additional terms are required for a faithful functional representation of actual limb darkening. Kopal (1949) took a seriesexpansion approach, and found the differences between quadratic and quartic approximations to be negligible at the level of accuracy with which he was concerned. Twocoefficient parameterizations subsequently became standard; in particular, a quadratic law of the form
(5) 
was widely adopted in the modern computational era (e.g., Manduca, Bell & Gustafsson, 1977; Wade & Rucinski, 1985), although a cubic version,
was advocated by van ’t Veer (1960). An alternative logarithmic law,
(6) 
was proposed by Klinglesmith & Sobieski (1970), and a squareroot law,
(7) 
by DíazCordovés & Giménez (1992).
For modern work, these limbdarkening laws suffer from various limitations. The obvious way to develop a single analytical expression that more accurately reproduces numerical limbdarkening results across a wide parameter space is simply to use more terms, a notion implemented by Claret (2000) who introduced a fourcoefficient fit in powers of ,
(8) 
Figure 2 illustrates the differing degrees to which the foregoing common functional forms reproduce the intensities computed here. While quantitative results depend on the chosen normalization and adopted figure of merit, the qualitative conclusions are robust: a linear limbdarkening law is a rather poor representation of detailed numerical results; a quadratic law fares little better; but the fourcoefficient law affords an improvement of up to an order of magnitude in precision over any standard twocoefficient option. While there has been some discussion in the literature as to which is the most appropriate limbdarkening ‘law’ to adopt for lightcurve analyses, and in particular under circumstances which permit empirical determination of limbdarkening coefficients (e.g., Southworth, 2008), there seems little justification for adopting anything other than a highorder fit such as eqtn. (8) if the principal purpose is to secure an accurate representation of modelatmosphere results.
4.3 Fitting procedure
For any given analytical form, there remains a choice of numerical technique for determining the limbdarkening coefficients. The most direct approach is simple leastsquares determination of coefficients from functional fits to computed values (with , the fitted value of , as an optional additional free parameter). A related approach, the integration method (Heyrovský, 2007), appears to offer no significant benefits (Claret, 2008).
A criticism that can be levelled against the leastsquares method is that subsequent integration of parameterized intensities may not accurately recover the flux, as it should (eqtn. 1). Alternative formulations in which which flux is explicitly conserved can be devised (e.g., Wade & Rucinski, 1985; van Hamme, 1993), giving rise to socalled flux conservation methods (cf., e.g., Claret, 2008, for a recent discussion). Such methods require some further ad hoc constraint; e.g., Wade & Rucinski (1985) set for the linear model.
For the most part, straightforward leastsquares fitting has been used here, to intensities evaluated at 20 angles.^{3}^{3}3For comparison purposes, the linear law has been modelled with and without as an optimized parameter, and with explicit flux conservation; Appendix B. Subsequent analytical integration of the fourcoefficient intensity model then reproduces directly computed fluxes to better than 0.1%. When fit in this manner, all twocoefficient models conserve flux to within 1%, while the linear model again fares poorly (Fig. 3).
Colour  Vega 9.40kK  Sirius 9.85kK  

