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Astronomy & Astrophysics manuscript no. (will be inserted by hand later) ⋆ Detailed analysis of Balmer lines in cool dwarf stars 1 1 2 1 1 3 P. S. Barklem , H. C. Stempels , C. Allende Prieto , O. P. Kochukhov , N. Piskunov , and B. J. O’Mara 1 Department of Astronomy and Space Physics, Uppsala University,Box 515, S 751-20 Uppsala, Sweden 2 2 McDonald Observatory and Department of Astronomy,University of Texas, Austin,TX 78712-1083, USA 0 3 Department of Physics, The Universityof Queensland,St Lucia, 4072, Australia 0 2 Received 30/11/01 / Accepted 28/01/02 n a J Abstract. An analysis of Hα and Hβ spectra in a sample of 30 cool dwarf and subgiant stars is presented using 1 MARCS model atmospheres based on the most recent calculations of the line opacities. A detailed quantitative 3 comparison of the solar fluxspectra with model spectra shows that Balmer line profile shapes, and therefore the temperature structure in the line formation region, are best represented under the mixing length theory by any 1 combination of a low mixing-length parameter α and a low convective structure parameter y. A slightly lower v effective temperature is obtained for the sun than the accepted value, which we attribute to errors in models 7 and line opacities. The programme stars span temperatures from 4800 to 7100 K and include a small number of 3 population IIstars. Effectivetemperatureshavebeenderivedusingaquantitativefittingmethodwith adetailed 5 erroranalysis.OurtemperaturesfindgoodagreementwiththosefromtheInfraredFluxMethod(IRFM)nearsolar 1 metallicitybutshowdifferencesatlowmetallicitywherethetwoavailableIRFMdeterminationsthemselvesarein 0 disagreement.ComparisonwithrecenttemperaturedeterminationsusingBalmerlinesbyFuhrmann(1998,2000), 2 who employed a different description of the wing absorption due to self-broadening, does not show the large 0 / differences predicted by Barklem et al. (2000b). In fact, perhaps fortuitously, reasonable agreement is found h near solar metallicity, while we find significantly cooler temperatures for low metallicity stars of around solar p temperature. - o r Key words.stars: atmospheres — stars: fundamentalparameters — convection t s a : v 1. Introduction influences the main continuum opacity source,changes in i the hydrogen abundance are hardly reflected in the lines’ X Hydrogen is by far the most abundant species in typi- strengths, and the strengths are much more weakly af- r cal stellar atmospheres.In late-type atmospheres,a small a fected by gravity and metal abundances than perturba- fraction of hydrogen atoms capture free electrons form- tions to the temperature. These properties attracted the ing H− ions, which dominate the continuum opacity. attentionofstellarspectroscopists,makingBalmerlinesa Hydrogenitselfisthemainopacitysourceforearlierspec- key feature in stellar classification schemes (e.g. Morgan tral types. The few lines that its simple atomic structure et al. 1943). makes in the spectrum have a very distinct sensitivity to theatmosphericpropertiescomparedtometallines.Inop- Detailed analyses of Balmer lines in late-type ticalstellar spectra,absorptionlines of the Balmer(n=2) stars are more recent (e.g. Gehren 1981; Fuhrmann seriesarecommonlyusedtostudyphotospheres.Thewell et al. 1993, 1994; van’t Veer-Menneret et al. 1998; populated lowest levels of the atom produce considerable Gardiner et al. 1999). These modern studies exploit opacity at the centre of the lines, and interactions with progress in theory and experiment on line broaden- charged ions, electrons and other hydrogen atoms result ing to infer stellar effective temperatures from Balmer in extended wings in high-density atmospheres. In late- lines. Vidal et al. (1970, 1973) developed a success- typedwarfs,thesewingsarebelievedtoformverycloseto ful unified theory to model the interaction of hydrogen LTE,in the deepest photosphericlayers.As mostprotons atoms with charged particles. Those calculations have are bound to electrons forming hydrogen, and hydrogen been recently superseded by Stehlé (1994) and Stehlé & Hutcheon (1999), who have computed Stark broad- ened line profiles including ion dynamic effects under the Send offprint requests to: P. S. Barklem, ⋆ Based on observations collected at the Isaac Newton model microfield method. A further important broaden- Telescope, La Palma, Spain, and McDonald Observatory, ing contributor is the collisions with neighbouring hydro- Texas, USA. gen atoms. Ali & Griem (1966) used the multipole ex- 2 P. S.Barklem et al.: Detailed analysis of Balmer lines in cool dwarf stars pansion of the resonance interaction potential in the im- 2. Observations and Reduction pactapproximationtocalculateline-widthsfromthispro- The majority of the targets were observed at the cess. Barklem et al. (2000a, 2000b, hereafter paper I and 2.5 m Isaac Newton Telescope (INT) at La Palma II respectively) have presented a self-broadening theory with the MUSICOS cross-dispersed echelle spectrograph accounting also for dispersive-inductive interactions and (e.g. Baudrand & Böhm 1992) fed by a fibre from the withoutuseofthemultipoleexpansion.Itwasshownthat Cassegrainfocus.Targetswere selectedinorderto obtain the new description of the self-broadening would have a a sample of dwarf and subgiant stars with a spread of largeimpactonthecomputedBalmerlineprofiles,inpar- temperature and metallicity, with emphasis on well stud- ticular when applied to derive effective temperatures for iedstars.Duetotheconstraintsofthetelescopeused,only metal-poor dwarf stars. a