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Atomic Diffusion in Star Models of Type Earlier Than G

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Atomic Diffusion in Star Models of Type Earlier Than G A&A 390, 611–620 (2002) Astronomy DOI: 10.1051/0004-6361:20020768 & c ESO 2002 Astrophysics Atomic diffusion in star models of type earlier than G P. Morel and F. Thevenin ´ D´epartement Cassini, UMR CNRS 6529, Observatoire de la Cˆote d’Azur, BP 4229, 06304 Nice Cedex 4, France Received 7 February 2002 / Accepted 15 May 2002 Abstract. We introduce the mixing resulting from the radiative diffusivity associated with the radiative viscosity in the calcula- tion of stellar evolution models. We find that the radiative diffusivity significantly diminishes the efficiency of the gravitational settling in the external layers of stellar models corresponding to types earlier than ≈G. The surface abundances of chemical species predicted by the models are successfully compared with the abundances determined in members of the Hyades open cluster. Our modeling depends on an efficiency parameter, which is evaluated to a value close to unity, that we calibrate in this study. Key words. diffusion – stars: abundances – stars: evolution – Hertzsprung-Russell (HR) and C-M diagrams 1. Introduction stars do not show low surface metallicities. This large helium depletion is not likely to be observed in main sequence B-stars, ff Microscopic di usion, sometimes named “atomic” or “ele- as supported by their spectral classification which is mainly ff ment” di usion, when used in the computation of models of based on the relative strength of the helium spectral lines. main sequence stars produces (see Chaboyer et al. 2001 for The large efficiency of the atomic diffusion in the outer lay- an abridged revue) a change in the surface abundances from ers results from the large temperature and pressure gradients. their primordial values and a shift of the evolutionary tracks to- For models of stars with types later than ≈G, i.e. M? ≈< 1:2 M , wards lower luminosities and temperatures (Morel & Baglin the external convection zone acts as a “reservoir” which in- 1999), consequently increasing the expected age of binaries hibits atomic diffusion. On the contrary, for stars with masses and clusters. For the Sun, the insertion of microscopic diffusion larger than ≈>1:4 M , the external helium and hydrogen convec- has spectacularly improved the agreement between the theoret- tion zones are so tiny that the reservoir is not large enough to ical and the “seismic” model (e.g. Christensen-Dalsgaard et al. be efficient. 1996). Indeed, for low mass stars and for population II stars, Such misleading physics in the calculation of microscopic ff atomic di usion eliminates, at least partially, discrepancies be- diffusion coefficients (e.g. Cox et al. 1989; Proffitt & Michaud tween observations and theoretical models (Morel & Baglin 1991; Michaud & Proffit 1993; Thoul et al. 1994) are very un- 1999; Lebreton et al. 1999; Cayrel et al. 1999; Chaboyer et al. likely owing to the high degree of sophistication of the method 2001; Salaris & Weiss 2001). based on the kinematical theory of gases (Burgers 1969) and ≈ For stellar models with mass larger than 1:4 M , the use also as seen by its success when applied to the Sun. Clearly, ff of microscopic di usion alone produces at the surface an im- a physical process which inhibits the efficiency of microscopic portant depletion of helium and heavy elements and a concomi- diffusion is lacking. tant enhancement of the hydrogen content. As an example, for Among physical processes acting against gravity, recent de- ff a2M model computed with atomic di usion, at the age of velopments were focused on: 20 Myr the surface mass fraction of hydrogen increased from X = 0:70 to X ≈> 0:98, while helium and heavy elements are – The radiative accelerations, which are efficient only in the strongly depleted. This strong depletion, which increases with case of chemical species with a large charge, e.g. iron. As the mass of the star, is not observed either by Varenne & Monier such “metals” have small relative abundances, the radiative (1999) in stars belonging to the Hyades open cluster or in OB accelerations are not efficient limiting the sedimentation of associations (Daflon et al. 2000) and in the Orion association helium and heavy elements as a whole (e.g. Alecian et al. (Cunha & Lambert 1994). 