is the correction to be added to columns 2–4 for energyintegrating detectors, and is the correction to be added to column 2 to obtain results for the 9.40kK Vega model. The and passbands are those recommended by Bessell (1990) for use in computing synthetic colours to be compared with observed colours.
5 Photometric matters
Broadband fluxes have potential utility in a variety of applications (effectivetemperature determinations, population synthesis, etc.). Thus while it is not the intention here to conduct a detailed confrontation of the new model fluxes with observations, a basic transformation of the synthetic photometry to observed magnitude systems may be of interest. Modern systems generally have dedicated programmes of flux calibration, so the focus here is on the Johnson and Strömgren systems.
5.1 Johnson system
5.1.1 Flux zeropoint
Full normalization of the synthetic photometry can be divided into two parts: absolute flux calibration, and differential (colour) calibration. The former aspect is traditionally bound to the absolute flux calibration of Vega, which has been addressed in detail by Hayes (1985) and Mégessier (1995), using primary measurements mostly made in the 1970s. Their key results can be summarized as pseudomonochromatic 5556Å fluxes:
respectively. These results can be used to provide a zeropoint calibration for the present grids by using a model matching Vega, and its observed band magnitude.
Both ‘the’ band magnitude, and the choice of model, are subject to uncertainty; widely adopted values are (Johnson et al., 1966), and K, , , kms (Castelli & Kurucz, 1994). Subsequent analyses have supported this effective temperature (e.g. Ciardi et al., 2001; Hill, Gulliver & Adelman, 2010), but the HST calibration programme uses a model with K, , , kms (Bohlin, 2007). Workers are therefore faced with choices in observed flux, observed magnitude, and adopted model, as well as in whether to adopt energyintegrating or photoncounting magnitudes. The situation is further complicated by the fact that Vega is a rapid rotator viewed poleon (Gray, 1988; Gulliver et al., 1991; Aufdenberg et al., 2006; Hill, Gulliver & Adelman, 2010), so that in principle no singletemperature model can give an accurate representation of the entire flux distribution (although for magnitude zeropoint calibration this is of minor importance, because the models are only used to scale between 5556Å and band fluxes).
The baseline adopted here is the Hayes calibration, 9.55kK model, Johnson et al. (1966) , and photoncounting magnitudes. For this system the band flux is , yielding
(9) 
where
is in
erg cm s Å and
the optional correction factors are
for the K model;
for the Mégessier (1995) flux;
more generally, ;
for energyintegrated photometry; and
.
5.1.2 Colourindex zeropoints
Differential normalization relies on relating the observed colours of a standard star, or stars, to model fluxes. A model colour index is formed from fluxes in two passbands and by
(10) 
where are model broadband fluxes and are constants. Equating the righthand side of eqtn. (10) with the observed (reddeningfree) colour index yields the required normalizing constant .
Vega has traditionally been adopted as the primary calibrating star, with increasinging use of Sirius as a supplementary standard since the work of Cohen et al. (1992). Once again, however, observed photometry is subject to both statistical uncertainties and deterministic changes (e.g., one might choose to define Vega to have colour indexes of zero, or to average values from an ensemble of standard stars), and the appropriate model parameters are also moot. Here the photometry compiled by Bessell, Castelli & Plez (1998) is adopted, as are the Sirius model parameters due to Kurucz which they quote: kK, , [M/H] = +0.5, kms (see also Castelli 1999). These parameters are in good agreement with other recent determinations (e.g., Hill & Landstreet 1993; van Noort 1998; Qiu et al. 2001).
The resulting normalizing constants are listed in Table LABEL:tab:zp. Agreement between the Vega and Sirius calibrations for the normalizing constants inferred from the specific set of observed colours adopted here is better for the 9.55kK Vega model, and the adopted values are the average of this model and the Sirius results.
5.1.3 Bolometric corrections
From the basic definitions of bolometric correction and magnitude, it is straightforward to show that the bolometric correction in some passband for a model at effective temperature is given by
(11) 
where
is 10 pc (expressed in the same units as the solar radius, ) and
is the magnitudenormalizing constant, which can be found for the Johnson passbands
from data summarized in the preceding two
subsections.^{4}^{4}4E.g.,
,
where numerical values for photoncounting photometry are from
Table LABEL:tab:zp and
eqtn. (9).
Bolometric corrections in the main Johnson bands are included for each model in the online listings, adopting . As a check on results, a solarlike model was calculated with K, (cgs), kms, , ; for this model, the (photoncounting) band bolometric correction is found to be , in excellent agreement with the empirical solar value of (Bessell, Castelli & Plez, 1998).
Band/  Vega  Vega  Sirius  

