We present grids of limb-darkening coefficients computed from non-local thermodynamic equilibrium (non-LTE), line-blanketed TLUSTY model atmospheres, covering effective-temperature and surface-gravity ranges of 15–55 kK and 4.75 dex (cgs) down to the effective Eddington limit, at 2×, 1×, 0.5× (Large Magellanic Cloud), 0.2× (Small Magellanic Cloud), and 0.1× solar. Results are given for the Bessell UBVRICJKHL, Sloan ugriz, Strömgren ubvy, WFCAM ZYJHK, Hipparcos, Kepler, and Tycho passbands, in each case characterized by several different limb-darkening ‘laws’. We examine the sensitivity of limb darkening to temperature, gravity, metallicity, microturbulent velocity, and wavelength, and make a comparison with LTE models. The dependence on metallicity is very weak, but limb darkening is a moderately strong function of log g in this temperature regime

Howarth, ID

Reeve, DC

UCL Discovery

MNRAS 456, 1294–1298 (2016) doi:10.1093/mnras/stv2631Limb-darkening coefficients from line-blanketed non-LTE hot-starmodel atmospheresD. C. Reeve and I. D. Howarth‹Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UKAccepted 2015 November 5. Received 2015 October 30; in original form 2015 October 6ABSTRACTWe present grids of limb-darkening coefficients computed from non-local thermody-namic equilibrium (non-LTE), line-blanketed TLUSTY model atmospheres, covering effective-temperature and surface-gravity ranges of 15–55 kK and 4.75 dex (cgs) down to the effectiveEddington limit, at 2×, 1×, 0.5× (Large Magellanic Cloud), 0.2× (Small Magellanic Cloud),and 0.1× solar. Results are given for the Bessell UBVRICJKHL, Sloan ugriz, Stro¨mgrenubvy, WFCAM ZYJHK, Hipparcos, Kepler, and Tycho passbands, in each case characterizedby several different limb-darkening ‘laws’. We examine the sensitivity of limb darkening totemperature, gravity, metallicity, microturbulent velocity, and wavelength, and make a com-parison with LTE models. The dependence on metallicity is very weak, but limb darkening isa moderately strong function of log g in this temperature regime.Key words: radiative transfer – stars: atmospheres.1 IN T RO D U C T I O NAn accurate description of limb darkening is essential to any in-vestigation involving the direct or indirect spatial resolution of astellar surface. Modelling of eclipsing-binary light-curves has beena major application historically, but studies of exoplanetary transitsand microlensing events, and interferometric observations of stellarsurfaces, have similar requirements.The advent of good-quality line-blanketed local thermodynamicequilibrium (LTE) model atmospheres (particularly from Kurucz’sATLAS codes; Kurucz 1979) made it possible to generate large gridsof synthetic spectra, and associated limb-darkening coefficients(LDCs), for wide ranges of stellar parameters (e.g. Wade & Rucinski1985; Claret 2000; Howarth 2011a, and references therein). Resultsfor non-LTE models are fewer, for reasons of computational com-plexity and expense, and available grids extend only to Teff  10kK (Claret & Hauschildt 2003; Claret, Hauschildt & Witte 2012,2013).Here, we report results from line-blanketed non-LTE models forthe OB-star regime. These may be of particular interest in the studyof extragalactic eclipsing-binary systems, where hot, luminous starspresent good opportunities for direct distance-scale determinations,in addition to measurements of basic stellar parameters (e.g. Har-ries, Hilditch & Howarth 2003; Hilditch, Howarth & Harries 2005;Bonanos et al. 2009; North et al. 2010). They should also haveapplicability in accurate photometric modelling of hot stars thatdepart from spherical symmetry, through rapid rotation (as in the E-mail: idh@star.ucl.ac.ukcase of Be stars; e.g. Townsend, Owocki & Howarth 2004), pul-sation, or surface inhomogeneities (e.g. Ramiaramanantsoa et al.2014; Buysschaert et al. 2015).2 M E T H O D2.1 ModelsOur