Astron. Astrophys. 335, 959-968 (1998)
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 6900 K still have their
initial velocities (when compared to the velocity distribution
observed in younger clusters) while stars cooler than
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.
Fig. 2. (top ) The size of the convective envelope () 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/ (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
( 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 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 6900 K
According to the observations, stars hotter than
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 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.
Fig. 3. Interior Li profile at the age of the Hyades for two different masses (1.55 and 1.45 ) and three different initial velocities (50, 100 and 150 km/s). The 1.55 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 model is strongly spun down and Li destruction is large for all values of the initial velocity.
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 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 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.
Fig. 5. Interior He profiles at the age of the Hyades, for the same models and rotation velocities as in Fig. 3.
Stars with 66006900
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 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.
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 6900 K conserves its global angular momentum during its main sequence life while for models with 6500 6900, angular momentum is lost at the surface. Interior redistribution of momentum takes place only through rotational mixing; the coolest model ( 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.
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
( 3.5 106K
instead of 2.5
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.
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 64006600
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 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
Stars cooler than 6400 K
The maximum of the Li abundance at 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
66006200 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 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.
© European Southern Observatory (ESO) 1998
Online publication: June 26, 1998