Colour  9.55kK  9.40kK  9.85kK  
is the correction to be added to the values in column 2 if adopting other sources of photometry;
is the correction to be added for energyintegrating detectors;
is the correction to be added for the Mégessier (1995) flux calibration.
Column 6 gives the corrections to be added for a kK Vega model, while the final column gives the corrections for Sirius photometry (baseline data from Crawford, Barnes & Golson 1970) and a 9.85kK model, where .
5.2 A Strömgren calibration
Flux calibration is readily carried out for the Strömgren uvby system in the same way as for the Johnson system. A potential complication is that for the intermediatewidth passbands of this system, individual spectral features may be significant; in particular, H (410nm) falls in the middle of the band. To check that the standard 20Å spectral sampling is adequate in these circumstances, highresolution synthetic spectra were computed from the Vega models. Strömgrenband fluxes from these synthetic spectra agree with the standard results to better than 0.1% for u, b, and y, and are only 0.5% smaller at v. Since the Balmer lines reach their greatest strength at earlyA spectral types, the standard sampling appears to be satisfactory.
Results are summarized in Table 3. Strömgren colours can be generated from the models with
where is the model flux in passband and the zeropoints are the last three entries in column 2 of Table 3.
Fabregat & Reig (1996) and Gray (1998) have previously provided Strömgren flux calibrations which are the results of convolving the uvby passbands with observed spectrophotometry,^{5}^{5}5The Gray vband calibration is modelbased. from Glushneva et al. (1992) and Taylor (1984), respectively (both nominally on the Hayes 1985 system). Comparison with their results, expressed as a zeromagnitude flux in units of erg cm s Å for consistency with their presentations, shows generally good agreement:
Band  FR  Gray  Atlas9.A10 

11.80  11.72  12.20  
8.69  (8.66)  8.73  
5.84  5.89  5.90  
3.69  3.73  3.72 
The 4% difference at u between spectrophotometric and model results may be a consequence of uncertainties in observed Vega magnitudes, which range over 10% in this passband (Perry, 1969; Barry, 1969; Johansen & Gyldenkerne, 1970; Crawford & Barnes, 1970); e.g., using Sirius as the calibrator yields a zeromagnitude uband flux of erg cm s Å.
6 Discussion
Much attention has been paid in the literature as to which is the best analytical representation of limb darkening, and which is the appropriate numerical technique for evaluating limbdarkening coefficients (see discussion and references in Sections 4.2, 4.3). While these are important details, the use of a highorder parameterization, which provides satisfactory results under most circumstances, renders such details academic.
However, the larger importance of the underlying model atmospheres has not generally been emphasized, perhaps because most analysts have used ‘offtheshelf’ modelatmosphere results. The new opacity sources introduced in the Castelli & Kurucz (2004) models show up directly in emergent fluxes and intensities, and the changes in treatment of convection also change the structures. These changes are particularly important at kK; for models at kK the differences between the current models, those of Castelli & Kurucz (2004), and the older Kurucz (1993) grids, are small, and result largely from revisions to adopted abundances.
As far as specific intensities are concerned, differences between the newer calculations and older results are illustrated by a comparison between the extensive listings provided by Claret (2000) and the Atlas9.A10 models presented here (Fig. 4). The changes can be much larger than any that arise through different fitting techniques, provided that the adopted functional form is adequate (Fig. 5).
We might expect further developments in Atlas models in future, but there is reason to hope that consequences may be modest in scale, at least in the short term; using the latest version of his opacitysampling Atlas12 code, Kurucz (personal communication) has kindly computed a solarabundance model at kK, , kms. Differences between old and new Atlas9 models are relatively large for these parameters, but further changes in the Atlas12 products are negligible for the present purposes.