relatively small number of metal-poor targets could be Assuming aproperunderstandingofthe line broaden- observed. The data consist of observations taken in May ing of the hydrogen lines, the most notable difficulty for 1999and January2000,using 2048 2048and1024 1024 the use of their wings as a temperature indicator is the × × pixel CCD arraysrespectively.The spectra have a resolu- fact that they are formed in very deep layers. The ther- tion of approximately R λ/δλ 30000, and signal-to- mal structure of the deepest optically transparent layers ≡ ≈ noiseratio(SNR)perpixeloftypicallybetterthan100at in late-type stars is significantly affected by convection. HαandHβ,reaching300forsomeofthebrightertargets. Simple modelling ofsurfaceconvectionis stillachallenge. The Hα and Hβ spectra have an averageSNR of 180 and Thecommonlyusedmixing-lengthformalismincorporates 150 respectively. The 1999 observations were obtained as unphysical parameters which are hard to connect with abackupprogrammeduringperiodsofhighcirrus.Wedo quantities that can be derived from observations or hy- notexpectthe subsequentscatteringtobe significantand drodynamical simulations and, therefore, are difficult to no attempt to remove the water vapour lines is made. constrain.Thisobstaclemakesflux-constanthomogeneous With this spectrograph,the Balmer lines arewellcen- models particularlyuncertain in these layers.Other theo- tredinthe orders;however,the broadwingsofthese lines ries thatdispense withthe free parametersin the mixing- canspanseveralorders,makingdeterminationofthecon- lengththeory(MLT)havealsobeenproposed(e.g.Canuto tinuum level a difficult process. This may be resolved by etal.1996;Canuto&Mazzitelli1991);however,Gardiner mergingseveralconsecutiveorders,butagoodnormalisa- et al. (1999) found that after adjusting the mixing-length tionisneeded.Flat-fieldingofMUSICOSspectratakenat parameter α, MLT performed similarly. Semi-empirical the INThaslongbeenaproblem,asthe internalflat-field modelling (e.g. Allende Prieto et al. 2001) does not of- lamps available at the Cassegrainfocus were designed for fer a viable solution, as the employed metal lines do not instrumentswithlowandmediumspectralresolution.The probe layers as deep as those where hydrogen lines are MUSICOS spectrograph is a high-resolution instrument, formed. These theoretical problems related to establish- andthe frontendofthe fibre is relativelysmallcompared ing a model atmosphere combine with observational con- to the entrance slits of the lower resolution instruments, straints.Balmerlinesrequirehighdispersionobservations thus missing a significant portion of the light from the with a large spectral coverage, and a predictable instru- lamps. The flat-field frames obtained during our observa- mental response, making possible a methodical and ac- tions were not satisfactory, seen for example by the fact curate continuum normalisation. In this situation, use of that the behaviour of the continuum of orders when di- Balmer lines as a part of spectroscopic analyses requires vided by the flat-field frames required a high-order poly- a careful assessment of all possible sources of error. nomial to fit them, rather than a low-order polynomial In this paper we investigate the impact of the afore- typical of the residual difference from the stellar flux dis- mentioned line broadening theory advances in the frame- tribution and the flat-field lamp. work of 1D model atmospheres, and attempt to iden- Forbroadlinesthepixel-to-pixelvariationsintheCCD tify those areas where improvement is most desirable. In responsefunctionarenotasimportantasfornarrowlines Sect. 2 we describe our observationsand reduction proce- and so instead of using traditional flat-fields, we con- dure. In Sect. 3 we describe how model spectra are com- structedartificialflat-fields whichneglectthesevariations puted, and put the atmospheric models in context with but reconstruct the blaze shape from stellar exposures. others. In Sect. 4 we introduce a method for quantita- The relation between the blaze shapes of the different or- tive comparison of observations and model spectra, and ders is in principle a smoothly changing function, andbe- subsequently an automated fitting procedure for deriving cause in general only a few orders contain broad spectral effective temperatures. In Sect. 5 we make a detailed sur- lines the blaze shape for these orders can be determined vey of possible errors and their effect on effective tem- byinterpolation.Thespectrallayoutofframestakenwith peratures.Applicationto the solarspectrumandthe pro- the MUSICOS spectrograph is very well adapted to this gramme stars is then presented. Finally in Sect. 6 the re- approach,because notonly is the number ofordersin the sults are compared with other work, and in Sect. 7 our framelarge(oforder50),but alsothe coverageinthe dis- conclusions are presented. persion direction is large (typically 60–100˚A),tracing the shape of the blaze function far out in its wings. Orders have significant overlap in wavelength, even at Hα where P. S.Barklem et al.: Detailed analysis of Balmer lines in cool dwarf stars 3 theoverlapissmallest,theamountdependingontheCCD Comparison of the McDonald and INT Procyon spectra array used. To determine the shape of the blaze function shows similar agreement, despite quite different instru- we used the fact that for our data set the extracted pro- mental setups. We will test this consistency later by ex- files contain only absorption lines. We then performed a aminingscatterintemperaturesderivedfromdifferentob- polynomialfittotheupperpointsineachextractedorder. servations. This