Gray model atmospheres are generally considered a reasonable approximation to make upon stars of mass greater than about 0.6 M-circle dot. Here we show that non-gray atmospheres can significantly affect evolutionary models, with masses up to 0.9 M-circle dot. The effect of including a non-gray atmosphere is strongest in the pre-main and post-main Sequence. This may have implications for the ages of the oldest globular clusters

Harris, GJ

Lynas-Gray, AE

Miller, S

Tennyson, J

English

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Stellar Evolution at low Metallicity: Mass Loss, Eplosions, CosmologyASP Conference Series, Vol. 353, 2006Henny J.G.L.M. Lamers, Norbert Langer, Tiit Nugis, Kalju AnnukThe role of non-gray model atmospheres in the evolutionof low mass metal poor stars.G. J. Harris, A. E. Lynas-Gray, J. Tennyson, and S. Miller.University College London, UK.Abstract. Gray model atmospheres are generally considered a reasonableapproximation to make upon stars of mass greater than about 0.6 M¯. Herewe show that non-gray atmospheres can signi¯cantly a®ect evolutionary models,with masses up to 0.9 M¯. The e®ect of including a non-gray atmosphere isstrongest in the pre-main and post-main Sequence. This may have implicationsfor the ages of the oldest globular clusters.1. Introduction.Stars with masses below about 0.85 M¯ have Main Sequence lifetimes in excessof 10 GYr. Many stars which have metallicity, colors and kinematics that suggestan age greater than 10 GYr have been observed. Numbered among these starsare the stars in old globular clusters, such as M 68, as well as isolated stars suchas HE0107-5240 (Christlieb et al. 2002). Not only do these stars provide us witha means to study stellar evolution at low and very low metallicity, but they alsoallow a stellar estimates for formation timescales of the Galaxy and a lower agelimit of the Universe.The most accurate age estimates for metal poor stars is achieved by meansof the ¯tting of theoretical isochrones to observed color magnitude diagrams(CMD). The accuracy of these ¯ts is dependent upon the accuracy of the physicsof the stellar models used to compute the isochrones. One potential avenues forthe improvement of the modeling of stellar evolution lies in the atmosphere,the model atmosphere provides boundary conditions for the equations of stellarstructure. Current low to intermediate mass stellar evolution models (Cariuloet al. 2004; Kim et al. 2002) in general use gray model atmospheres. However,Saumon et al. (1994), and later Bara®e & Chabrier (1996) Bara®e et al. (1997)and Chabrier & Bara®e (1997), performed stellar evolution computations usingnon-gray atmospheres. These works clearly demonstrate that the gray approxi-mation is invalid for Tef <5000 K, which corresponds to 0.8 or 0.6 M¯ dependingupon metallicity. In this work we show that using non-gray models has a sig-ni¯cant e®ect upon the e®ective temperature and luminosity of low metallicitymodel stars of 0.9 M¯ both undergoing pre-Main Sequence collapse and on thered giant branch.2. Stellar evolution.We have recently developed a stellar evolution code which makes use of a non-gray model atmosphere to provide the boundary conditions for the equationsof stellar structure. This code is based upon the stellar evolution code cesam8384 G.J. Harris, A.E. Lynas-Gray, J. Tennyson, and S. Miller(Morel 1997), for which new opacity and equation of state subroutines have beenwritten and a non-gray model atmosphere added. cesam is a 1D low and inter-mediate mass stellar evolution code, which is capable of computing evolution forpre-Main Sequence to the Helium °ash. The non-gray model atmosphere codewhich is used is based upon marcs (Gustafsson et