calculations make use of line-blanketed TLUSTY models (Hubeny& Lanz 1995); these assume plane-parallel, hydrostatic atmo-spheres, relaxing the approximation of LTE. Two extensive grids ofstructures are available, based on these models: OSTAR2002 (Lanz& Hubeny 2003; 27.5–55.0 kK) and BSTAR2006 (Lanz & Hubeny2007; 15.0–30.0 kK), each extending over a range of surface gravi-ties from 4.75 dex (cgs) to the effective Eddington limit, at severalmetallicities.We computed emergent intensities from the atmospheric struc-tures with moderately dense wavelength sampling (median inter-val ∼0.4 Å) at 20 angles, approximately equally spaced over therange 0.001–1 in μ ≡ cos θ , where θ is the emergent angle mea-sured from the surface normal. The calculations were made usingHubeny’s SYNSPEC code,1 with the accompanying line lists aug-mented with data from Kurucz’s GFALL compilation,2 to extend cov-erage of (near-)IR wavelengths.1 http://nova.astro.umd.edu/Synspec49/synspec.html2 http://kurucz.harvard.edu/linelists/gfall/C© 2015 The AuthorsPublished by Oxford University Press on behalf of the Royal Astronomical SocietyNon-LTE limb-darkening coefficients 1295Table 1. Summary of models; sampling in gravity is at steps of 0.25 dex. Metallicity codings are identified in columns 7–11 (e.g. the L grid corresponds toZ/Z = 0.5); ξ is the microturbulence; N is the number of models. The high-ξ models summarized in the final row of the table are intended to correspond toB supergiants, and do not extend to main-sequence gravities (cf. Lanz & Hubeny 2007); in view of the weak sensitivity of results to metallicity, we computedonly solar-metallicity LDCs for these models.Grid Teff log g (min) (max) ξ Z/Z coding N(kK) [Teff(lo)] Teff(hi)] (km s−1) —————————————————–OStar02 27.5–55.0 @ 2.5 3.00 4.00 4.75 10 2 1 0.5 0.2 0.1 345BStar06 15.0–30.0 @ 1.0 1.75 3.00 4.75 2 C G L S T 817BStar06 15.0–30.0 @ 1.0 1.75 3.00 3.00 10 G 49Results were used to generate broad-band specific intensitiesappropriate for photon-counting detectors (such as CCDs),Ii(μ) =∫λIλ(μ) φi(λ) λ dλ∫λφi(λ) λ dλ,where φi(λ) is the response function for passband i. We employedthe response functions adopted by Howarth (2011a) for the BessellUBVRICJKHL, Sloan ugriz, Stro¨mgren ubvy, WFCAM ZYJHK,Hipparcos, Kepler, and Tycho passbands.2.2 CharacterizationParticularly for photometric applications, limb darkening is, inpractice, conveniently characterized by ad hoc analytical ‘laws’representing the specific intensities (or monochromatic radiances),thereby reducing the data description to a small set of LDCs.While numerous limb-darkening laws have been proposed (cf.,e.g. Claret 2000; Howarth 2011a), current observations (particularlyof exoplanetary transits) are generally capable of usefully constrain-ing only one- or two-parameter forms. The analytical solution for amodel atmosphere in which the source function is linear in opticaldepth isIλ(μ) = Iλ(1) [1 − uλ(1 − μ)] , (1)and use of this linear limb-darkening law persists in the contextof light-curve modelling, although it often gives a rather poor rep-resentation of model-atmosphere emergent intensities. A quadraticlaw,Iλ(μ) = Iλ(1)[1 − u1,λ(1 − μ) − u2,λ(1 − μ)2] (2)offers greater flexibility, and two coefficients represent the practicallimit for empirical determinations of LDCs (within limits; cf., e.g.Howarth 2011b; Mu¨ller et al. 2013). The most accurate algebraicrepresentation of model-atmosphere results in routine use is the4-parameter law introduced by Claret (2000),Iλ(μ) = Iλ(1)[1 −4∑k=1ak,λ(1 − μk/2)], (3)which is able to fit intensity profiles across a wide parameter spacewith good accuracy.2.3 ResultsThe 1211 models for which we have computed emergent intensitiesare summarized in Table 1. For each model we calculated LDCs foreach ‘law’ (equations 1–3) by least-squares fit of the given functionsto Ii(μ) over the range 0.05 ≤ μ ≤ 1.00,3 with and without