method fails if there are no continuum points available due to the presence of a broad hydrogen line in the 3. Models and Synthetic Spectra order.Asurfacefitoftheneighbouringordersisthenused todetermine the artificialflat-fieldatanyproblematicor- In this work we employ 1D LTE plane-parallel model at- ders. The regions for interpolation are selected interac- mospheres from the MARCS code (Asplund et al. 1997 tively, which may include other problematic regions such version, see this reference and those therein for details). as strong metal lines or molecular bands. This procedure The adopted version of MARCS employs opacity sam- can also be used if the frames are first divided by a tra- pling techniques, except in the infrared where opacity ditional flat-field exposure, as the flat-field blaze-function distribution functions are deemed sufficiently accurate. is still smoothly changing. On the larger CCD fringing This circumvents problems of opacity distribution func- at Hα made this procedure necessary for the rectification tionsnamelyallowingfreechoiceofindividualmetalabun- of this line. However, for other cases we did not divide dances(cf.Fuhrmannetal.1997).Convectionismodelled the exposures as we found the variation of the orders was underMLTwhichisparameterisedintermsofanumberof in general smoother. Orders are extracted and scattered parameters,themostimportantofwhichformodelatmo- light corrected using standard IRAF echelle package rou- spheresandourdiscussionsarethemixing-lengthl,which tines, before normalisationas described above using IDL. is usually expressed in units of the pressure scale height Finally the wavelength calibration is determined, orders H as the mixing-length parameter α l/H , and the p p ≡ are merged together weighted by the SNR at the wave- structure parameter y describing the temperature struc- length in the order, and wavelengths are corrected to the ture within convective elements (see Henyey et al. 1965 the stellar rest frame. The method is illustrated in Fig. 1 for details). A logτ (τ is Rosseland mean optical ross ross for one of the more difficult cases (low SNR 120 and depth) grid suitable for computation of Balmer lines was ≈ broad line). chosen.The depth gridextends into logτ =2 and the ross For Procyon, in addition to spectra from grid is finest in the region where these lines are formed INT/MUSICOS, a very high quality spectrum has 1<logτ <0.6. ross − been obtained by Allende Prieto et al. (2002) with the Models are computed with scaled-solar photospheric McDonald Observatory 2.7 m telescope and the 2dcoudé abundances from Anders & Grevesse (1989) except for spectrograph (Tull et al. 1995) for a large spectral region and most importantly the iron abundance where a “low” including Hβ. The spectra have SNR of about 1000 valueoflog(N /N )+12=7.50isadoptedinagreement Fe H at Hβ, and a resolution of approximately 2 105 (see with the meteoritic value (e.g. Grevesse & Sauval 1998). × Allende Prieto et al. 2002 for details). The same contin- A microturbulence of 1.5 km s−1 was used in all cases uum placement techniques described for INT/MUSICOS for computing the models. The MLT parameters used were applied to these observations. were α = 1.5, y = 0.076 the default MARCS setting, ThespectrumofHD103095(Gmb1830)wasacquired and α = 0.5, y = 0.5 following Fuhrmann et al. (1993). on 28 and 29 April 1990 with the McDonald 2.7 m tele- Hereafter we will use MARCS to refer to the former and scope and the Coudé spectrograph.The observations em- MARCS05 to the latter.The reasonsfor these choicesare ployed a conventional grating in first order and a colour discussed later. Each grid was also recomputed with en- filter to eliminate light from higher orders. The detec- hancementofthealpha-elementsby0.4dex,whichwewill tor was a 800 800 pixel CCD. The dispersion is about denoteindiscussionswithan“a”(e.g.MARCS05a).When × 0.12 ˚A/pixel and the SNR was a few hundred. This spec- referring specifically to the grids with scaled-solar abun- trum wasprocessedusing IRAF being normalisedusing a dances we will use an “s”.The model grids are computed second-order polynomial fit. withspacingsof100KinT ,0.1dexinlogg and0.5dex eff Comparison of reduced spectra from different expo- in [Fe/H], within which we interpolate specific models. suresofthe sametargets,revealedsimilarresultstothose Synthetic model flux spectra are computed as de- ofFuhrmannetal.(1993),namelyaninternalconsistency scribed in paper II. In short, spectra are computed as- ofapproximately1%ofthe continuumflux,dependenton sumingLTEusingSYNTH(Piskunov1992),Starkbroad- SNR. Most of this error probably stems from the deter- ening is described by the model-microfield method calcu- mination of the continuum placement. The error can be lations of Stehlé & Hutcheon (1999) and self-broadening reduced through co-adding a number of independent ob- calculations of paper II are used. Radiative broadening is servations (Fuhrmann et al. 1993). Direct comparison of included,asisanestimateoftheheliumcollisionalbroad- moon spectra with the FTS atlas (Kurucz et al. 1984) ening(basedonrescalingofBarklem&O’Mara1997),al- degraded to appropriate resolution shows agreement at though both effects are small. We will refer to this as the the level of 0.5–1%, although small differences in wave- STEHLE+BPO broadening recipe. For comparison with length calibration make the direct comparison difficult. previous work we also do calculations substituting the 4 P. S.Barklem et al.: Detailed analysis of Balmer lines in cool dwarf stars Fig.1. An example