1989; Turcotte et al. 1998a). There is no possibility of observing helium in main se- – The turbulent mixing generated by the shear created by quence stars of intermediate mass to check its depletion; such the differential rotation between the solar convection zone and the radiative core (Richard et al. 1996; Gabriel 1997; Send offprint requests to:P.Morel, Brun et al. 1998). It has been recently claimed (Schatzman e-mail: [email protected] et al. 2000) that the shear instability invoked to produce this Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20020768 612 P. Morel and F. Th´evenin: Atomic diffusion in star models of type earlier than G turbulent mixing only concerns a narrow region and may be not as efficient as expected. – The rotation-induced mixing accounting for the transport of angular momentum explains well the hot side of the Li dip in the Hyades (Talon & Charbonnel 1998). – The use of a hypothetical mixed reservoir with an ad-hoc adjusted size (e.g. Turcotte et al. 1998b). – The refinement of the physical description. In the calcu- lations of G to A type star models (Turcotte et al. 1998a, 1998b; Richer et al. 2000), the local mean Rosseland opac- ity is updated at each time step, according to local changes in the density and temperature, but also in the genuine mass fractions of chemical species. In some favorable situ- ations the authors have obtained the formation of an “iron” mixed convective zone that inhibits the efficiency of the atomic diffusion. Even with this noticeable improvement of the modeling, several difficulties remain; in particular the models do not reproduce the abundances of star mem- Fig. 1. Profiles of kinetic radiative viscosity νrad (heavy full), kinetic ff bers of the Hyades open cluster as observed by Varenne & molecular viscosity νmol (heavy dashed), hydrogen di usivity dH (dot- ff Monier (1999); their Figs. 5 to 8 for elements from C to Ba dash-dot-dash) and helium di usivity dHe (dotted) with respect to ra- ≤ only reveal a mild depletion by a factor ≈3 for 6600 K ≤ dius in the radiative zones of a 1 M (R 0:61 R )andina1:8 M stellar models at age 700 Myr. The kinematic molecular and radiative T ff ≤ 7800 K. Also the theoretical strong depletions pre- e viscosities and the diffusivities are in c.g.s. (cm2 s−1) units. dicted by Turcotte et al. (1998b) for effective temperatures larger than 6600 K are not observed. – The influence of the stellar mass-loss described phe- and ions. In an astrophysical plasma with radiation, the kine- nomenologically (Chaboyer et al. 1999) in the modeling of matic viscosity ν has two components: Procyon A. ν = νmol + νrad: The kinematic molecular viscosity, ν , results from energy In this exploratory work, we focus on the role of the radia- mol exchanges between thermal collisions leading to excitation and tive diffusivity generated by the photon-ion collisions that is ionization of atoms and ions. Thus, the thermal collisions gen- not presently taken into account in the microscopic diffusion erate a “molecular mixing” resulting from the molecular diffu- coefficients. As we shall see, the photon-ion collisions are an sivity associated with ν . The kinematic molecular viscosity efficient physical process that inhibits the large sedimentation mol is approximated by (Schatzman 1977): of helium and heavy elements in outer layers of main sequence ≈ 5 star models with types earlier than G. T 2 1 + 7X ν ' 2:21 × 10−15 ; (1) In Sect. 2 we present a simplified phenomenological model mol ρ 3lnΛ ff of the radiative di usivity resulting from photon-ion collisions. for a mixture of hydrogen and helium, assuming that the heavy Section 3 briefly describes the theoretical framework used for elements have a negligible effect. T is the temperature, ρ is the constructing stellar evolution models. In Sect. 4 we estimate the density, X is the hydrogen mass fraction and ln Λ is the so- radiative diffusivity parameter connecting the radiative kinetic ffi ff called Coulomb logarithm (e.g. Michaud & Pro t 1993). With viscosity to the radiative di usivity; our investigation is based the more elaborate description based on the kinetic theory of mainly on observed abundances of C, O, Fe and Mg in stars gases (Burgers 1969), the molecular diffusion has two com- belonging to the Hyades open cluster as reported by Varenne & ponents: the isotropic one, being the molecular diffusivity that Monier (1999). Illustrations are given in Sect. 5 and, in Sect. 6, generates a mixing and the anisotropic one, responsible for the we summarize and discuss the main results of the present in- gravitational sedimentation in stars. vestigation. The kinematic radiative viscosity νrad arises because pho- tons deposit their momentum in the fluid element into which they are absorbed. Thus, the radiative collisions generate a “ra- 2. Radiative diffusivity diative mixing” resulting from the radiative diffusivity asso- ciated with νrad. In the optically thick medium the kinematic Apparently, the radiative diffusivity, first used by Baglin radiative viscosity is expressed as (Thomas 1930; Mihalas & (1972), has not been introduced for intermediate stellar mass Weibel-Mihalas 1984, p. 461-472): models, although implemented in the evolution of models of 4 aT 4 larger mass (Heger et al. 2000). According to the standard ν ≡ ; rad 2 (2) description of the micro-physics of gases (e.g. Burgers 1969; 15 cκρ Mihalas & Weibel-Mihalas 1984, Chap.
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