Finally, the limitations of the present work should be recognised. First, although the Kurucz/Castelli ODFs used here represent a significant improvement over older compilations, they are still inevitably incomplete. In particular, there are a number of indications that Atlas9 models begin to show significant disagreement with observations for effective temperatures 4kK (e.g., Bessell, Castelli & Plez 1998; Bertone et al. 2008), presumably because of the increasing importance of missing molecular opacities. Models at the coolest effective temperatures should be used with caution.
Secondly, the synthetic photometry can be expected to show not only the successes in reproducing observations reported for similar models by Bessell, Castelli & Plez (1998) and Castelli (1999), but also the failures; note that Bessell, Castelli & Plez (1998, cf. their Appendix E.3.1) find that their modelled colours (i.e., using passbands) required an ad hoc scaling of to bring them into agreement with observations.
Thirdly, it must be owned that the new models presented here fare no better than old ones in matching the handful of apparently reliable empirical determinations of stellar limb darkening (e.g., Heyrovský, 2007; Claret, 2008, 2009), although such determinations are usually limited to estimates of coefficients for linear or, occasionally, quadratic limbdarkening laws for stars other than the Sun (and such laws are poor representations of model results). Common simplifying modelatmosphere approximations for scattering phase functions will be a factor in this; although such approximations may be appropriate for structure and integratedflux calculations, they have obvious weaknesses for evaluating intensities (or polarizations). Addressing these issues is a matter for future study.
7 Recommendations
Practically all modern studies in the optical region use photoncounting detectors, and so models appropriate to that context will normally be used. The investigator is still faced with choices in model convection, microturbulence, and abundance (Table LABEL:tab:Grids), and in limbdarkening parameterization (equations 4–8). If, as is frequently the case, there are no results from detailed analyses to suggest otherwise, it’s reasonable to adopt the appropriate global abundance system (solar, LMC, or SMC), kms, . The careful investigator might well conduct sensitivity tests to examine the consequences of other parameters (including varying and ); in particular, for effective temperatures less than 8kK, it would be prudent to examine the effects of models computed at .
Other than for comparison with empiricallydetermined limbdarkening laws (normally linear, but increasingly for quadratic coefficients), there is little merit in adopting anything other than a fourcoefficient limbdarkening parameterization (equation 8) if seeking an accurate representation of modelatmosphere intensities. The computational cost compared to simpler formulae is trivial in anything other than MonteCarlo calculations, and even there the overhead should be entirely tolerable.
Acknowledgments
I gratefully acknowledge the contribution made by R.L. Kurucz in making his codes open source and his datasets freely available, and the work carried out by F. Castelli and her colleagues in porting them to run on gnulinux systems. I thank Susana Barros, Mike Bessell, Antonio Claret, Steve Fossey, Urtzi Jauregi, Bob Kurucz, Luca Sbordone, Barry Smalley, and the referee for helpful correspondence and comments.
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Appendix A Adopted MagellanicCloud Abundances
Baseline solar abundances were taken from Asplund, Grevesse & Sauval (2005). Cloud abundances were adopted, in order of preference, from: FLAMES results (Hunter et al. 2007, Trundle et al. 2007, as summarized by Evans et al. 2008); Garnett (1999); and Russell & Dopita (1992, applying their differential values to Asplund solar abundances). In each case, samples of earlytype stars or Hii regions were analysed, so the abundances are representative of Population I. For elements not studied in these sources, solar abundances were used, adjusted by the median metal offsets from Russell & Dopita (1992): (LMC) and dex (SMC). These global adjustments are in adequate agreement with morerecent discussions (e.g., Hill, Andrievsky & Spite, 1995; Hill, 1997; Luck et al., 1998; Venn, 1999; Mokiem et al., 2007).
Element  LMC  SMC  Element  LMC  SMC  Element  LMC  SMC  Element  LMC  SMC  