al. 1975), for which, again,new equation of state and opacity subroutines have been written. The resultingcode we refer to as NG-ELMS.2.1. Atmospheric equation of state and opacity.The computation of a non-gray model atmosphere requires a monochromaticopacity. In turn to compute opacity it is essential to know the chemical andionic composition of the gas. We have developed a new equation of state whichwe refer to as LoMES to model low metallicity gases, currently the equation ofstate accounts for 15 elements H, He, Li, C, N, O, Ne, Na, Mg, Al, Si, S, Ca,Ti, Fe and electrons. The ¯rst 2 ionization states and cations are considered foreach metal. In total LoMES computes the number densities of 136 ions, atomsand molecules, which are all listed in table 1. Due to the temperature limitationof the molecular partition functions used by LoMES, LoMES is able to solvechemical equilibrium between 1000 and 10 000 K. LoMES has been successfullyused to compute opacities for zero metallicity stars (Harris et al. 2004a) andhelium rich white dwarfs (Harris et al. 2004b).Table 1. The molecular, atomic, and ionic species accounted for in LoMES.e¡ H2 H H+ H¡ H+2 H¡2 H+3 HeHe+ HeH+ Li Li+ Li¡ Li++ C C+ C¡C++ N N+ N¡ N++ O O+ O¡ O++Ne Ne+ Ne¡ Ne++ Na Na+ Na¡ Na++ MgMg+ Mg¡ Mg++ Al Al+ Al¡ Al++ Si Si+Si¡ Si++ S S+ S¡ S++ Ca Ca+ Ca¡Ca++ Ti Ti+ Ti¡ Ti++ Fe Fe+ Fe¡ Fe++CO CO+ CN CN+ C2 C+2 CH CH+ OHOH+ NH NH+ NO NO+ O2 O+2 N2 N+2CS TiO MgH MgH+ SiH SiH+ FeH MgO AlOSiO SiO+ SO SO+ CaO CaO+ FeO NaH AlHAlH+ HS HS+ CaH SiS S2 S+2 NS NS+TiS MgS Al2 Si2 TiN Na2 LiH HCN H20CO2 CH4 C2H2 C3 NH3 Fe(OH)2 SiH4 SiC2 Si2CMgOH Mg(OH)2 H2S Al2O AlOH Ca(OH)2 CaOH NaCN NaOHTiO2Many of the species in table 1 are directly and indirectly important in the com-putation of opacity. For example H¡ is often the dominant source of continuousopacity, whilst molecules such as H2O, TiO and HCN make extremely importantcontributions to opacity via their line absorption. Other species such as H+3 andHeH+ are important electron donors at low metallicity, thereby indirectly a®ectopacity. We refer to our low metallicity opacity subroutine as LoMO. LoMOaccounts for H, H¡, He, He+ and H+2 bound-free continuous absorption, H, H¡,Non-gray Model-Atmospheres 85He, He¡, He+, H¡2 and H+2 free-free absorption and H2-He, H2-H2 and H-Hecollision induced absorption. LoMO also accounts for H2, H and He Rayleighscattering and Thomson scattering by electrons. Further details of the data usedto account for these scattering and absorption processes can be found in Harriset al. (2004a). To account for molecular bands for each molecule a mean opacityis computed over a small frequency range and tabulated as a function of temper-ature and frequency, this allows the rapid computation of opacity at runtime.The approximation of using mean opacities is valid as molecular bands consistof many week lines which result in a near continuous opacity over the band. Themolecular species for which we compute opacity tables are H2O (Barber et al.in preparation), HCN (Harris, Polyansky, & Tennyson 2002), CO (Goorvitch1994), CN (J¿rgensen & Larsson 1990), CH (J¿rgensen et al. 1996), TiO (Plez1998), OH, NH (Kurucz et al. 1995) and FeH (Dulick et al. 2003).2.2. Input physics.The results of stellar evolution computations are dependent upon the inputdata. Our high temperature opacities are computed by interpolation upon theOPAL Rosseland mean opacity tables of Iglesias & Rogers (1996), the OPALequation of state (Rogers & Nayfonov 2002) is used for high temperatures. Forlow temperatures (> 9000 K) we use the data computed by LoMES and LoMO,at intermediate temperatures (9000{11 000 K) we interpolate between the OPALdata and the data from LoMES and LoMO. We use