ofthe continuum determinationprocedure,for the Hβ line in HR 5447,obtainedon 29 May 1999 with INT and MUSICOS. The upper panelshows the extractedorderwhich is centredonHβ. The smoothlines show the fit to the continuum for nearby orders where full lines are determined from the spectra, and dashed lines were determinedby interpolationfromthe full lines.The thicker dashedline is the continuumcorrespondingto the plotted spectral order. Note the pixel number is arbitrary as the ends of orders have been removed to avoid poor behaviour of the polynomial fits. The rapid variation of the blaze function across an order is a characteristic of the Littrow configurationused inMUSICOS. The lowerpanels showthe finalspectrum(with best fit modelspectrum) alongwith anestimateofthe SNRperpixelin the continuum whichalsoservestoshowthe positionofeachorder.Theupwardly convex regions are the centres of the orders, and the concave parts the overlapping ends of the orders which have higher formal SNR due to contributions from the two overlapping orders. For clarity the region corresponding to the ordercentred onthe Balmer line (i.e. that spectrumshownin the top panel)is shaded, with the areasof overlapwith the adjacent orders shaded darker. Note the cosmic ray hit in the upper panel has been removed. Ali & Griem (1966) resonance broadening theory for the 3.1. Comparison to other models self-broadeningcalculations,everythingelseremainingthe same.ThiswillbereferredtoastheSTEHLE+AGbroad- ToputtheMARCS05modelswhichwillbeusedinmostof ening recipe. In our implementation of the Ali & Griem this work in context, we now compare them differentially theory we only consider the broadening of the 2p state, with some other models. To compare the models particu- sincethe2p–ndtransitiondominates(seediscussioninpa- larly as regards computed Balmer line profiles, we made perII).NotethatthisdiffersfromFuhrmannetal.(1993) comparisonsbyfindingtheTeff’softheMARCS05Balmer where both the 2p and np state are included. line profiles which best match profiles produced by other models using least-squares minimisation. This should indicate approximately the difference in T that would be eff found from the input model relative to MARCS05 for the givenline. STEHLE+BPOrecipe wasusedto produceall profiles and only the wings are considered. P. S.Barklem et al.: Detailed analysis of Balmer lines in cool dwarf stars 5 Table 1. Differences between other solar models, in Table 2. Differences between MARCS05 and MUNICH terms of effective temperatures from Balmer lines, of our models in terms of effective temperatures derived from MARCS05 models. Balmer lines. Model ∆Teff(Hα) ∆Teff(Hβ) Model ∆Teff(Hα) ∆Teff(Hβ) (K) (K) Teff/logg/[Fe/H] (K) (K) GS −81 −196 5200/4.2/0.0 +6 +3 (Grevesse & Sauval1999) 5600/4.2/0.0 +9 +11 HM −114 −192 6000/4.2/0.0 +21 +13 (Holweger & Mu¨ller 1974) 6400/4.2/0.0 +39 +1 MISS −132 −250 (AllendePrieto et al. 2001) 5200/4.6/0.0 +3 −2 KOVER −24 −214 5600/4.6/0.0 +6 +1 (Kurucz1993, see paper II) 6000/4.6/0.0 +16 −1 KNOVER +73 −10 6400/4.6/0.0 +29 −6 (Kurucz1993, see paper II) 5200/4.2/−1.0 −8 −15 5600/4.2/−1.0 −1 −11 6000/4.2/−1.0 +14 −7 6400/4.2/−1.0 +28 −6 First we compared with other common solar models, both semi-empirical and Kurucz (1993) theoretical flux- 5200/4.2/−2.0 +2 −8 constant models. Table 1 shows the difference in temper- 5600/4.2/−2.0 +2 +1 ature of the MARCS05 model needed to match the in- 6000/4.2/−2.0 +11 +1 put profile (i.e. Teff(best match) 5777) for the first two 6400/4.2/−2.0 +31 +2 − Balmerlines.Largedifferenceswithsemi-empiricalmodels are seen, which produce much weaker Balmer lines than the MARCS05 solar model. When comparing with solar observations the semi-empirical models produce lines too weakandtheMARCS05toostrongwhenSTEHLE+BPO is employed (see paper II and Sect. 5.2). The differences with Kurucz models in Table 1 stem from different MLT parameters and inclusion of overshoot in the KOVER model.Weshouldpointoutthatafterpublicationofpaper II itwasrealisedthatthe MARCSmodelusedinthatpa- per was in fact computed with α=1.25 and y =0.5, the same as the Kurucz models, not α=1.5 and y =3/(4π2) as stated, hence the reasonable agreement of the profiles with KNOVER. One should note that despite the roughagreementofHβ temperaturesfromKNOVERand MARCS05, the line shapes are markedly different, which was true of Hβ in all cases. The differences with the Munich group models (see Fuhrmann et al. 1997, hereafter MUNICH) are of Fig.2. Differences between T–τ structures of MARCS05 interest to understand the discrepancies between the and MUNICH models, for T = 5200, 5600, 6000, and eff Teff values from this work and Fuhrmann (1998, 2000). 6400 K (full, dotted, dashed and dot-dashed lines respec- Comparisons were made for a range of stellar temper- tively), all for logg = 4.2. The upper and lower panels atures and metallicities and the results are shown in show [Fe/H]=0.0 and [Fe/H]= 2.0 cases respectively. Table2.Alpha-enhancementisincludedforallmodelsbe- − low solar metallicity. A single variation of gravity is also shown. We see that the effect on Hα is generally larger andcanbeashighas40Kforhotterstars.Formostcases the correctionsarepositive,whichindicates ahotter tem- peraturewouldbe foundusingMARCS05thanMUNICH 4. Fitting Method models.Fig.2comparestemperaturestructuresfromboth modelgrids.Weseethatintheupperregionsoftheatmo- Quantifying the comparison between a synthetic profile spheres, MARCS05 models typically have slightly hotter and an observedprofile, allows us to automate the fitting andlesssteepT–τ structures.We note thatfor the