ROTATING LOW-MASS STELLAR MODELS WITH ANGULAR MOMENTUM REDISTRIBUTION

ROTATING LOW-MASS STELLAR MODELS WITH ANGULAR MOMENTUM REDISTRIBUTION Luiz Themystokliz Sanctos Mendes Electronics Engineering Department, Universidade Federal de Minas Gerais Belo Horizonte (MG), Brazil luizt@cpdee.ufmg.br Luiz Paulo Ribeiro Vaz Physics Department, Universidade Federal de Minas Gerais Belo Horizonte (MG), Brazil vaz@fisica.ufmg.br Francesca D Antona Osservatorio Astronomico di Roma Monte Porzio Catone, Italy dantona@coma.mporzio.astro.it Italo Mazzitelli Istituto di Astrofisica Spaziale CNR, Rome, Italy aton@hyperion.ias.rm.cnr.it Abstract The evolution of rotating low-mass stars has been followed with a new version of the ATON 2.0 stellar evolution code (Ventura et al. 1998, which includes both rotation and internal angular momentum redistribution, coupled with surface angular momentum loss. Rotation was implemented according to Kippenhahn s equipotential surfaces method, while angular momentum transport in radiative zones was implemented according to the general framework established by Zahn (1992) and co-workers (e.g. Talon et al. 1997, Maeder & Zahn 1998) based on the assumption of stronger turbulent transport in the horizontal than in the vertical direction. The computed models cover both the pre-main sequence and main sequence phases. Particular results for an one solar mass model are presented. 1

2 1. Introduction For more than a decade there has been a growing concern that internal angular momentum transport plays an important role in the mixing of the chemical constituents of the stars, affecting their surface abundances. As part of the ongoing effort to improve the macro- and micro-physics of the ATON 2.0 stellar structure code (Ventura et al. 1998), we present here new and improved results concerning the introduction of internal redistribution of angular momentum and the associated rotational mixing. 2. Implementation details As it is well known since Eddington (1925) and Von Zeipel (1924a, 1924b), rotation causes a thermal imbalance in stars which in turn drives meridional circulation currents. Even if the initial rotation state is the one corresponding to rigid body rotation, those circulation currents will alter the internal angular velocity profile leading to differential rotation, which in turn can trigger a number of hydrodynamical instabilities in the almost non-viscous stellar plasma, resulting in turbulent motions. Both dynamical (Solberg-Hoiland and dynamical shear) and secular (meridional circulation, Goldreich-Schubert-Fricke instability, and secular shear) instabilities have been considered in our calculations. The physics to compute those instabilities and their associated diffusion coefficients is the same customarily adopted by other researchers in the field (e.g. Endal & Sofia 1978, Heger et al. 2000), except regarding the secular shear instability for which we adopted the more modern treatment presented by Maeder (1997). Internal angular momentum transport has been modeled according to the general framework established by Zahn (1992) and later pursued by other authors (e.g. Urpin et al. 1996, Talon et al. 1997, Maeder & Zahn 1998), which is based on the assumption of stronger turbulent transport in the horizontal than in the vertical direction. Under that framework, the internal angular momentum transport in radiative zones is given by the advection-diffusion partial differential equation [ ] ρr 2 Ω = 1 t 5r 2 r (ρr4 ΩU) + 1 [ r 2 ρν v r 4 Ω ] r r (1) where U(r) is the meridional circulation velocity and the other symbols retain their usual meanings. In equation (1), the total diffusion coefficient ν v is computed as the sum of the individual diffusion coefficients computed for each hydrodynamical instability considered.

ROTATING LOW-MASS STELLAR MODELS 3 3. Input physics The ATON 2.0 code has many updated and modern features regarding the physics of stellar interiors, of which a full account can be found in Ventura et al. (1998). Below we highlight some of these features: most up to date OPAL (Rogers & Iglesias 1993) opacities, supplemented by those of Alexander & Ferguson (1994) for lower (T < 6000K) temperatures; diffusive mixing and overshooting; convection treated under both mixing length theory or the Full Scale of Turbulence (FST) from Canuto & Mazzitelli (1991, 1992) and Canuto et al. (1996). The structural effects of rotation were already included in the ATON code (Mendes et al. 1999) by using the Kippenhahn & Thomas (1970) method with the improvements brought by Endal & Sofia (1976) regarding the calculation of the potential function. Angular momentum losses in the star s external layers due to magnetized stellar winds are also taken in account, and enter the PDE advection- diffusion equation as a boundary condition at the surface. We adopted the prescription used in Chaboyer et al. (1995) with a wind index n = 1.5, which reproduces well the Skumanich (1972) law v t 1/2 : ( ) J R 2 n ( ) M n/3 t = K ω 1+4n/3, ω < ω crit, (2) R M ( ) J R 2 n ( ) M n/3 t = K ωω 4n/3 crit R M, ω ω crit. (3) where ω crit introduces a critical rotation level at which the angular momentum loss saturates. The constant K is usually calibrated by requiring that the model surface velocity match the current solar rotation rate at the equator. 4. Results for the 1 M model We have run a series of low-mass stellar models with internal angular momentum redistribution, from which we now discuss the specific results for the 1 M case. The 1 M model was run with a chemical composition of Y = 0.271, Z = 0.0175 and under the mixing length framework for convection. The initial angular momentum was taken from the Kawaler (1987) relations for low-mass stars, and corresponds to J 0 = 1.566 10 50 (cgs units) for the case of 1 M. The chosen rotation law corresponds to local angular momentum conservation

