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Astron. Astrophys. 335, 959-968 (1998)

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4. The efficiency of rotational mixing in F-stars

4.1. Constraints from the observed surface rotation in galactic cluster F stars

As can be seen in Fig. 1, the stars of the Li dip are peculiar as far as their rotational history is concerned. From the observational data, one may conclude that the physical processes responsible for the surface velocity are different or operate with different time scales when one goes to lower effective temperature: stars hotter than [FORMULA] 6900 K still have their initial velocities (when compared to the velocity distribution observed in younger clusters) while stars cooler than [FORMULA] 6400 K have been very efficiently spun down at the age of the Hyades (700 Myr).

This behavior is linked to the variation of the thickness of the external convection zone (Fig. 2 top). Indeed, the hottest stars have only a very shallow surface convection zone which is not an efficient site for magnetic generation via a dynamo process. The coolest stars have a deeper surface convection zone, thus sustaining a strong magnetic field which spins down the outer layers efficiently. The rapid diminution with increasing effective temperature in the efficiency of magnetic braking observed for stars of the Li dip is a clear signature of the rapid decrease in mass of the envelope convection zone in stars of the corresponding effective temperature. Let us note that the diminution of the moment of inertia of the convective envelope as the effective temperature increases implies an even more drastic change in the magnitude of the magnetic torque than the variation of the surface velocities indicate.

[FIGURE] Fig. 2. (top ) The size of the convective envelope ([FORMULA]) at the age of the Hyades is plotted as a function of the stellar effective temperature. The filled dots represent standard models (i.e., non rotating); the open dots correspond to rapidly rotating models (150 km/s, not spun down at the age of the Hyades). Fast rotators have lower effective temperatures and deeper convective envelopes than the standard models. (bottom ) Rotational velocities of our models at the age of the Hyades (filled dots; see Table 1). Also shown are the measured projected velocities times 4/[FORMULA] (open dots).

We calculate Li destruction in models of different stellar masses within the theoretical framework described in Sect. 3. We use the statistical study of rotation velocities in the Hyades performed by Gaigé (1993) in order to estimate the spin down associated to stars of different masses: we take an initial velocity of 100 km/s that corresponds to the mean velocity of hot stars ([FORMULA] K) and that is consistent with velocities of cooler stars measured in younger clusters. The resulting velocity at the age of the Hyades (Fig. 2 bottom) corresponds to the average value for stars of a given effective temperature. We also calculate the dispersion expected in the Li abundances from different rotational histories using the [FORMULA] values from Gaigé's study. The corresponding initial velocities are 50 and 150 km/s. The main characteristics of the stellar models, together with the rotation velocities and the lithium and beryllium depletion factors at different ages (corresponding to the ages of the clusters shown in Figs. 6 and 7) are given in Table 1.


Table 1. Characteristics of the stellar models, rotation velocities and lithium and beryllium depletion factors at the ages of the considered galactic open clusters. The models are computed with [Fe/H] of the Hyades (+0.12)

4.2. The depletion on the blue side of the Li dip

Stars hotter than [FORMULA]6900 K
According to the observations, stars hotter than [FORMULA]6900 K are not slowed down by a magnetic torque. Since Eq. (1) admits a stationary solution, those stars soon reach a regime with no net angular momentum flux, in which meridional circulation and shear turbulence counterbalance each other. The weak mixing resulting from these processes is just sufficient to counteract the effect of microscopic diffusion, except in the slowest rotators in which its signature is visible (cf. Fig. 3 top and Fig. 4 top for the shape of the diffusion coefficients in the 1.55[FORMULA] model). In our calculations of microscopic diffusion, only gravitational settling was included whereas in these stars, lithium may be supported by radiative acceleration (Michaud 1986). In improved radiative force calculations, Richer et al. (1997) showed however that the actual force on Li critically depends on the helium and metals abundance; in the dip region, gravitational settling of helium is needed for lithium to be supported against gravity. As can be seen in Fig. 5, the mixing strongly slows down the diffusion of helium in the external layers of the star. This may lead to a significantly reduction of the radiative acceleration on Li on the hot side compared to previous estimations. In any case, the main effect is that turbulence considerably reduces microscopic diffusion, and leads only to small variations in the surface abundances.

