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Model atmospheres of magnetic chemically peculiar stars. A remarkable strong-field Bp SiCrFe star HD 137509

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Model atmospheres of magnetic chemically peculiar stars. A remarkable strong-field Bp SiCrFe star HD 137509
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    a  r   X   i  v  :   0   8   0   6 .   1   2   9   6  v   1   [  a  s   t  r  o  -  p   h   ]   7   J  u  n   2   0   0   8 Astronomy & Astrophysics  manuscript no. shulyak  c  ESO 2008June 7, 2008 Model atmospheres of magnetic chemically peculiar stars A remarkable strong-field Bp SiCrFe star HD137509 D. Shulyak  1 , O. Kochukhov 2 , and S. Khan 3 , 4 1 Institute of Astronomy, Vienna University, Turkenschanzstrasse 17, 1180 Vienna, Austria 2 Department of Astronomy and Space Physics, Uppsala University, Box 515, 751 20, Uppsala, Sweden 3 Physics and Astronomy Department, University of Western Ontario, London, ON, N6A 3K7, Canada 4 Institute for Computational Astrophysics, Saint Mary’s University, 923 Robie Street, Halifax, B3H 3C3, Nova Scotia, CanadaReceived  /   Accepted ABSTRACT Context.  In the last few years we have developed stellar model atmospheres which included e ff  ects of anomalous abundances andstrong magnetic field. In particular, the full treatment of anomalous Zeeman splitting and polarized radiative transfer were introducedin the model atmosphere calculations for the first time. The influence of the magnetic field on the model atmosphere structure andvarious observables were investigated for stars of di ff  erent fundamental parameters and metallicities. However, these studies werepurely theoretical and did not attempt to model real objects. Aims.  In this investigation we present results of modelling the atmosphere of one of the most extreme magnetic chemically peculiarstars, HD137509. This Bp SiCrFe star has the mean surface magnetic field modulus of about 29kG. Such a strong field, as well asclearly observed abundance peculiarities, make this star an interesting target for application of our newly developed model atmospherecode. Methods.  We use therecent version of theline-by-line opacity sampling stellar model atmosphere code LL models ,which incorporatesthe full treatment of Zeeman splitting of spectral lines, detailed polarized radiative transfer and arbitrary abundances. We comparemodel predictions with photometric and spectroscopic observations of HD137509, aiming to reach a self-consistency between theabundance pattern derived from high-resolution spectra and abundances used for model atmosphere calculation. Results.  Based on magnetic model atmospheres, we redetermined abundances and fundamental parameters of HD137509 usingspectroscopic and photometric observations. Thisallowedustoobtain abetter agreement betweenobserved andtheoretical parameterscompared to non-magnetic models with individual or scaled-solar abundances. Conclusions.  We confirm that the magnetic field e ff  ects lead to noticeable changes in the model atmosphere structure and should betaken into account in the stellar parameter determination and abundance analysis. Key words.  stars: chemically peculiar – stars: magnetic fields – stars: atmospheres – stars: individual: HD137509 1. Introduction The atmospheric structure of magnetic chemically peculiar (CP)stars deviates from that of normal stars with similar funda-mental parameters due to unusual chemistry, abundance inho-mogeneities and the presence of strong magnetic field. Thesee ff  ects are not considered in the standard model atmospherecalculations, possibly leading to errors in the stellar parame-ter determination and abundance analysis. To circumvent thislong-standing problem of stellar astrophysics, we have devel-oped a new line-by-line opacity sampling model atmospherecode LL models  (Shulyak et al. 2004). Using this tool in the se-ries of recent papers, we investigated in detail the e ff  ects of anomalous Zeeman splitting (Kochukhov et al. 2005), polarizedradiativetransfer(Khan & Shulyak2006a) andinclinationof themagnetic field vector (Khan & Shulyak 2006b) on the modelstructure, energy