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Astron. Astrophys. 341, 181-189 (1999)

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4. Results of the calculations

In Fig. 1 solutions of the nonstationary problem are presented for the [FORMULA] model. The surface rotational velocity [FORMULA] km s-1 has been chosen as large as those observed in the most rapidly rotating OB-stars. It corresponds to [FORMULA] s-1 and to the ratio of the centrifugal acceleration to the gravity on the stellar equator [FORMULA]. The solid line in Fig. 1a gives the steady-state asymptotic solution (Eq. (12 )). One can see that in this particular case the nonstationary solutions approach the stationary one very quickly (numbers along the lines show their respective ages in years). The MS life-time for the [FORMULA] model is [FORMULA] years, so, for [FORMULA] s-1 it takes only about 1% of the MS life-time for the [FORMULA] star to settle in the steady-state rotation. This "relaxation" time can be roughly estimated a priori as the ratio [FORMULA] (see below) which is approximately proportional to [FORMULA] (Eqs. (1-2 )) for a fixed stellar mass. Obviously, the more rapidly rotating stars reach the state of asymptotic rotation quicker than stars rotating slower.

[FIGURE] Fig. 1a-d. Approaching the steady-state rotation by the [FORMULA] star rotating with surface velocity [FORMULA] km s-1 (corresponds to [FORMULA] s-1). The asymptotic stationary solution is plotted with the solid line in panel a . Numbers along the lines in panel a give ages in years of the respective angular velocity profiles. Panels b , c and d show the evolution of the meridional circulation velocity, of the total diffusion coefficient and of the ratio of the time-scale for mixing chemical elements, defined as [FORMULA], to the star's MS life-time, respectively. In all the panels lines of the same type have the same age. In this and in the next figures where the coordinate [FORMULA] or [FORMULA] is used as abscissa it always starts on the left approximately from the convective core border or, more precisely, from the point where [FORMULA] (see text)

In panels b and c the evolution of the meridional circulation velocity and of the total diffusion coefficient are shown. For the initial uniform rotation law the quantity U changes sign near the stellar surface. This is a well-known result being due to the negative term [FORMULA] in brackets in the first row of expression (2) (Gratton 1945; Öpik 1951). In the classical description with the uniform rotation law applied, the meridional circulation currents consist of two zones, the mass of the outer zone being much smaller than that of the inner zone (Pavlov & Yakovlev 1978). Slow mixing across the "quiet" layer separating these zones can occur only as a purely microscopic diffusion process (Charbonnel & Vauclair 1992). In the past there were even some speculations that the quiet layer could prevent the radiative envelope of a single rotating MS star from being fully mixed (Vauclair 1988; Leushin et al. 1989). However, in Zahn's scheme, as our calculations show (see also Urpin et al. 1996; Talon et al. 1997), the position of the quiet layer shifts towards the stellar surface as the rotation profile evolves. As a result, the outer zone gradually (and, for a large enough value of [FORMULA], rather quickly) becomes smaller in mass and then disappears completely. Simultaneously with these changes a characteristic value of U (say, in the middle of the envelope) decreases by a factor of about 50. As long as the rotation profile stays close to the initial flat one, [FORMULA] (i.e. meridional circulation) remains the main contributor to the total diffusion coefficient. On the other hand, for [FORMULA] approaching the asymptotic distribution, [FORMULA] (turbulent diffusion) begins to play a dominant role. Finally, the sum [FORMULA] is found to decrease near the convective core border by [FORMULA] dex compared to the case of uniform rotation (Fig. 1 c).

In Fig. 1d we have plotted the ratio [FORMULA] against the relative mass coordinate, the mixing time being here defined as [FORMULA]. We see that even for the rotation profiles lying very close to the flat one, [FORMULA] exceeds [FORMULA] in the greater part of the inner envelope. This result strongly contrasts with the classical (Eddington-Sweet) estimate of the mixing time

[EQUATION]

where [FORMULA] is the Kelvin-Helmholtz time. For our [FORMULA] ZAMS model [FORMULA] years, so, for [FORMULA] and [FORMULA] years we get [FORMULA]! A main reason for such the large difference in the mixing time-scales (Zahn's scheme versus the classical description) is the horizontal erosion responsible for the factor [FORMULA] appearing in (8) which considerably reduces the product [FORMULA], the classical estimate of the mixing rate. It is this factor that causes a substantial diminution of [FORMULA] even for the nearly uniform rotation.

It should be noted that in real evolving stellar models the advection and diffusion will never reach a stage where they exactly balance each other so as to make the time derivative in (4) equal to zero. Therefore, it would be more correctly to consider [FORMULA] not as a time to achieve a stationary regime (which is never met) but instead as a characteristic time for the change of the inner rotation distribution.