cooler process through minimisation of the chosen statistic, and models,particularlyatlowmetallicity,thetemperatureis also provides a statistical measure of the goodness-of-fit. hotter throughout the entire atmosphere. This further allows the subjectivity associated with such 6 P. S.Barklem et al.: Detailed analysis of Balmer lines in cool dwarf stars fitting,usuallydonebyinspection,tobeshiftedaswewill One shouldalso be sure that windows areappropriate for demonstrate. We employ a reduced χ2 statistic, namely the spectral resolution of the observations, and this will be discussed in more detail shortly. N 2 χ2 = 1 fi−Fi (1) When comparing model spectra with observations in N M (cid:18) σ (cid:19) this way, we employ the same spectral windows, or a − Xi=1 i slightly reduced subset, for all stars. Rejection of some where N is the number of wavelength points, M is the windows is necessary in specific cases, such as the fast- numberoffreeparameters(hereone,namelyTeff),fiisthe rotating early F stars where metal lines are very broad synthetic residual flux, Fi the observed residual flux, and and encroach on these unblended regions, or if an atmo- σi =1/SNR. Forthe value ofσi =1/SNRwe estimateda spheric line by chance falls within a window in the refer- constantaveragevalue of the SNR (in the continuum) for enceframeofagivenstar,orifawindowhappenstofallin the given Balmer line observation. Despite the fact that thelinecore.Thismethodrestrictsthehumaninteraction theformalSNRofourobservationsvariesacrossthelineas inthefittingprocess,oncetheinitialwindowsaredecided, showninFig.1,theregionswithhighestSNRarethoseat to simply this choice of rejecting some windows. By fol- the end ofthe orderswhere continuumplacement maybe lowing this approach we attempt to avoid the situation less certain. Therefore these should not be given a higher where one must distinguish between noise fluctuations in weight,anda constantaveragevalue is moreappropriate. the spectra and possible weak metal lines, which can be For our case of Balmer lines, blending lines must be particularly difficult in metal-poor stars with lower SNR accounted for. Calculations including the metal lines are observations. Thus in essence, the subjectivity involved impractical as current spectral line databases have not in the usual fitting by visual inspection is shifted to this yet reached a level of completeness or accuracy where all choice of windows. observed lines in the solar spectrum in the Balmer line In cool stars, where self-broadening is important, one regions are well accounted for (see Fig. 9 in paper II). must be cognizant that both descriptions of this process Furthermore,suchanapproachwouldrequireaniterative used in this work employ the impact approximation. The procedure with more free parameters. extreme limit of validity for the self-broadening in the A much simpler approach is to attempt the identifi- STEHLE+BPO recipe is given by cation of spectral windows across the line which are ex- pectedto be freefromblendsinthe starsofinterest.This T(2+α)/4 0.145α ∆λ =3.654 10−7λ2 (2) wasdonebyusingthehighresolutionFTSsolarspectrum max × √σ104 (Kurucz et al. 1984) and the best of our spectra of hotter and cooler stars of around solar metallicity (namely where the result ∆λmax is in ˚A, with λ the central wave- Procyon and HR 8832) to identify windows which would lengthin˚A,T thetemperatureinK,σ104 thecross-section be free from obvious blending across most of our sample. in atomic units for v =104ms−1 andα the dimensionless High resolution spectra of a metal-rich K dwarf would be velocity parameter (not to be confused with the MLT α particularly useful in this respect, as this perhaps repre- parameter). Typical values for the sun are discussed in sents the hardestcase,but to our knowledge suchspectra paper II. The region of validity varies through the stellar arenotavailable.Wemaketheassumptionthatunblended atmosphere, but based on the assumption that the line spectralwindows do in fact existand that we see them in wings are predominantly formed in the region of the at- the abovementioned high quality spectra.This is reason- mosphere around optical depth unity, a reasonable esti- ableforthesunandProcyonatleastatHαandHβ based mate for a given stellar profile can be obtained by set- on examination of the spectra, but may for other stellar ting T =Teff.Spectralwindowsfallingoutsidethis region parameters or higher Balmer lines be questionable. Once of validity are rejected. For consistency, even when the these windows had been identified we considered other STEHLE+AG recipe is used we use the same limit as for desirable qualities of the windows as a set. Firstly the STEHLE+BPO, although a calculation based on the Ali Balmer line cores, which are formed high in the atmo- & Griem (1966) cross-section would give a slightly larger sphere innon-LTE areexcluded,only the pressurebroad- region of validity. ened wings are considered. We attempted to give approx- If errorsin the data are normally distributed, minimi- imatelyequalweighttoallpartsofthe line wings,despite sationoftheχ2 statisticprovidesthemaximumlikelihood anaturaltendencyfortheinner-wingstoshowlessblends estimate of the model parameters, in our case T . Our eff duetosaturationbythehydrogenline.Wenotethatifthe errors will in fact be Poissonian, approaching a normal windows are evenly distributed acrossthe line wings, and distribution at high SNR. A more robust (insensitive to a least-squares fitting is employed such as in this work, departures from normal distribution) statistic to better the procedure is quite