the NACRE (Angulo et al.1999) compilation of nuclear reaction rates for temperatures below 50£106 K,and the Caughlan & Fowler (1988) rates for higher temperatures. Conductiveopacity is computed by interpolating on the tables of Hubbard & Lampe (1969).Mixing length theory is used for convection.3. Preliminary models.Our preliminary models are non-rotating, with no element di®usion and arecalculated in the mass range 0.7 to 0.9 M¯. In order to compare with the Y2models of Kim et al. (2002) we adopt a hydrogen mass fraction of X=0.76997and a metal mass fraction of Z= 10¡5. Figure 1 shows a plot of our grayand the Y2 0.8 M¯ models, we have used di®erent values of the mixing lengthparameter in our calculations. The importance of the adopted value of themixing length parameter is clear. Possible reasons for the other di®erencesbetween our calculations and those of Y2, are likely to be due in part to thelack of di®usion within our models and the di®erences in input physics. Forthe purposes of this work we adopt a value of ® =2.0 for the mixing lengthparameter.86 G.J. Harris, A.E. Lynas-Gray, J. Tennyson, and S. Miller 0 0.5 1 1.5 2 2.5 3 3.64 3.66 3.68 3.7 3.72 3.74 3.76 3.78 3.8 3.82 3.84log(L/Lsun)log(Tef)a=1.5000a=1.7432a=2.0000Y2 (Yi et al 2001) a=1.7432Figure 1. Stellar evolution tracks on the HR diagram, for the 0.8 M¯ modelof Y2 (Kim et al. 2002) and our gray models computed with values of themixing length parameter of 1.5, 1.7432 and 2.0.Non-gray Model-Atmospheres 87Figure 2 shows the evolutionary tracks on the HR diagram of stars of 0.7, 0.8and 0.9 M¯, computed with gray and non-gray model atmospheres. The non-gray models show a signi¯cant di®erence in the gradient of the Hayashi trackand the red giant branch. In general on the red giant branch non-gray modelsare cooler than gray models for a given luminosity. It is envisaged that thisdi®erence in e®ective temperature will result in di®erent color magnitudes, andthus may have an e®ect upon the ¯tting of globular cluster CMDs.The core hydrogen burning lifetimes of these models are given in table 2, wede¯ne the end of core hydrogen burning as the point at which the hydrogenmass fraction at the core falls below 10¡5. Overall the di®erences betweenthe hydrogen burning lifetimes of the gray and non-gray models are small, butincrease with decreasing mass, this indicates that the gray approximation haslittle e®ect upon main sequence evolution. The small e®ect of the non-grayatmosphere on the conditions in the core is supported by ¯gure 3, which showsthe evolution of the Central temperature and pressure. Along the main sequencethere is no signi¯cant di®erence in central temperature and pressure betweengray and non-gray models, however on the pre and post Main Sequence thereare small di®erences in temperature between gray and non-gray models.Table 2. Core hydrogen burning lifetimes for gray and non-gray models [Gyr].M [M¯] Gray non-gray0.7 21.305 21.3930.8 13.260 13.2570.9 8.738 8.7394. Conclusion.We present new low and high temperature equation of state and opacity sub-routines. These subroutines have been incorporated with the non-gray stellaratmosphere code marcs into the stellar evolution code cesam to create a newstellar evolution code ng-elms. The non-gray atmosphere is used to providethe outer boundary conditions for the equations of stellar structure. ng-elmshas been used to compute both gray and non-gray preliminary stellar evolutionmodels with Z= 10¡5 and with masses between 0.7 and 0.9 M¯.Acknowledgments. We thank Bob Barber for providing us with his waterlinelist prior to publication, and Prof. B. Gustafsson and Dr. P. Morel formaking available versions of marcs and cesam. The UK Particle Physics andAstronomy