2  He  10.88  10.85  24  Cr  5.43  5.06  46  Pd  1.39  1.09  68  Er  0.63  0.33  
3  Li  0.75  0.45  25  Mn  5.06  4.88  47  Ag  0.64  0.34  69  Tm  
4  Be  1.08  0.78  26  Fe  7.23  6.93  48  Cd  1.47  1.17  70  Yb  0.78  0.48  
5  B  2.40  2.10  27  Co  4.62  4.32  49  In  1.30  1.00  71  Lu  
6  C  7.73  7.37  28  Ni  6.02  5.82  50  Sn  1.70  1.40  72  Hf  0.58  0.28  
7  N  6.90  6.50  29  Cu  3.91  3.63  51  Sb  0.70  0.40  73  Ta  
8  O  8.35  7.98  30  Zn  4.20  4.00  52  Te  1.89  1.59  74  W  0.81  0.51  
9  F  4.26  3.96  31  Ga  2.58  2.28  53  I  1.21  0.91  75  Re  
10  Ne  7.60  7.20  32  Ge  3.28  2.98  54  Xe  1.97  1.67  76  Os  1.15  0.85  
11  Na  6.97  5.78  33  As  1.99  1.69  55  Cs  0.77  0.47  77  Ir  1.08  0.78  
12  Mg  7.06  6.72  34  Se  3.03  2.73  56  Ba  1.94  1.23  78  Pt  1.34  1.04  
13  Al  6.07  6.27  35  Br  2.26  1.96  57  La  1.06  0.84  79  Au  0.71  0.41  
14  Si  7.19  6.79  36  Kr  2.98  2.68  58  Ce  1.45  1.26  80  Hg  0.83  0.53  
15  P  5.06  4.76  37  Rb  2.30  2.00  59  Pr  0.41  0.11  81  Tl  0.60  0.30  
16  S  6.70  6.30  38  Sr  2.46  1.32  60  Nd  1.66  1.47  82  Pb  1.70  1.40  
17  Cl  5.01  4.95  39  Y  1.88  1.60  83  Bi  0.35  0.05  
18  Ar  6.20  5.90  40  Zr  2.20  1.93  62  Sm  0.98  1.13  
19  K  4.78  4.48  41  Nb  1.12  0.82  63  Eu  0.22  0.25  90  Th  
20  Ca  5.83  5.63  42  Mo  1.62  1.32  64  Gd  0.82  0.52  
21  Sc  2.62  2.31  65  Tb  92  U  
22  Ti  4.63  4.31  44  Ru  1.54  1.24  66  Dy  0.84  0.54  
23  V  4.10  3.56  45  Rh  0.82  0.52  67  Ho  0.21 
Appendix B Online material
New opacitydistribution functions using the Asplund, Grevesse & Sauval (2005) solar abundances are provided online, at metallicities [M/H] = , together with the LMC and SMC ODFs. In each case, ODFs are given sampled at the standard ‘big’ and ‘little’ wavelength sampling intervals required for use with Atlas9, and for microturbulent velocities of 0, 1, 2, 4, and 8 kms, together with the corresponding Rosselandopacity files. These are binary files in the format required by Atlas9.
Limbdarkening and flux data products are organized into directories corresponding to the grids listed in Table LABEL:tab:Grids (for both new models, and those of Castelli & Kurucz 2004). Within a given directory, there are three plaintext files for each model, with names of the form t05000g40.<ext> where the base identifies and , and the extension identifies the file content.
Files with .flx extensions contain listings of physical fluxes at 1221 wavelengths, from 90Å to 160m (with 20Å sampling through the optical). Fluxes are tabulated in erg cm s Å as a function of wavelength (in Å).
Limbdarkening coefficients are provided in files with extensions .ucP (for photoncounting detectors; Section 4.1) and .ucE (for energyintegrating detectors). The .ucE files list broadband limbdarkening coefficients for Strömgren and Johnson systems (including and passbands for computing synthetic colours), while the .ucP files give broadband limbdarkening coefficients for all 30 passbands listed in Section 4.1, plus [energyintegrated] bolometric coefficients. The data given for each passband may be illustrated by band results for a Vega model (kK, , [M/H], kms, ):
BessellV 5467.7 5.61840E+07 2.08372E+07 0.207 4coeff +4.73679E01 +5.53030E01 5.01375E01 +1.56613E01 +2.75266E04 7.09332E04 quadratic +2.25617E01 +3.71954E01 +1.56919E02 4.19850E02 quad  FC +2.74255E01 +3.01889E01 +1.82965E02 6.25047E02 square root +2.25306E02 +6.82355E01 +2.77154E03 +6.52661E03 logarithmic +6.36370E01 +3.25055E01 +5.45636E03 1.48612E02 linear  2 +1.04831E+00 +5.92481E01 +3.77170E02 9.16531E02 linear  1 +5.25808E01 +4.55362E02 1.09348E01 linear  FC +4.25199E01 +7.68496E02 2.08951E01
Here the first line lists

the passband,

effective wavelength (in Å),

the broadband physical flux (in erg cm s Å),

the broadband surfacenormal intensity (in erg cm s Å sr), and

the bolometric correction (for Johnson passbands only).
Subsequent lines list coefficients for the limbdarkening laws summarized in Section 4.2; the final two numbers in each row are the r.m.s. and maximum differences between input and modelled values of .
The ‘linear  FC’ and ‘quad  FC’ results use flux conservation in place of least squares, with the fitted value anchored to the model in the first case, and additionally to in the second (cf. Wade & Rucinski 1985). The ‘linear  1’ listing is for the standard (singleparameter) leastsquares fit to eqtn. (4), while the coefficients in the ‘linear  2’ listing are for a model with free, i.e.,