distribution, hydrogen line profiles, photomet-ric colors and the magnitude of bolometric corrections for a gridof model atmospheres with di ff  erent e ff  ective temperatures andmetallicities. For the first time we were able to obtain new re-sults applying direct and self-consistent modeling of all e ff  ects Send o  ff   print requests to : D. Shulyak,e-mail:  denis@jan.astro.univie.ac.at mentioned above and to answer the question how does the mag-netic field act at di ff  erent temperatures and what one could ex-pect if the magnetic field is ignored in calculations of model at-mosphereof magneticCP stars. It was shownthat the strengthof the magnetic field is the key characteristic controlling the mag-nitude of the magnetic field e ff  ects and the polarized radiativetransfer should be taken into account. In contrast, the orienta-tion of the magnetic field vector does not have much influenceon any of the observed stellar characteristics and, thus, can besafety ignored in the analysis routines.So far our models with the magnetic field e ff  ects includedhave been developed and applied only in the context of purelytheoretical studies. Here we make the first attempt to model theatmosphereofa real star. In this work we use the LL models  stel-lar model atmosphere code to investigate the atmospheric struc-tureofthestar,takingintoaccountindividualchemicalcomposi-tion, anomalous Zeeman splitting and polarized radiative trans-fer.HD137509 (HIP76011, NN Aps) is a B9p SiCrFe chemi-cally peculiar star with a strong reversing longitudinal field andvariable lines of Si ii  and iron-peak elements (Mathys 1991;Mathys & Lanz 1997). Kochukhov (2006) (hereafter Paper I) has detected resolved Zeeman split lines in the spectrum of   2 D. Shulyak et al.: Model atmospheres of magnetic chemically peculiar stars HD137509, showing that this star is characterized by a non-dipolar magnetic field geometry with a mean surface fieldstrengthofabout29kG.Thisis thesecond-largestmagneticfieldever found in a CP star (the first place is occupied by the well-knownBabcock’s star, Preston (1969)). The atmosphericparam-eters, T  e ff   =  12750 ± 500 K   and log g  =  3 . 8 ± 0 . 1, were derivedin Paper I using theoretical fit to the observed H  β  and H γ   lineprofiles based on the ATLAS9 (Kurucz 1993a) model with en-hanced metallicity, [  M  /  H  ]  = + 1 . 0. The appearance of such astrongmagneticfield andthe presenceof outstandingabundanceanomaliesinferredin PaperI allow us to use HD137509as a testgroundfortheapplicationofthenewgenerationmagneticmodelatmospheres.In the next section we briefly describe the techniques em-ployed to construct magnetic model atmospheres. Results of thecalculations for HD137509 are presented in Sect. 3. Main con-clusions of our study are summarized in Sect. 4. 2. Model atmosphere calculations To calculate magnetic model atmospheres we used the cur-rent version of the LL models  code, srcinally developed byShulyak et al. (2004). The LL models  is a LTE, 1-D, plane-parallel, hydrostaticstellar model atmosphere code for early andintermediate-typestars. The direct line-by-linecalculationof thebound-bound opacities implemented in this code allows one toaccount for arbitrary individual and stratified stellar abundancepatterns and include various e ff  ects caused by the magneticfield. The VALD database (Piskunov et al. 1995; Kupka et al. 1999) was used as a source of atomic line parameters. The ex-tracted lines were subjected to a preselection procedure insideLL models , with the default criterion  α line /α cont  >  1% in at leastone atmospheric layer allowing to select only those lines thatsignificantly contribute to the opacity (here  α line  and  α cont  arethe line center and continuumopacity coe ffi cients, respectively).The numerical experiments showed that this criterion is su ffi -cientforaccuraterepresentationofthe energydistributionandT-P structure of the models. Decreasing this value leads to signifi-cant increase of the numberof selected lines but withoutproduc-ing noticeable changes in overall blanketing (see Shulyak et al.2004). The preselection procedure was performed twice permodelatmospherecalculation:atthefirstiterationandattheiter-ation