similar to matching the equivalent account for outliers might be of benefit. width of the wings of the line, and the information on For each star we computed a grid of Hα and Hβ pro- the line shape is not biased towards any part of the line. filesforagivenmodelgridat10KintervalsinT around eff Furthermore,a reasonablenumber of windows need to be an initial guess based on literature values. The χ2 statis- employedsoastoavoidthepossibilitythatstatisticalfluc- ticwasalwaysfoundtobe wellbehavedvaryingsmoothly tuations in the observations affect the fitting procedure. with T , approximately parabolic in form with only a eff P. S.Barklem et al.: Detailed analysis of Balmer lines in cool dwarf stars 7 Fig.3. The fitting method for solar Hα and Hβ profiles, here showing the best fit found with MARCS05 and STEHLE+BPO which corresponds to the parameters in Table 5. The shaded regions show the windows used for determining the χ2 statistic. The full vertical lines show the estimated limit of validity of the impact approximation. Note the windows outside this region are rejected and thus no residual is plotted. Fig.4. As in Fig. 3 for ProcyonHα and Hβ profiles. 8 P. S.Barklem et al.: Detailed analysis of Balmer lines in cool dwarf stars single minimum (no other local minima). We therefore The results for differentresolutions using the eventual searched this grid for the best fitting profile, and finally set of spectral windows are incorporated in Table 3. A performed a polynomial fit to the χ2 statistic values near brief survey of different resolutions indicates that a spec- the minimum and determined the predicted minimum in tral resolution of at least R 50000 is more appropriate ≈ ordertoimprovetheresolutionofthefittingto1K,while for gaining both T and information on the line profile eff saving computing time, a procedure which was verified shape from Hβ for solar-type stars. by direct calculations. Figs. 3 and 4 show the best fits for the sun and Procyon from this method, including our 5. Results final chosen spectral windows. Despite obtaining observations of Hγ of reasonable quality for some stars, this 5.1. Error Estimates line is not considered in this work due to the problem of choosing suitable windows and the limited validity of the Due to the complicated interplay of several mechanisms self-broadening calculations. inthe formationofhydrogenBalmerlines,estimating the sources of error and their behaviour with stellar parame- Inusingsuchamethodthereisanaturaltendencythat tersissimilarlycomplicated.Fuhrmannetal.(1993,1994) a chosen window may not be entirely free from blends, discussmanyoftheseeffects.Here,wehavetakenaquan- as wings of nearby lines may encroach on the apparent titative approach to the errors. By making a reasonable window or unseen weak blends exist. One must therefore guess of the error in a number of possible sources, we expect that the fitting method may very slightly over- havesurveyedhowtheseerrorstranslateintoerrorsinT estimate T in a systematic manner. This is likely of all eff eff throughχ2comparisonsofmodelprofiles.Forconsistency, such fitting, automated or not. the samewindows as willbe usedfor the observationsare used, and test calculations employing the whole line profile indicate this does not introduce much error, which is 4.1. Influence of spectral resolution also a confirmation that our windows do not introduce significant bias to certain parts of the line profile. Our It is important to understand the effect of spectral reso- error estimates are summarised in Table 4. Test calcula- lution on our results. Sufficient resolution is needed such tions showed that these errors do vary with MLT param- thatasuitablenumberofwindowsbetweenblendinglines eters. For example, for the T =6000 K and [Fe/H]= 2 are available. Once again, this is true not just of our fit- eff − case,Table 4 indicates Hα to be quite sensitive to gravity ting method but any approach. There must exist a limit- while Hβ is rather insensitive. Test calculations for this ingresolutionforwhichallwindowswillbeaffectedbythe case with MARCSa models in place of MARCS05a find instrumental broadening of metal lines, but above which the sensitivity to gravityof Hα and Hβ to be roughly the thereisatleastoneunaffectedwindow,andthislimitdif- same,a 0.1dex changecorrespondingto about 34K for fers with star, and spectral region. For our purposes, a ± bothlines.Therefore,ourresultsaresomewhatdependent single small spectral window is insufficient, since as dis- on our choice of MLT parameters. cussedabove,atthe minimum,we wouldliketo havesev- Differences between the often used Vidal et al. (1973, eralspectralwindowsevenlydistributedacrosstheBalmer hereafter VCS) and Stehlé (1994) calculations perhaps line. give an indication of the magnitude of the error in the Ideally, windows should be chosen such that for the Stark broadening calculations, and these differences are test cases the derived effective temperature does not dif- reportedinTable4.Testcalculationsintroducinga5%er- fer between the high resolutionspectra andthe resultob- rortotheStarkbroadeningfoundresultsofsimilarmagni- tainedwiththesamespectradegradedtotheresolutionof tudes.VCScalculationsalwaysresultedinastrongerpro- our observations R 30000,or at least introduces only a fileandthereforeacoolertemperature.Stehlé’s(1994)cal- ≈ small error (say <10 K). This requirement was quite eas- culationsforprotonperturbersareknowntohavereason- ily fulfilled for Hα in the sun. However, for Hβ this was able agreement with experiment, better than VCS where problematic. It proved difficult to find