Astron. Astrophys. 333, 629-643 (1998)

1. Introduction

The different processes which drive the angular momentum evolution of young low-mass stars are contraction, internal evolution (formation and increase of the radiative core, retreat of the convective zone), interaction with surrounding environment (accretion disk, Bouvier et al. 1993), and angular momentum loss through magnetic winds (Schatzman, 1962).

The stellar contraction and the apparition of the radiative core in the inner parts are at the origin of a density gradient inside the star and hence, local conservation of angular momentum leads to a large velocity gradient with radius. Convective motions enforce a rigid rotation in the envelope and a solar-type wind applies a magnetic braking to the envelope. At ZAMS ages the core then rotates far faster than the envelope. On the other hand, helioseismology tells us that the Sun rotates as a solid-body down to at least (Gough, 1991). Tomczyk et al. (1996) also suggests that the radiative core rotates as a solid-body. To conciliate differential rotation at ZAMS ages inferred from contraction, and quasi-solid rotation at the age of the Sun, it is necessary to suppose redistribution of angular momentum in stellar interiors.

Several processes of angular momentum transport have been proposed to reproduce the surprising rotational profile of the Sun (see Zahn 1996 for a review, and references below). Endal and Sofia (1978) treated angular momentum transfer induced by various instabilities. They introduce two different kind of instabilities: dynamical instabilities whose characteristic time scales are shorter than the evolutionary time scale of the star: convection, dynamical shear and Soldberg-Hoiland instability; and secular instabilities, whose mixing time-scales are comparable to evolutionary time-scales: Eddington circulation (1925), secular shear and Goldreich-Schubert-Fricke instability (Fricke 1968, Goldreich and Schubert 1967). They suppose that convection induces solid-body rotation in the envelope, and that angular momentum transfer in the radiative core can be treated as a pure diffusive process. Pinsonneault et al. (1989) used the same equations of diffusion than Endal & Sofia in a model of angular momentum evolution. They find that the rotational profile is mainly dependent of angular momentum transport in the radiative parts. They manage to reproduce the flat rotational profile down to a radius of 0.4 - 0.5 , and find that a very efficient diffusive process is necessary to reproduce the low velocities of the inner parts.

More recently Chaboyer et al. (1995), used the description of angular momentum transport by Endal & Sofia. and investigated the evolution of rotation and lithium depletion. The model leads to a velocity in the inner parts of the Sun an order of magnitude too large than the observed value. Another problem arises from lithium abundances observations. They find that rotational mixing is necessary to explain lithium depletion in the Sun and young clusters, but they cannot reproduce the lithium dispersion observed in young clusters.

Zahn (1992) suggested that in addition to diffusive processes, meridian circulation, driven by solar-type wind, was as the origin of core angular momentum loss (see also Tassoul and Tassoul in a series of papers, 1995 and references therein). They treated meridian circulation as an advection process. Under the assumption that angular velocity is a function of depth only, it is equivalent to an "hyper diffusion" process. But Matias & Zahn (1997) found that this process was not efficient enough to reproduce solar rotational profile at the age of the Sun.

Mestel, Tayler and Moss (1988) suggested that a primordial magnetic field would penetrate the core and enforce nearly uniform rotation along the field lines. Charbonneau and McGregor (1993) studied different poloidal field geometries and compared the velocity evolution inferred to a solar-type star spin-down on the main-sequence. Very different rotational evolutions are inferred from different field geometries. They found that a poloidal magnetic field geometry in which the magnetic field is restricted to the radiative parts leads to a velocity braking on the main sequence in agreement with the observations and is also in agreement with the internal rotational profile of the Sun. But there is no proof that such a configuration is stable and remains long enough to lead to an efficient braking.

Another process has been suggested to be at the origin of angular momentum extraction: internal (or gravity) waves (Press, 1981, Schatzman 1993). These waves, produced in the radiative zone by turbulent motions in the convective zone, would lead to an efficient braking of the core. Calculations suggest that angular momentum extraction in the solar interior would occur over a time scale of 10⁷ yr (Zahn et al. 1997, Kumar & Quataert 1997).

In the past years, different models were developed to model the different processes that rule angular momentum evolution during PMS and MS.

McGregor & Brenner (1991) introduced a simple parameterized model of redistribution of angular momentum between the core and the envelope, both supposed to rigidly rotate to explain early MS evolution of solar-type stars. They found that a coupling time-scale of 10⁷ yr was consistent with rapid spin-down of rapid rotators on the ZAMS.

Li & Collier Cameron (1993) investigated rotational evolution from ZAMS ages for solar-type stars. They supposed that the convective envelope applied a magnetic torque upon the - rigidly - rotating core: - . They found that only a weak coupling, characterized by a low value of the exponent , and a large value of the ratio of the coupling time-scale to the braking time-scale was required to fit the observations of rapid rotators both in the Pleiades and Hyades clusters.

In another paper Collier Cameron and Li (1994) investigated the spin down of ZAMS stars without core-envelope decoupling - solid-body rotation - and found that appropriate Weber-Davis wind model combined with a simple linear dynamo which saturates at high rotation rates was also consistent with observations from ZAMS ages. They also introduce a mass-dependence of the saturation rate : higher masses require higher saturation rates.