whenthe temperaturecorrectionat eachmodel atmospherelayer is less than few tens of K. The remaining model iterationsare performedwith the later line list, thus ensuring a consistencyof the preselected lines and the model structure obtained. Thespectrum synthesis for the magnetic model atmosphere calcula-tions was carried out in the range between 100 and 40000Å,with a constant wavelength step of 0.1Å.Generally,duetothevariationofmagneticfieldvectoracrossthe visible stellar surface, one has to compute a number of localmodel atmospheres for each appropriately chosen surface gridelement using individual values of the strength and inclinationof the magnetic field. Then, the total flux coming from the starshould be obtained by integration of the radiation field intensitycorresponding to individual surface zones. However, as it wasshown by Khan & Shulyak (2006b), the inclination of the mag- netic field vector does not significantly influence the structure of magnetic models and the resulting energy distribution, i.e. theanisotropy e ff  ects can be neglected in the magnetic model at-mosphere calculations. Consequently, we assume the magneticfield vector to be perpendicular to the atmosphere normal and,according to the results of Paper I, adopt the field strength of  Table 1.  Element abundances of HD137509 compared to thesolar values (Asplund et al. 2005). The second and the thirdcolumnsgive,respectively,abundancesderivedusing thescaled-solar abundance model atmosphere (Paper I) and using modelwith  T  e ff   =  13750K, log g  =  4 . 2 with magnetic field included(see Sec. 3.4). Abundances are given in the logarithmic scalelog(  N  el /  N  total ). Element t12750g3.8 (Paper I) t13750g42 SunHe  <  − 3 . 50  <  − 3 . 50  − 1 . 10Si  − 3 . 73  − 3 . 58  − 4 . 53Fe  − 3 . 19  − 3 . 00  − 4 . 59Cr  − 4 . 20  − 3 . 90  − 6 . 40Ti  − 4 . 54  − 4 . 20  − 7 . 14Ca  − 7 . 93  − 7 . 50  − 5 . 73Mg  − 5 . 71  − 5 . 50  − 4 . 51 29 kG. The approach for calculating magnetic models used hereis equivalent to the one described in Sect. 2 in Khan & Shulyak(2006a).For all models presented in this study we assume a force-free configuration of the surface magnetic field. This agreeswith the results of multipolar modeling of magnetic topology(Kochukhov 2006) and implies that possible modification of thehydrostatic equilibrium by the Lorentz force (see Shulyak et al.(2007) and references therein) is absent in HD137509. Thus, alldi ff  erences in the pressure structure between magnetic and non-magnetic models that we obtain are caused only by additionalopacity in the Zeeman components.The necessity to perform polarized radiative transfer calcu-lations over a wide wavelength range makes the models withmagnetic field very computationally expensive. To reduce com-putational costs, we used the following approach. Fixing funda-mental parameters and abundance pattern we calculated a non-magnetic model. After this model is converged, a model withmagnetic field is iterated using the temperature-pressure struc-ture of the initial non-magnetic model as a first approximation.The abundances of seven chemical elements for HD137509were derivedin Paper I and are listed in Table 1 second column).Due to the abnormal weakness of the He lines, the value of theHe abundance should be considered as an upper limit only. Onecan note the strong overabundanceof Si, Fe, Cr and Ti, whereasCa and Mg are underabundantby more than 1 dex relative to thesolar chemical composition. We assume solar abundances fromAsplund et al. (2005) for all other elements. Using available ob- servations it is not possible to derive abundances of elementsother than those listed in Table 1 with good accuracy due tocomplex line shapes and strong blending by iron-peakelements.Thereare no usablelines of intrinsically abundantelements suchas C, N and Ne, while the abundance of O that could in princi-ple be determined from the IR triplet (7772–7775 Å) is unreli-able due to large NLTE e ff  ects and instrumental artifact in theUVES spectrum of HD137509 in this region. According to ourrecentinvestigation(Khan & Shulyak2007),C, N,andObelongto the groupof elementsthat producesonlya small changein themodel structure of A and B stars even if their abundances are re-duced by 1 dex relative to the solar values. Finally, to ensurethe consistency between the model atmosphere and the abun-dancepatternofthestar,weconstructedstellar atmospheremod-els with these individual abundances including magnetic field.Then we try to assess relative importance of this sophisticatedapproach.  