In Fig. 2 the same parameter distributions as those shown in Fig. 1 are presented for the [FORMULA] model rotating with surface velocity [FORMULA] km s-1 ([FORMULA] s-1). We have also performed calculations for [FORMULA] km s-1 ([FORMULA] s-1).

[FIGURE] Fig. 2a-d. The same as in Fig. 1 but for the [FORMULA] star with [FORMULA] km s-1 ([FORMULA] s-1)

A comparison of the numerical results for the two chosen M values and for the two different rotational velocities for the [FORMULA] model led us to the following rough estimate of the relaxation time:

[EQUATION]

This time is much shorter than the mixing time [FORMULA] defined above because the angular momentum transport by meridional circulation is not affected by the horizontal erosion (Eq. (4 )). It is the difference in the rates of chemical mixing and of angular momentum transport that is considered as the main advantage of Zahn's new scheme, particularily, for the interpretation of a rather quick spin-down of the Sun and solar-like stars accompanied by a much slower depletion of their surface Li abundance (Zahn 1997).

If we take the mass-radius relation of Beech & Mitalas (1994) [FORMULA] valid for MS stars in the mass range [FORMULA], then [FORMULA] and, consequently, in the first approximation [FORMULA]. This relation holds (within an uncertainty factor of [FORMULA]) in our numerical calculations. For example, the [FORMULA] model has [FORMULA] years (the Kelvin-Helmholtz time increases with decreasing M on the MS), and for [FORMULA] and [FORMULA] s-1 the calculations give [FORMULA] and [FORMULA] years, respectively (compared to [FORMULA] and [FORMULA] years for the [FORMULA] model rotating with [FORMULA] s-1).

The meridional circulation velocity and the total diffusion coefficient are found to scale approximately as [FORMULA], for example, for [FORMULA] s-1 distributions of [FORMULA] versus [FORMULA] go about 0.6 ([FORMULA]) above the curves plotted in Fig. 2c for [FORMULA] s-1.

From Figs. 2c and 2d we infer that there is no a big difference (if one does not consider very deep layers adjacent to the convective core) between a state of rotation close to the uniform one and that approaching the asymptotic regime with respect to their ability (or, better to say, disability) to mix chemical elements. In particular, complete mixing of the radiative envelope is reached in neither case. Therefore, searching for an additional angular momentum transport mechanism (for instance, internal gravity waves; Zahn et al. 1997; see also the next section), which could support a state of nearly uniform rotation will hardly help to speed up mixing of chemical elements unless this new mechanism can effectively mix them itself. But it is important to note that even in the most unfavourable considered case of [FORMULA], [FORMULA] s-1 and [FORMULA] years the turbulent diffusion (we have mentioned above that for rotation close to the steady-state one [FORMULA] is much smaller than [FORMULA]) turns out to be fast enough for some mixing, as diffusion-like abundance profiles in the envelope are built up and the surface abundance of 14 N is altered significantly by the end of the star's MS life (Fig. 3). To plot Fig. 3 we solved Eqs. (7 ) by the method and with the input physics described in Denissenkov et al. (1998, the nuclear kinetics network for 26 nuclides) for the fixed temperature and density distributions taken from our present ZAMS models. Note that the surface N abundance has begun to increase only after some delay time required for the diffusion wave to reach the surface (Fig. 3b, see also Talon et al. 1997).

[FIGURE] Fig. 3a and b. Panel a : Abundance distributions ([FORMULA], [FORMULA] is the atomic mass number) of the main CNO nuclides and of He in the radiative envelope of the [FORMULA] star by the end of its MS life are shown for two cases: without mixing (thin lines) and with mixing by turbulent diffusion (the meridional circulation plays an unimportant role here) induced by rotation with [FORMULA] km s-1 (thick lines). The diffusion coefficient profile for the calculations with mixing was taken from the nonstationary solution at [FORMULA] years (Fig. 2a). Panel b : Increase of the surface N abundance with time in the calculations with mixing. The abundance of N begins to grow after some delay time which is required for the diffusion wave to reach the surface

Fig. 4 demonstrates that shortly after the rotation profile has begun to evolve, all the consistency criteria (14-17 ) are fulfilled, and only during the very early evolution we meet a situation when [FORMULA], but we ignored that.

[FIGURE] Fig. 4. An illustration of the fact that the consistency criteria (14-17) are fulfilled shortly after evolution of the rotation profile has begun

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© European Southern Observatory (ESO) 1999

Online publication: November 26, 1998
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