Keppens et al. (1995) modeled evolution from T Tauri phase to MS. Their model treated angular momentum loss by a stellar wind, disk-locking, and angular momentum transport from the radiative interior to the convective envelope using McGregor & Brenner description. They found that a short coupling time of 10⁷ yr and a dynamo saturated law for velocities larger than = 20 were necessary to explain the large spread of velocities among young clusters solar type stars, and rapid spin-down of rapid rotators on the ZAMS. They studied the evolution of a distribution at T Tauri ages with various assumptions in the initial velocity distribution, coupling time scale, and disk-lifetimes distribution. They found that the large velocity spread in young clusters could only be explained by an initial bimodal velocity distribution - the consequence of a bimodal disk lifetime distribution - and a mass spread, with 0.8 and 1 stars in equal proportions. But at the age of Alpha Per and the Pleiades, observed proportion of slow rotators in the velocity range 0 - 10 km are 30% and 50% (respectively), while the model gives fractions lower than 5% in both clusters.

Barnes and Sofia (1996) focused on the existence of a population of ultra fast rotators among the young clusters Alpha Per and the Pleiades. They computed evolutionary models from T Tauri phases using a Kawaler-type braking law (see Sect.  3.2). They find that the Skumanich braking law () does not allow the existence of rapid rotators and investigated the effect of two different braking laws. The first one supposes a saturation of the momentum loss leading to a braking scaling as , and the second suggests a change of the magnetic configuration from a dipolar form during pre-main sequence to a more solar form on the MS, and thus leads to a braking law scaling as . Comparisons with the observations of ultra-fast rotators tend to favor the first hypothesis, but they conclude that a combination of the two phenomena could be a better description of angular momentum evolution during pre-main sequence and main sequence phases. They also found that lower mass models require lower saturation thresholds.

In a recent paper, Krishnamurthi et al. (1997) investigated PMS and MS angular momentum evolution and compared solid-body models to models with internal differential rotation. They use the same diffusive processes as Chaboyer et al. (1995) for the treatment of angular momentum transport in the radiative parts. They find that a saturated braking law, with a mass-dependent value of the saturation rate is convenient to explain the mass dependence of the ultra-fast rotators (UFR's) phenomenon on the ZAMS. They conclude that the solid-body model requires a too large proportion of disks surviving longer than 20 Myr and thus, cannot reproduce the large proportion of slow rotators in young clusters, and that differential rotation is more convenient to reproduce the distributions of rotational velocities in young open clusters for masses in the 0.5 - 1.2 range. They compare the fraction of slow rotators observed in Alpha Per and the Pleiades and find that there is a larger fraction of slow rotators in the latter. But they make their statistic with uncomplete sample and unresolved (Allain et al. 1997, Queloz et al. 1997a). They find that different disk-lifetimes distributions are required to fit the velocity distributions in different clusters and discuss the possibility of cluster-to-cluster intrinsic variations (e.g. of the initial conditions). They put a lower limit on the characteristic time-scale for core-envelope coupling of 70 to 100 Myr to explain the existence of a large proportion of slow rotators in young clusters.

Bouvier et al.(1997b) modeled the angular momentum evolution of stars in the mass range 0.5-1.1 , during the PMS and MS phases, with the assumptions of solid-body rotation, disk-locking and saturated wind braking law. They explore the evolution of a population of stars that appear in the T Tauri phase with a Gaussian-like period distribution. They show that solid rotation with a mass-dependent saturation rate, and a disk lifetime distribution which is a function of log(t), could reproduce the observed velocity distributions at different ages, and different masses. But the model did not try to reproduce the fraction of very slow rotators at ZAMS ages (with velocities lower than 5 km ), nor its evolution on the main sequence. Finally, this model requires that 10% of the stars are still coupled to their disk at an age of 20 Myr, and that the maximum disk-lifetime be 40 Myr.

In this paper, I retain the same hypothesis of disk-locking but replace the solid-body assumption by a core-envelope decoupling hypothesis. I use the prescription by McGregor and Brenner: whatever may be the physical process of angular momentum transport in the radiative interior of the stars, it is supposed that the core rotates as a solid-body and that the exchanges between the radiative core and the convective envelope are controlled by a characteristic time-scale called coupling time, , introduced as a free parameter of the model. The coupling time is supposed to be constant all over the evolution of the star, from T Tauri phase, to the age of the Sun.

The aim of this paper is to investigate the effect of differential rotation on the angular momentum evolution of young low-mass stars. By this work, and in the light of new observational constraints, I wish to bring new insight on angular momentum transport processes, and especially on time scales involved in this processes.

In Sect. 2, I describe the constraints recent observations shed on rotation. In Sect. 3, I briefly describe the model and the different assumptions. In Sect. 4, I test the effects of the different parameters. and in Sect. 5, I investigate the specific effect of the coupling time, and present the evolution of a star of different masses, and different disk life times, for different coupling times.

© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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