D. Shulyak et al.: Model atmospheres of magnetic chemically peculiar stars 3 -6 -5 -4 -3 -2 -1 0 1 log τ  5000 050010001500    ∆    T ,   K ind - scaled(ind+mag) - scaled -6 -5 -4 -3 -2 -1 0 1 log τ  5000 -0.25-0.2-0.15-0.1-0.050    ∆    l  o  g   P   g  a  s  ,    d  y  n   /  c  m ind - scaled(ind+mag) - scaled Fig.1.  Di ff  erence in temperature ( upper panel ) and gas pressure( bottom panel ) of the models calculated with individual abun-dances (solid line) and with individual abundances + magneticfield (dashed line) with respect to the reference model atmo-sphere computed with scaled-solar abundances ([  M  /  H  ]  = + 1).For all models  T  e ff   =  12750K and log g  =  3 . 8 is adopted. 3. Results 3.1. Model structure  The e ff  ect of the peculiar abundance pattern and magnetic fieldon the temperature-pressure structure of the model atmosphereof HD137509 is shown in Fig. 1. The model with individualabundancesexhibitsa decreaseoftemperaturein thesurfacelay-ers, while the layers close to the photosphere (log τ 5000  ≈  0) areheated compared to the reference scaled-solar abundance model([  M  /  H  ]  = + 1 . 0). The e ff  ect of magnetic field is, therefore, con-siderably di ff  erent compared to a change in metal abundance.The inclusion of magnetic field leads to heating of the surfacelayers due to additionalopacity in the Zeemancomponents.ThisoccursbecauseahighlinedensityinthedominantUVpartofthestellarspectralenergydistributionmakesitpossiblefortheback-warming e ff  ect to occur even in surface layers which becomeopaque at these wavelengths (Kochukhov et al. 2005). The tem-perature distribution of the magnetic model in the surface layerstends to be close to that of the scaled-solar model. At the sametime, magnetic field appears to influence the temperature struc-ture more e ffi ciently in deep atmospheric layers compared to thenon-magnetic model with individual abundances.Both magnetic and non-magneticmodels show a decrease of the gas pressure throughoutthe atmosphere compared to the ref-erencemodel.However,themodelwiththemagneticfieldshowsa steeper depth dependence of the pressure di ff  erence relative tothe model where only individual abundances are taken into ac-count. In our hydrostaticmodels the decrease of the gas pressureis caused by the respective increase of the radiative pressure due 4830 4840 4850 4860 4870 4880 48900.20.40.60.81   r  e  a   t  v  e  u  x UVES observationst12750g38 [M/H]=+1t12750g38 individualt12750g38 individual+magnetic   4830 4840 4850 4860 4870 4880 4890 λ, Α -20246    ∆   Ι    λ    /   Ι    λ  ,   % Fig.2.  Observed and calculated H  β  line profiles.  Upper panel :thick line – UVES observations (Paper I), thin line – the log g  = 3 . 8 model with solar-scaled abundances [  M  /  H  ]  = + 1, dashedline – the model with individual abundances, dotted line –the model with both individual abundances and the magneticfield.  Bottom panel : di ff  erence (in per sent) between modelwith only individual abundances (solid line) and with individualabundance + magnetic field (dotted line) relative to the scaled-solar abundance model ( T  e ff   =  12750K, log g  =  3 . 8 for allmodels).to enhanced opacity for models with individual abundances andmagnetic field. Note that the model with the magnetic field in-cluded exhibits a noticeable increase of the radiative pressure inthe outer atmospheric layers comparedto other models. This be-haviour of the radiative pressure and corresponding changes inthe overall temperature-pressurestructure of the magnetic atmo-sphere are relevant for modern studies of the radiatively-drivenchemical di ff  usion in the atmospheres of magnetic stars (e.g.,Alecian & Stift 2007).At this point it is di ffi cult to predict the total e ff  ect of themagnetic field on the