a set of windows iondynamicsareneglected.However,oneshouldconsider which fulfilled both our criteria of a T in reasonable that while protons dominate in hot stars, in cooler stars eff agreement with the high resolution result, and a reason- the perturbing ions progressively become more a mix of ablecoverageofthewholelineprofileshape.Thewindows protonsandheavierionisednuclei.Self-broadeningforhy- eventually used for Hβ were chosen as a compromise be- drogen is less well studied than Stark broadening, no ex- tweenboth criteria,whichresultedina temperaturefrom perimental results currently existing. Based on error esti- the degradedspectra of 17 K hotter than the result using matesforhydrogenbroadeningofmetallinesusingsimilar high resolution spectra in both stars. This error is con- theoreticaltechniquesweestimatetheerrorataround5% sidered acceptable, in view of gaining some information (e.g. Barklem & O’Mara 2001), though we note that the on the fit of the line shape, though we expect this error interactionisfundamentallydifferentduetotheresonance to increase in cooler stars. Because of this, and quite sig- interaction.Asthebroadeningbyheliumisonlyestimated nificant extra blending, Hβ will not be employed for the here we adopta ratherlargeerrorbar of50%.Typicaler- coolest stars in our sample. rors in the gravity and metallicity are estimated at 0.1 P. S.Barklem et al.: Detailed analysis of Balmer lines in cool dwarf stars 9 Table 3. Derived T values for the sun and Procyon using different or degraded observations. Results are for eff MARCS05s models. Note all R and SNR values are approximate. Observation R SNR SNR Teff Hα Teff Hβ Hα Hβ (K) (K) Sun Kitt Peak FTS atlas >3×105 3000 3000 5733 5723 — 50000 3000 3000 5735 5729 — 30000 3000 3000 5739 5740 INT/MUSICOS (co-added) 30000 375 240 5743 5748 — (single exposure) 30000 170 110 5721 5711 Procyon McDonald/2dcoudé 2×105 — 1000 — 6474 — 30000 — 1000 — 6481 INT/MUSICOS (co-added) 30000 250 160 6538 6487 — (single exposure) 30000 120 80 6498 6532 dex. Possible interdependency of errors makes their cor- Hα andHβ respectively,conservativevalues basedon the rect combination non-trivial, thus we chose a somewhat tests we performed in Sect. 4.1 (see Table 3). Totals are ad hoc procedure. The errors discussed to this point are showninTable4forthecasewherethereisnoerrordueto grouped into two categories, broadening and stellar pa- convectionor abundancescatter,andthe best caseobser- rameters,andthesetwoerrorsarecombinedinquadrature vations. Our final errors will account for these additional supposing they are independent and random. The totals factors on a case by case basis. for each category are found by simply summing. These fixederrorsaresub-totalledinTable4.Toestimateerrors 5.2. The sun for a given line we interpolate in these fixed values which are then combined with the errorsourcesdiscussed below Thesunisthemostimportanttestcasesincehighquality which vary from case to case. observations are available and the stellar parameters are As discussed in Sect. 2 observational errors are esti- wellknown.We follow Fuhrmannet al.’s (1993) approach mated at approximately 1%, and so the estimated errors and consider the MLT parameters as free and attempt to in Table 4 corresponding to a 0.5% shift in the contin- calibratethematleastapproximatelyusingthesolarcase. uum placement represent perhaps a best case which we Wesayapproximatelysincetheappropriateconvectiveef- assume for our best spectra. In computing errors we in- ficiency for representing Balmer lines must be expected terpolate in these values, and scale the result depending to vary across the HR diagram (e.g. Ludwig et al. 1999). on the quality (SNR) of the observations employed. We BothFuhrmannetal.(1993)usingsolarobservations,and assume 1.5% for our lowest quality spectra. In Sect. 5.2 Steffen & Ludwig (1999) using hydrodynamical simula- we will see for our adopted model grid there is a discrep- tions ofthe sun, havefound that a low value ofα 0.5 is ≈ ancy of order 50–60 K in the solar T value with the needed to represent Balmer lines, both with y set at 0.5. eff known value. Assuming negligible error in observations, We note that Steffen & Ludwig (1999) have shown using gravity and metallicity, approximately half of this can be 2Dhydrodynamicalmodelsthat,atleastforflux-constant explainedby ourestimates ofthe errordue to broadening MLTmodels,“differentmean[temperature]stratifications theorygiveninTable4.Theremainderofthediscrepancy are needed to represent different spectroscopic properties canmostlikelybe attributedto the models.Thus,forthe of an inhomogeneous stellar photosphere” and therefore solar case we estimate a conservative model error of or- weshouldnotexpecttheconvectiveefficiencyfromobser- der 40 K. It must be anticipated that this error will vary vationsofHαandHβ whichwillbeinvestigatedheretobe withstellarparameters,butitisdifficulttoestimatehow. thesameasthatrequiredtorepresentotherobservational The error due to possible variationin MLT parameters is quantities, even other Balmer lines. estimated from the difference between T values derived This work differs from Fuhrmann et al.’s (1993) in eff with MARCS and MARCS05 models, which is added to that, in addition to the mixing-length parameter α, the this 40 K. This above procedure amounts to using the convective structure parameter y is allowed to vary, and sun to approximately calibrate our error bars. The effect the STEHLE+BPO broadening recipe is employed (see of deviation from the solar or alpha-enhanced abundance Fuhrmann et al. 1993 for details of their broadening patterns (i.e. abundance scatter) is estimated by half the recipe).We haveinvestigatedthe solarcase withthe FTS difference between the results obtained from MARCS05s solar atlas (Kurucz et al. 1984) using a number of mod- and MARCS05a model grids. We also include a fitting els where the MLT parameters α and y have been var- error, including resolution effects, of 20 K and 40 K for ied within reasonable limits. Two approaches are taken. 