[FIGURE] Fig. 3. Interior Li profile at the age of the Hyades for two different masses (1.55 and 1.45 [FORMULA]) and three different initial velocities (50, 100 and 150 km/s). The 1.55 [FORMULA] model is not spun down and conserves its initial angular momentum; only the slower model feels the effect of microscopic diffusion, while rotation induced mixing dominates below the convection zone of the fastest models. The 1.45 [FORMULA] model is strongly spun down and Li destruction is large for all values of the initial velocity.

[FIGURE] Fig. 4. Interior profile of the diffusion coefficient at the age of the Hyades and for an initial velocity of 100 km/s. The wide full line represents the total diffusion coefficient, the thin full line, the turbulent diffusion coefficient (cf. Eq. 3) and the dashed line, the effective diffusion coefficient (cf. Eq. 4). The 1.55 [FORMULA] model is in the asymptotic regime, and it is turbulence which dominates the transport of chemicals whereas the transport of angular momentum by turbulence and meridional circulation is of equal magnitude and opposite signs. For the 1.45 [FORMULA] model, while the transport of angular momentum is done mainly by the circulation, the transport of chemical is dominated by the former only close to the surface.

[FIGURE] Fig. 5. Interior He profiles at the age of the Hyades, for the same models and rotation velocities as in Fig. 3.

Stars with 6600[FORMULA]6900
In this effective temperature range, a weak magnetic torque slows down the outer layers of the stars. As the magnetic torque increases with diminishing temperatures, meridional circulation has much more angular momentum to transport to the surface, leading to a larger destruction of Li (cf. Fig. 3 bottom and Fig. 4 bottom for the shape of the diffusion coefficients in the 1.45[FORMULA] model). Again, Li is transported by meridional circulation, shear turbulence and microscopic diffusion. Let us note however that here, the gap is not shaped by element segregation, as in Michaud (1986) but by the properties of braking and angular momentum transport. Furthermore, since here Li is destroyed and not barely hidden below the convection zone, no dredge-up of this element is expected due to the deepening of the external convection zone when the stars will leave the main sequence. This is in agreement with the absence of Li detection in Balachandran's (1995) sub-giants.

The predictions for the lithium abundance at different ages are given in Figs. 6 and 7, and compared to observations in galactic clusters. We also show predictions for models with [Fe/H]=-0.15 in Fig. 7, in order to take into account the metallicity differences between the various clusters. At the age of the Hyades, rotational mixing described in Sect. 3 perfectly explains the shape of the blue side of the Li dip, as well as the observed dispersion. This clearly indicates that, in this effective temperature range, the process which participates to the transport of angular momentum in the Sun is not yet efficient.

[FIGURE] Fig. 6. Comparison of the models (filled dots and triangle, pluses and minuses) with the observations in the Hyades (open dots and triangles). The model with [FORMULA] 6900 K conserves its global angular momentum during its main sequence life while for models with 6500 [FORMULA] 6900, angular momentum is lost at the surface. Interior redistribution of momentum takes place only through rotational mixing; the coolest model ([FORMULA] 6450 K) is computed assuming solid body rotation. The filled dots correspond to numerical calculations performed with an initial velocity of 100 km/s, the pluses to an initial velocity of 150 km/s, and the minuses to initial velocity of 50 km/s. The corresponding rotational velocities at the age of the Hyades are shown in Fig. 2 (see also Table 1). Numerical calculations are performed using a value of [Fe/H]=+0.12, i.e., that of the Hyades.

[FIGURE] Fig. 7. Shape of the Li dip at the age of different galactic clusters. Symbols are as in Fig. 6. Filled squares denote calculations made for a metallicity of [Fe/H]=-0.15 and an initial velocity of 100 km/s. Typical observational error bars are shown when available.