spectral characteristics of the star. In twodi ff  erent atmospheric regions (surface and photospheric layers),the magnetic field influences the model structure in a di ff  er-ent manner, so that the properties of magnetic atmosphere of HD137509 are not equivalent to a usual non-magnetic modelwith a di ff  erent set of fundamental parameters. One can expectthat spectral features which are selectively sensitive to eithertemperature or pressure may show di ff  erent behavior. In the fol-lowing section we examine the overall e ff  ect of the individualabundancesandmagnetic field on the hydrogenBalmer line pro-files and metallic line spectra.  4 D. Shulyak et al.: Model atmospheres of magnetic chemically peculiar stars 4310 4320 4330 4340 4350 4360 43700.20.40.60.81   r  e  a   t  v  e  u  x UVES observationst12750g38 [M/H]=+1t12750g38 individualt12750g38 individual+magnetic  γ  4310 4320 4330 4340 4350 4360 4370 λ, Α -20246    ∆   Ι    λ    /   Ι    λ  , Fig.3.  Same as in Fig. 2 but for the H γ   line. 3.2. Hydrogen line profiles and metallic line spectra  The introductionof peculiar abundancestaken fromPaper I (seeTable 1) to the model atmosphere calculations for HD137509,together with the magnetic field, leads to noticeable changes inthe hydrogen line profiles. In Figs. 2 and 3, we compare theo- retical profiles for di ff  erent model atmospheres with those ob-served for HD137509. Synthetic profiles were calculated usingthe S ynthmag  program (Kochukhov 2007). This code incorpo-rates recent improvements in the treatment of the hydrogen lineopacity (Barklem et al. 2000) and takes into account magneticsplitting of hydrogen lines. However, possible modification of the Stark broadening by magnetic field (Mathys et al. 2000) isneglected. The observed profiles of hydrogen lines are extractedfrom the UVES spectrum of HD137509 described in Paper I.One can see that in order to retain acceptable agreement be-tween observations and theory, it is necessary to increase thelog g  value from 3 . 8 derived in Paper I to 4 . 0. Note, that mostof the changes in the hydrogen line profiles are due to anoma-lous abundances adopted for the star (mainly due to extreme Heunderabundance).Nevertheless, the changes in the temperature-pressure structure of the atmosphere produced by the magneticfield are also important and has to be incorporated to the modelin order to retrieve an accurate estimate of the gravitational ac-celeration.The metallic line spectra are not particularly sensitive to thechanges in the atmospheric model structure associated with theinclusion of the magnetic field. Fig. 4 shows the observed andsynthetic profiles of some prominent spectral lines of silicon.All theoretical spectra were calculated using the S ynthmag  codewith the same abundances and the homogeneous surface mag-netic field distribution, characterized by    B   =  29 kG, but us-ing two di ff  erent model atmospheres: one model is only withindividualabundancesand another one includes both anomalous   F e   3   4  1  2  2 .  0  2  5   P  F e   2   4  1  2  2 .  6  6  8   P  F e   3   4  1  2  2 .  7  8  0   P  F e   2   4  1  2  4 .  7  8  7   P  F e   2   4  1  2  6 .  3  0  0   P  C r   2   4  1  2  7 .  0  5  7   P  S  i   2   4  1  2  8 .  0  5  4   P  F e   2   4  1  2  8 .  7  4  8   P  T  i   2   4  1  2  9 .  1  6  1   P  S  i   2   4  1  3  0 .  8  7  2   P  S  i   2   4  1  3  0 .  8  9  4   P  F e   1   4  1  3  2 .  0  5  8   P  C r   2   4  1  3  2 .  4  1  9   P  S   2   4  1  3  2 .  9  8  4   P  F e   2   6  3  4  5 .  9  3  1   P  S  i   2   6  3  4  7 .  1  0  9   P  F e   2   6  3  4  8 .  0  9  5   P  F e   2   6  3  4  9 .  6  0  1   P  C r   2   6  3  5  2 .  0  6  5   P  F e   2   6  3  5  7 .  0  9  3   P  F e   2   6  3  5  7 .  1  6  2   P  F e   2   6  3  6  2 .  4  7  4   P  F e   2   6  3  6  2 .  8  4  2   P  F e   2   6  3  6  4 .  1  3  8   P  F e   2   6  3  6  7 .  4  1  3   P  F e   2   6  3  6  7 .  8  2  9   P  F e   2   6  3  6  9 .  0  2  0   P  F e   2   6  3  6  9 .  4  6  2   P  S  i   2   6  3  7  1 .  3  7  1   P  F e   2   6  3  7  1 .  7  2  1   P  F e   2   6  3  7  2 .  4  1  9   P  F e   2   6  3  7  5 .  