Ii(1) asa free parameter in the fit. We also computed flux-conserving least-squares coefficients for the linear and quadratic laws (followingHowarth 2011b). Results are presented online (q.v. Appendix A).3 D I S C U S S I O N A N D C O N C L U S I O N SFig. 1 shows the general form of the limb darkening at represen-tative values of temperature, gravity, abundance, and wavelength;and illustrates the sensitivity to technical details of the differentmodels (OSTAR2002, BSTAR2006, and line-blanketed LTE modelsfrom Howarth 2011a). These aspects are reviewed across a broaderparameter space, but in a coarser manner, in Fig. 2, which showsthe linear LDC for a range of models (equation 1, I(1) fixed; seealso fig. 1 of Howarth 2011b).As well as the obvious dependences on temperature and passband(generally, smaller limb-darkening at longer wavelengths), there isa fairly strong dependence on gravity in the OB-star regime (Fig. 2;Fig. 1, panels g, h), although the sensitivity to metallicity is ratherweak (panels e, f).Differences in limb-darkening between Kurucz (LTE) and TLUSTY(non-LTE) models turn out to be reasonably small (Fig. 1, panelsa–c). As expected, the BSTAR2006 models at ξ = 10 km s−1 arein excellent agreement with their OSTAR2002 counterparts in theirlimited overlap range, and both show only small differences fromthe ξ = 2 km s−1 B-star models (Fig. 1i).Finally, it should be borne in mind that the LDCs presentedhere are all based on hydrostatic, plane-parallel model structures.Real O stars, and B supergiants (at least), have substantial stellarwinds, which raises the question of the applicability of our results.Lanz & Hubeny (2003) address this issue at some length, and con-clude that TLUSTY models give a satisfactory representation of mostspectral lines in the UV–IR regime (a conclusion that applies afortiori to the continuum), and that line blanketing is the more im-portant consideration. We therefore expect our broad-band LDCsto provide a reasonable description of nature, other than when thecontinuum forms in dynamical layers or where spherical exten-sion is significant (e.g. in Luminous Blue Variables). These circum-stances probably occur only outside the log g domain covered by thegrids.3 We discarded Ii(10−3), as the intensity can drop off rapidly at the extremelimb (cf. Fig. 1), ‘throwing’ the fit. This has no important consequences formost applications.MNRAS 456, 1294–1298 (2016)1296 D. C. Reeve and I. D. HowarthFigure1.Representativeintensitydistributionstoillustratethesensitivityofcomputedlimbdarkeningtotemperature(compare,e.g.panelsa–c),gravity(panelsg,h),microturbulence(d,i),abundance(e,f),andwavelength(JohnsonB,I C,andLbandsthroughout).Modelsareidentifiedbysource[B,Oarenon-LTETLUSTYBSTAR2006andOSTAR2002results;KdenotesATLAS(‘Kurucz’)LTEcalculations,fromHowarth2011a],microturbulence(2or10kms−1 ),abundance(C,G,SforZ/Z =2,1,0.2),effectivetemperature(inkK),andlogg(cgs).Continuousanddashedcurvesareparametrized(4-coefficient)characterizations,withdots(and‘+’symbols)showingindividualcomputedI(μ)values.(Thecentrepaneldoescorrectlydisplayresultsoftwodifferent,butalmostindistinguishable,models.)SeeSection3fordiscussion.MNRAS 456, 1294–1298 (2016)Non-LTE limb-darkening coefficients 1297Figure 2. Linear (one-parameter) LDCs for selected models, to illustratelog g, ξ , and metallicity dependences. Results are labelled by Johnson pass-band and log g (additional results are available at intermediate gravities); ‘G’and ‘S’ models have Z/Z = 1 and 0.2, respectively; ‘K’ identifies LTE(Kurucz) results from Howarth (2011a). B-band results for B-star models atlog g = 2.5 and 3.0 are shown for both ξ = 2 and 10 km s−1(‘normal’ and‘supergiant’ values).AC K N OW L E D G E M E N T SWe thank Ivan Hubeny for his support of TLUSTY and SYNSPEC. Ourreferee, Sergio Simo´n-Dı´az, made several useful suggestions thatimproved the presentation.R E F E