Research Council (PPARC) is thanked for ¯nancial support.88 G.J. Harris, A.E. Lynas-Gray, J. Tennyson, and S. Miller-0.5 0 0.5 1 1.5 2 2.5 3 3.65 3.7 3.75 3.8 3.85log(L/Lsun)log(Tef)0.9 Msun0.7 MsunGreynon-greyFigure 2. Stellar evolution tracks on the HR diagram, for both gray andnon-gray models of 0.7, 0.8 and 0.9 M¯.Non-gray Model-Atmospheres 89 14 15 16 17 18 19 20 21 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5log(pressure [dyn/cm2])log(time [Myr])0.9 Msun0.7 MsunGreynon-Grey 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5log(T [K])log(time [Myr])0.9 Msun0.7 MsunGreynon-GreyFigure 3. Evolution of the central temperature and pressure, for both grayand non-gray models of 0.7, 0.8 and 0.9 M¯.ReferencesAngulo, C., Arnould, M., Rayet, M., et al., 1999, Nuclear Physics A, 656, 3.Bara®e, I., & Chabrier, G., 1996, in from stars to galaxies, eds. C. Leitherer, U. Fritze-von-Alversleben, & J. Huchra (ASP conference series) vol. 98, P. 209Bara®e, I., Chabrier, G., Allard, F., & Hauschildt, P., 1997, A&A, 327,1054.Cariulo, P., Degl'Innocenti, S., & Castellani, V., 2004, A&A, 421, 1121.Caughlan, G. R., & Fowler, W. A., 1988, Atomic Data and Nuclear Data Tables, 40,283.Chabrier, G., Bara®em I., 1997, A&A 327, 1039.Christlieb, N., Bessell, M. S., & Beers, T. C., et al., 2002, Nature, 419, 904.Dulick, M., Bauschlicher, C. W., Jr., Burrows, A., Sharp, C. M., Ram, R. S., & Bernath,P., 2003, ApJ, 594, 651.Goorvitch D., 1994, ApJS, 95, 535.Gustafsson, B., Bell, R. A., Eriksson, K., & Nordlund, º A., 1975, A&A, 42, 407Harris, G. J., Polyansky O. L., & Tennyson J., 2002, ApJ, 578, 657Harris, G. J., Lynas-Gray, A. E., Miller, S., & Tennyson, J., 2004a, ApJ, 600, 1025Harris, G. J., Lynas-Gray, A. E., Miller, S., & Tennyson, J., 2004b, ApJ, 617, L143.Hubbard, W. B., & Lampe, M., 1969, ApJS, 18, 297.Iglesias, C. A. and Rogers, F. R., 1996, ApJ, 464, 943.J¿rgensen, U. G., & Larsson, M., A& A, 238, 424.J¿rgensen, U. G., Larsson, M., Iwamae, A., & Yu, B., 1996, A& A, 315, 204.Kim, YC., Demarque, P., Yi, S. K., & Alexander D.R., 2002, ApJS, 143, 499.Kurucz, R. L., 1995, in Laboratory & astronomical high resolution spectra, eds. A. J.Sauval, R. Blomme, & Grevesse (ASP Conference series) vol. 81, p. 583.Morel, P., 1997, A& ASS, 124, 597.Plez, B, 1998, A& A, 337, 495.90 G.J. Harris, A.E. Lynas-Gray, J. Tennyson, and S. MillerRogers, F. J., & Nayfonov, A., 2002, ApJ, 576, 1064.Saumon, D., Bergeron, B., Lunine, J. I, Hubbard, W. B., & Burrows, A., 1994, ApJ,424, 333DiscussionMaeder: Could you use the electron densities you are obtaining to also improvethe calculation of the screening factors for nuclear reactions in low mass stars?Harris: Our new equation of state is computed only for low temperatures (T<11000 K). It therefore has no e®ect upon the interior high temperature regions,for which we use the OPAL equation of state.Gustafsson: You demonstrate clearly that the e®ects of a non-grey treatment areimportant for the real giant stage. But, are not the uncertainties in convectioneven more signi¯cant there? Also, one may speculate that a proper treatment ofconvection, with 3D HD and radiative transfer would further increase the e®ectsof non-greyness.Harris: Yes, the uncertainties in mixing length theory are large. As we showby changing the mixing length parameter. However, accounting for non-greye®ects changes the gradient of the red giant branch on the HR diagram, usingdi®erent values of the mixing length parameter changes the position but not thegradient. A proper treatment of convection is still highly desirable.Gregory Harris

The role of non-gray model atmospheres in the evolution of low mass metal poor stars

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