7  9  2   P Fig.4.  Influence of magnetic model structure on the profiles of selected spectral lines for  T  e ff   =  12750K, log g  =  4 . 0 models.Double line – UVES observations, thin line – spectra calculatedusing the non-magnetic atmospheric model, dashed line – withthe magnetic field included.abundances and the magnetic field. For both models the funda-mental parameters are  T  e ff   =  12750K, log g  =  4 . 0. The calcu-lated spectra were convolved with  v sin i  =  20 kms − 1 rotationalbroadeningto allow comparisonwith observationaldata. The ef-fects of the magnetic field in the model atmosphere distort dif-ferent lines in di ff  erent ways. Some of the spectral features con-sidered here are not sensitive to the magnetic model atmospheree ff  ects at all (forexample,Si ii  4130.872Å andSi ii  4130.894Å),while other lines show a noticeable discrepancy between thespectra computed for magnetic and non-magnetic atmospheres(e.g., Ti ii  4129.161Å). The average di ff  erence between the lineprofiles obtained from the magnetic and non-magnetic modelswas foundto be about few per sent forthe spectra already broad-ened byrotation.Due to a verylarge magneticbroadening(com-parable to the rotational Doppler e ff  ect), the e ff  ect on uncon-volved spectra is roughly of the same magnitude for all spec-tral lines considered. These results are in a good agreement withour previous investigation (Kochukhov et al. 2005). However,the present results are more accurate because polarized radiativetransfer was taken into account for both the model atmosphereand the spectrum synthesis calculations. 3.3. Energy distribution  Generally, for self-consistent modelling of stellar atmospheres,the following observables must be reproduced simultaneously:hydrogen line profiles, energy distribution and metallic linespectra. Among these, the flux distribution is especially sensi-tive to the overall energy balance in the stellar atmosphere overa wide range of optical depths. So, this observable is extremely  D. Shulyak et al.: Model atmospheres of magnetic chemically peculiar stars 5 useful for determining basic stellar parameters and is prefer-able in this respect to broad- and intermediate-bandphotometricdata. The role of energy distributions is even more important forchemically peculiar stars because most of the widely used pho-tometric calibrations are based on observations of normal starsand, hence, may not be applicable to stars with strong magneticfields and unusual abundances (Khan & Shulyak 2007).Figure 5 illustrates the e ff  ects of individual abundances andthe magnetic field on the energy distributions for the modelsof HD137509 with  T  e ff   =  12750K, log g  =  3 . 8. It is evidentthat both the model with individual abundances and the modelcombining the e ff  ects of unusual chemical composition and thestrong magnetic field exhibit flux redistribution from the UV tovisualwavelengthregion.Themostsignificante ff  ectis foundforthe model with the magnetic field included, which is in agree-ment with our previous results.Due to the increased level of flux in the visual region, themodels with magnetic line blanketing are characterized by ananomalous bolometric correction. We find that the di ff  erencein this parameter with respect to the computations done forsolar chemical composition reaches  ∆ BC  =  0 . 1 mag for themodel with individual abundance, whereas  ∆ BC  ≈  0 . 15 magfor the model with individual abundance and magnetic field.Thus, the anomaly in the bolometric correction is quite substan-tial and has to be taken into account in the determination of ab-solute luminosity and comparison with evolutionary models of CP stars (e.g., Kochukhov & Bagnulo 2006), yet, even for suchan extreme star as HD137509, we cannot confirm the reality of  ∆ BC  >  0 . 2 mag proposed by Lanz (1984). Unfortunately, no suitable observed energy distribution forHD137509 which could be used for verifying theoretical mod-els is available in the literature. One can possibly use the flux-calibrated spectrum from the UVES Library 1 . The descriptionof the reduction of these data can be found in Bagnulo et al.