10 P. S.Barklem et al.: Detailed analysis of Balmer lines in cool dwarf stars Table 4. Estimated errors in T (in K) corresponding to introduced errors in the input physics, stellar parameters eff or observations for MARCS05 models, with MARCS05a models used for [Fe/H]= 1 and 2 cases. Results are given − − for various T at both solar and low metallicity (quoted as T /[Fe/H]). All cases use logg = 4.2, except the sun eff eff where logg = 4.44. Errors are introduced (amounts discussed in the text) and the corresponding changes in derived T observed. Signs show the direction of the change, for the corresponding change in the input physics or stellar eff parameter.Errorsareusually symmetric,but if not the largestis quoted.We emphasise these errorsvary with chosen MLT parameters. For a given stellar spectrum the continuum error is interpolated from the listed values and scaled depending on the SNR, and the values of ∆T (convection) and ∆T (abundance scatter) are computed case by case eff eff and given in Table 5. Introduced Sun 5000/0.0 6000/0.0 7000/0.0 5000/−1.0 6000/−1.0 7000/−1.0 5000/−2.0 6000/−2.0 7000/−2.0 error Hα/Hβ Hα/Hβ Hα/Hβ Hα/Hβ Hα/Hβ Hα/Hβ Hα/Hβ Hα/Hβ Hα/Hβ Hα/Hβ Stark-broadening VCS −5/−15 −3/−10 −7/−18 −19/−28 −1/−7 −7/−17 −18/−26 0/−3 −8/−17 −17/−25 Self-broadening ±5% ∓14/∓7 ∓10/∓6 ∓13/∓6 ∓8/∓3 ∓15/∓13 ∓23/∓11 ∓11/∓3 ∓23/∓20 ∓38/∓15 ∓12/∓4 He-broadening ±50% ∓6/∓4 ∓5/∓3 ∓6/∓3 ∓4/∓2 ∓6/∓7 ∓10/∓6 ∓5/∓2 ∓10/∓11 ∓17/∓8 ∓6/∓2 logg ±0.1dex — ∓6/∓6 ∓9/∓3 ∓3/±8 ∓13/∓12 ∓25/∓7 ∓3/±11 ∓33/∓17 ∓52/∓11 ∓5/±11 [Fe/H] ±0.1dex — ±19/−2 ±7/∓17 ∓25/∓31 ±35/±20 ±17/∓7 ∓14/∓15 ±16/∓7 ±6/∓4 ±6/±6 Sub-total ±25/±26 ±31/±21 ±31/±34 ±42/±51 ±53/±42 ±58/±37 ±38/±40 ±59/±48 ±86/±43 ±37/±35 Continuum ±0.5% ±38/±29 ±36/±23 ±29/±19 ±34/±21 ±33/±25 ±38/±23 ±35/±21 ±38/±25 ±56/±26 ±36/±21 Model ←−40+|∆Teff(convection)|−→ Abundancescatter ←−|∆Teff(abundancescatter)|−→ Fitting&resolution ←−20/40−→ Total(bestcase) ±69/±72 ±65/±65 ±61/±69 ±70/±79 ±76/±75 ±82/±71 ±69/±73 ±83/±79 ±111/±76 ±68/±70 Firstly, we derive T considering it a free parameter for since an increase in α can be compensated by a decrease eff a given MLT α, y set and record the χ2 value. Secondly, in y, further strengthening Steffen & Ludwig’s (1999) as- wesettheeffectivetemperatureat5777Kandsurveythe sertion that Balmer lines should not be seen as evidence χ2 valueacrossthedifferentMLTparameters.Theresults for low efficiency of solar-type convection. aredisplayedinFig.5,whereallχ2 values arenormalised On the assumption that the FTS observations em- so the best possible fit for a given line has χ2 = 1. Steps ployed are of high accuracy in comparison with the in χ2 of 0.5, from 1 to 20, are plotted (except in one case model spectra, one interprets this as a shortcoming of whereweusesteps of1forclarity).Differencesbelowthis either the model atmospheres or the broadening the- level correspond to only small differences in the line pro- ory, though most likely a combination of both. In pa- files and are probably meaningless. The contour at 1.1, per II using the same broadening theory we found that where it exists, is also shown to illustrate the plateau at the semi-empirical model of Holweger & Mu¨ller (1974), χ2 =1. which matches limb-darkening better than MARCS mod- It is seen that no single parameter set shows itself els (Blackwell et al. 1995), in fact gives lines too shal- to be especially preferred on the basis of agreement with low.Thereforeitiscertainlyfeasiblethatthisdiscrepancy line shape and T . A simple approximate picture is that couldbeduetomodels.Basedontheseresultswedecided eff agreement with the line shape indicates correct temper- to adopt the same parameters as Fuhrmann et al. (1993) ature structure in the line formation region, and agree- namely α=0.5 and y=0.5 as our default values since this ment with T indicates correct temperature at the line parameter combination reproduces the line shapes and eff formation region, although the reality is somewhat more therefore the temperature stratification in the line form- complex. The line profile shapes for both Hα and Hβ ing regionas wellas anyotherchoice.As we seein Fig.3, are best fit anywhere on the plateau at low α and low the fit to Hβ is not perfect showing differences of order y with a T some 50 K lower than the accepted value. If 1%withatrendacrossthe line,againindicatingsomede- eff we force T = 5777 K the fits to line profile shapes for ficiency in either models or line broadening. Although a eff bothHα andHβ arequite poor,typically being too deep. lowervalueofy isperhapsmorejustifiedphysicallyonthe Furthermore, a single α,y combination will not give the basis of the diffusion approximation(Henyey et al. 1965), correct T for both lines, seen by the fact that the loci adopting these parameters will make comparisons with eff of α,y values giving the correct solution (thick lines) do Fuhrmann(1998,2000)morestraightforward.Inanycase, not overlap. More generally one can see from Fig. 5 that all calculations will be repeated with α=1.5 and y=0.076 Balmer lines do not well constrain the MLT parameters, as a test of sensitivity to this choice. The error in T in- eff