R E N C E SBonanos A. Z. et al., 2009, AJ, 138, 1003Buysschaert B. et al., 2015, MNRAS, 453, 89Claret A., 2000, A&A, 363, 1081Claret A., Hauschildt P. H., 2003, A&A, 412, 241Claret A., Hauschildt P. H., Witte S., 2012, A&A, 546, A14Claret A., Hauschildt P. H., Witte S., 2013, A&A, 552, A16Harries T. J., Hilditch R. W., Howarth I. D., 2003, MNRAS, 339, 157Hilditch R. W., Howarth I. D., Harries T. J., 2005, MNRAS, 357, 304Howarth I. D., 2011a, MNRAS, 413, 1515Howarth I. D., 2011b, MNRAS, 418, 1165Hubeny I., Lanz T., 1995, ApJ, 439, 875Kurucz R. L., 1979, ApJS, 40, 1Lanz T., Hubeny I., 2003, ApJS, 146, 417Lanz T., Hubeny I., 2007, ApJS, 169, 83Mu¨ller H. M., Huber K. F., Czesla S., Wolter U., Schmitt J. H. M. M., 2013,A&A, 560, A112North P., Gauderon R., Barblan F., Royer F., 2010, A&A, 520, A74Ramiaramanantsoa T. et al., 2014, MNRAS, 441, 910Townsend R. H. D., Owocki S. P., Howarth I. D., 2004, MNRAS, 350, 189Wade R. A., Rucinski S. M., 1985, A&AS, 60, 471S U P P O RT I N G IN F O R M AT I O NAdditional Supporting Information may be found in the online ver-sion of this article:Appendix. http://www.mnras.oxfordjournals.org/lookup/suppl/doi:10.1093/mnras/stv2631/-/DC1Please note: Oxford University Press is not responsible for thecontent or functionality of any supporting materials supplied bythe authors. Any queries (other than missing material) should bedirected to the corresponding author for the article.A PPENDIX A : O N-LINE MATERIALResults are given in two summary files (one for linear and quadratic forms and one for 4-coefficient fits), and for individual models, identifiableby file name; e.g., the file BG20000g425v02.LDC is from the Bstar06, G-abundance (‘Galactic’, Z/Z = 1) grid, for Teff = 20 kK, log g =4.25, microturbulence ξ = 2 km s−1. Each individual model file contains a series of blocks giving results for different passbands; for example,BG20000g425v02.LDC startsThe first line lists(i) the passband identifier (here Bessell’s characterizations of the Johnson U and B passbands used in computing U − B colours);(ii) the effective wavelength (in Å);(iii) the broad-band physical flux Fλ (=4πHλ, in erg cm−2 s−1 Å−1);(iv) the broad-band surface-normal intensity Iλ(1) (in erg cm−2 s−1 Å−1 sr−1)(multiply values by 10−2 to obtain results in units of J m−2 s−1 nm−1). Subsequent lines list coefficients for the limb-darkening laws discussedin Section 2.2; ‘FCLS’ indicates values obtained by flux-conserving least squares. The first value is ˆIλ(1)/Iλ(1), the ratio of the fittedsurface-normal intensity to the model-atmosphere value, followed by the set of limb-darkening coefficients.MNRAS 456, 1294–1298 (2016)1298 D. C. Reeve and I. D. HowarthThe last four columns are indicators of the fit quality, being the r.m.s. and maximum O−C values of Iλ(μ)/Iλ(1) (where ‘O’ and ‘C’ referto model-atmosphere and analytical-fit values); the value of μ for which the absolute value of the mismatch is greatest; and the percentageerror in the flux obtained by integrating the analytical represention of the intensity to obtain the first moment of the radiation field (Eddingtonflux, Hλ = Fλ/4π ).The summary files extract and compile the LDC information in a format intended to be self-explanatory, and amenable to processing bysimple text-manipulation tools (e.g., UNIX grep, awk, and cut commands). The Summary1.LDC file of linear and quadratic coefficients,for example, beginsThe Summary2.LDC file has corresponding results for the coefficients in eqtn. 3.All files are available through CDS.This paper has been typeset from a TEX/LATEX file prepared by the author.MNRAS 456, 1294–1298 (2016)

Limb-darkening coefficients from line-blanketed non-LTE hot-star model atmospheres

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Limb-darkening coefficients from line-blanketed non-LTE hot-star model atmospheres

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