(2003). Despite the fact that the UVES pipeline provides a userwith the spectra calibrated in relative units, these data can notbe used for comparisons with models in a wide spectral rangedue to the flux calibration uncertainties (S. Bagnulo, privatecommunication) that could a ff  ect the shape of the energy dis-tribution. Nevertheless, the relative shape of the UVES spectrawithin small spectral regions (300–500 Å) is reasonably well-determined.Therefore,in Fig. 6we comparerelativefluxes(nor-malized at  λ  =  5000Å) of the magnetic and non-magnetic mod-els of HD137509with the UVES spectrum in two short regions.Note, that both theoretical models presented in the figure werecalculated with the abundances from Paper I. Thus, the di ff  er-ence between the two model fluxes is entirely due to the mag-netic field, which is responsible for producing complex spectralfeatures clearly seen in the observed spectra. In particular, theZeeman splitting influences the strength of the line absorptionaround  λ  =  5200Å, which is associated with the well-knowndepression in the spectra of CP stars and its photometric char-acteristic ( ∆ a  photometry, discussed below). Note, that for thecomparisonof fluxes in the UV region(upperpanel in Fig. 6) wehad to shift the energy distribution produced by non-magneticmodel alongthe  y -axis to match the observeddata. This is a con-sequence of the substantial di ff  erence in the fluxes close to theBalmer jump where the model with the magnetic field shows aprominentflux excess comparedto the non-magneticmodel (seeFig. 5).The only other possibility would be to use the flux cali-brated spectra of HD137509 obtained by the Far Ultraviolet 1 http://www.eso.org/uvespop 3300 3400 3500 3600 3700 λ, [Α] 1.31.41.51.61.7   n  o  r  m  a   l   i  z  e   d   f   l  u  x UVES observationst12750g40 individual + magnetict12750g40 individual 4800 4900 5000 5100 5200 5300 5400 λ, [Α] 0.60.811.2   n  o  r  m  a   l   i  z  e   d   f   l  u  x UVES observationst12750g40 individual + magnetict12750g40 individual Fig.6.  Comparison between the observed and calculated fluxesof HD137509 in the near UV (upper panel) and visual region(lower panel). Gaussian smearing with resolution  R  =  1500 isapplied to all the spectra to provide a better view. In the upperplot the fluxes for the non-magnetic model are shifted along thevertical axis to match the observed spectrum.Spectroscopic Explorer 2 (FUSE) mission. However, these datawere obtained in a very short wavelength range (910–1185Å),which makes it impossible to use these spectra for accurate de-termination of fundamental stellar parameters and testing theo-retical models. For example, the overall radiative energy emit-ted inside 910–1185Å region is about 1 . 6% of the total fluxfor the model with  T  e ff   =  12750K and about 2 . 5% for the T  e ff   =  13750K model, which is unimportant for overall radia-tive energy balance in the atmosphere of HD137509. Moreover,the slope of the energy distribution in this region remains thesame for both models mentioned above. This means that, basedon FUSE data, it is impossible to distinguish models di ff  erentbyas much as 1000 K. Well-calibrated energydistribution coveringa wide spectral regions is needed. 3.4. Photometric colors  As a next step, we have calculated a grid of model atmosphereswith di ff  erent e ff  ective temperatures ( T  e ff   =  12250K–13750K, ∆ T   =  250K) and gravities (log g  =  3 . 8–4 . 2,  ∆ log g  =  0 . 2) toassess the ability ofourmodels to reproducethe observedphoto-metric properties of HD137509. To investigate the role of mag-netic field, we have included in the grid both magnetic and non-magnetic models with individual abundances. We also consid-ered the reference scaled-solar abundances model from Paper I.The theoretical colors were calculated following the proce-dure outlinedin Kochukhov et al. (2005). We use modified com- puter codes by Kurucz (1993a), which take into account trans- mission curves of individual photometric filters, mirror reflec-tivity and a photomultiplier response function. The synthetic ∆ a values were computed with respect to the theoretical normal- 2 http://fuse.pha.jhu.edu/
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