We computed the effect of rotational mixing on beryllium, which burns at a slightly higher temperature than lithium ([FORMULA] 3.5 [FORMULA] 106K instead of [FORMULA] 2.5 [FORMULA] 106K). At the age of the Hyades, the surface beryllium abundance has diminished by as much as a factor 5 in the center of the dip. This is in agreement with the observations by Boesgaard & Budge (1989) in the Hyades and by Stephens et al.(1997) in field stars, as can be seen in Fig. 8. However, the comparison with Stephens' data is marginal because of the inhomogeneity in metallicity and evolutionary status of their sample. New observational data, with modern detectors, would be necessary for beryllium in more stars of the Li dip.

[FIGURE] Fig. 8. Logarithm of the lithium to beryllium ratio as a function of the effective temperature. The predictions are given with the same symbols as in Fig. 6. The asterisks and inverted triangles correspond to the observations and upper limits in the Hyades (Boesgaard & Budge 1989), and the square and triangles to observations and upper limits in field stars with [Fe/H] between +0.09 and -0.39 (Stephens et al. 1997)

4.3. The red side of the Li dip

Stars with 6400[FORMULA]6600
On the red side of the dip, the magnetic torque strengthens as the convective zone grows (Fig. 2). If we assume there that all the momentum transport is assured by the wind-driven meridional circulation, and if we keep the same parameters that explain the Li abundances on the blue side of the dip as well as the chemical anomalies in more massive stars, we obtain too much lithium burning compared to the observations in this cool effective temperature domain (cf. Fig. 6, model with [FORMULA] 6550 K). Even though a different calibration of the free parameters may lead to the observed lithium abundances, it would not change the internal rotation profile which is known to be inconsistent in the solar case.

We rather propose that the red side of the dip corresponds to a transition region where some other physics for angular momentum transport, which is known to be present in the Sun, starts to become efficient. In that case, the magnitude of both the meridional circulation and shear turbulence is reduced, as well as the Li depletion by rotation-induced mixing. We suggest that this increase of efficiency is linked to the growth of the surface convection zone (see Fig. 2 top). This efficient transport mechanism for angular momentum could be due to gravity waves or to a magnetic field in the radiative interior (see references given in the first part).

A complete description of the efficient mechanism for the transport of angular momentum is required in order to calculate self-consistently the Li destruction in this region, and has not been attempted here.

Stars cooler than [FORMULA] 6400 K
The maximum of the Li abundance at [FORMULA] 6400 K indicates that the "extra" process reaches there its full efficiency, which must be sufficient to lead to solid body rotation in the Sun (at the solar age). We stress once more that a self-consistent calculation of Li destruction requires a description of that mechanism. It is however possible to estimate the lower limit of Li destruction by calculating the meridional circulation present in solid body rotators (filled triangles in Figs. 6, 7 and 8); this must be viewed only as an upper limit on the Li abundance for stars of this domain.

Two important remarks have to be done on the transition region.

Firstly, it was shown by Balachandran (1995) that the location of the Li dip in different clusters depends on the effective temperature on the ZAMS and is thus independent of metallicity. Our explanation is entirely consistent with this observation 1 since for low mass stars, the size of the external convection zone is directly related to the effective temperature. It is then the former which controls magnetic activity (and thus, the spin-down) as well as the onset of the "efficient" mechanism for angular momentum transport.

Secondly, the measures of Li abundances in the Hyades stars of 6600[FORMULA]6200 exhibit a large dispersion while it is much smaller in cooler stars. This can be again linked to different rotational histories of stars of the same mass. Indeed, when these are young, the fast rotators have a larger convective zone than their slower counterparts (see Table 1 and Fig. 2 top). Therefore, in the early stages meridional circulation could be lower than expected, leading to a smaller Li destruction. Even though the magnitude of this change of [FORMULA] may seem small, it is not negligible compared to the size of the transition region. A detailed description of angular momentum transport by the "efficient" process is however required in order to quantify the magnitude of this effect.

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998