4 in radiative regions and rigid body rotation in convective ones (Mendes et al. 1999). Fig. 1 shows the angular velocity profile along the star s radius for six representative evolutionary ages, three of them in the pre-main sequence phase and the three others in the main sequence. The respective locations at the evolutionary path in the H-R diagram are shown in Fig. 2, labeled A through G. We can easily see that during most of the pre-main sequence phase (points A through C) the star s contraction is much more effective than the redistribution mechanisms, giving rise to a rapidly rotating radiative core. By the time the star is near the zero-age main sequence (point D), it is now slowly contracting and the redistribution mechanisms have already taken over. From this point on the angular momentum transport, coupled to wind-driven surface angular momentum loss, forces the star to spin down while smoothing out its internal angular velocity profile. At point G, which indicates the current solar age, the diffusion coefficients associated to hydrodynamical instabilities have very low values and the surface angular momentum loss leads to higher rotation rates at the star s core. Fig. 3 shows the total diffusion coefficient computed for the 1 M model at the same ages as in Fig. 1. -3.5 C -4 D B -4.5 A -5 E -5.5 F -6 0 0.2 0.4 0.6 0.8 1 G Figure 1. Angular velocity profile as a function of radius for the 1 M model with J = J 0, at seven different ages. A (black): log t = 6.5; B (red): log t = 7.0; C (magenta): log t = 7.5; D (light blue): log t = 8.0; E (yellow): log t = 8.5; F (blue): log t = 9.0; G (green): log t = 9.7.

ROTATING LOW-MASS STELLAR MODELS 5 2 1.5 1 0.5 0-0.5 3.8 3.75 3.7 3.65 3.6 3.55 Figure 2. Evolutionary track for the 1 M model. The points labeled A to G correspond respectively to the same ages as in Fig. 1. Figure 3. Total diffusion coefficient at the radiative core. Ages and colors as defined in Fig. 1.

6 It is very interesting to see that the model s angular velocity profile at the current solar age resembles very well the corresponding solar one from the surface down to 0.4 R obtained through helioseismology, which corresponds to nearly solid body rotation (Fig. 4). 700 600 500 400 300 200 100 0 0.2 0.4 0.6 0.8 1 Figure 4. Solar rotation curve as inferred from p-mode frequency splittings (top picture; from Kosovichev et al. 1997), as opposed to the internal rotation curve for the 1 solar mass model (lower picture), at the current solar age. The dotted line indicates the surface velocity at the 0 degrees latitude, as taken from the top picture.

ROTATING LOW-MASS STELLAR MODELS 7 This result shows that internal angular momentum redistribution, coupled with angular momentum losses at the surface due to stellar winds, can be very effective in smoothing the internal rotation profile. It is worth mentioning that other low-mass stellar models in the literature, that also include internal angular momentum redistribution (e.g. Chaboyer et al. 1995), resulted in very steep angular velocity profiles in the radiative interior, which led many researchers to look for other angular momentum transport mechanisms such as gravity waves (e.g. Talon & Zahn 1998). It should be noted that the main reason of obtaining a lower surface rotation rate for our models, when compared to the sun, can be attributed to the fact that we adopted the value of the constant K (see eqs. 2 and 3) given by Chaboyer et al. (1995), which was obviously calibrated for their models and not ours. Work is in progress to calibrate it properly according to our models. As for the rotational mixing, we found that internal transport of angular momentum contributes for a higher depletion of the light elements such as lithium, in accordance with earlier results from other researchers e.g. Pinsonneault et al. (1990), though this effect is barely noticeable for the 1 M model as it can be seen from Fig. 5. However, we note that in our models the direct effect 3.5 2.30 2.25 3.0 2.20 7.00 7.25 7.50 log [Li/H] 2.5 2.0 6 7 8 log age Figure 5. Lithium depletion (in the scale N[Li] = 12 + log N[Li] ) as a function of age for N[H] the 1 M model. Dotted line: non-rotating model. Solid line: rotating model with no angular momentum transport. Dashed line: rotating model with angular momentum transport.

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