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

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3. Description of the model

3.1. Evolutionary models

The evolutionary models for 1, 0.8, 0.6 and 0.5  [FORMULA] have been computed by Forestini (1994). I refer to Bouvier (1997b) for details of the computation of the models. In Fig. 1 I present the evolution of the moment of inertia of both the core and the envelope for each model during the pre-main sequence phase of the star, and in Fig. 2 the evolution of the radii of the star and of the core (left panel), and the variation of the core's mass (right panel).


[FIGURE] Fig. 1. Evolution of the moment of inertia for the core (right panel) and the envelope (left panel). Solid line is for 1 [FORMULA], dashed for 0.8 [FORMULA], dotted for 0.6 [FORMULA] and dotted-dash for 0.5 [FORMULA]

[FIGURE] Fig. 2. Evolution of the star and core radius (left panel), and evolution of the mass variation of the core (right).

3.2. Hypotheses

The model is based on 3 assumptions: internal differential rotation, disk-locking and solar-type wind braking.

H1 differential rotation: it is assumed that the star rotates as a solid-body as long as it is completely convective. After the radiative core develops, the two zones will both rotate as solid-bodies with different angular velocities [FORMULA] (for the core) and [FORMULA] (for the envelope). Angular momentum exchanges between the two zones will then occur. Decoupling hypothesis is a reasonable compromise between solid-body models, and more physical models assuming local conservation of the angular momentum and transport processes in the radiative core.

The exchanges considered here have been described by MacGregor (1991) who suggested that a quantity [FORMULA] of angular momentum is extracted from the (fast) rotating core and is transfered to the (slower) envelope, and is defined by:

[EQUATION]

Where [FORMULA] and [FORMULA] are the moment of inertia for the radiative and convective zones respectively:

[EQUATION]

The quantity [FORMULA] is transfered over a time-scale [FORMULA], called coupling time, introduced as a free parameter of the model. If [FORMULA] is short, the transfer of [FORMULA] would be almost instantaneous and would equilibrate [FORMULA] and [FORMULA], leading to a quasi solid-body rotation.

Two main assumptions of the model presented here are thus that angular momentum exchanges are controlled by a coupling time, and that this coupling time has a fixed value and is not a function of mass, or core and/or envelope velocity, or any other parameter. This is probably a spurious assumption as there are some theoretical evidences that angular momentum transport depends on various physical characteristics of the star (core and/or envelope rotation, mass, depth of the convective zone...). In the case of angular momentum transfer by diffusive mechanisms, transport is a function of the velocity gradient inside the star. Krishnamurthi et al. (1997) modeled the angular evolution of young low-mass stars using angular momentum transport by hydrodynamics mechanisms and their time-scale for transport depends on the rotation rate (it is long for slow rotators and short for rapid rotators). Meridian circulation is induced by the angular momentum loss at the star's surface, and is thus a function of the surface velocity. In the case of internal waves extraction process, Zahn et al. (1997) found that the angular momentum flux depends linearly on the differential rotation.

It is the purpose of this paper to provide constraints on the coupling time, and eventually find a relationship with other stars parameters.

As the core grows, a quantity of material [FORMULA], contained in a thin shell at a radius [FORMULA] and with a velocity [FORMULA] becomes radiative, (see Fig. 2)and the amount of angular momentum which is transfered from the envelope to the core during the time interval dt is:

[EQUATION]

H2 Disk-locking: it is supposed that during the phase when the star accretes material from its surrounding disk, its surface rotational period remains constant. The theoretical basis for this assumption is the magnetic interaction between the star and its surrounding disk. Magnetic field lines shred the disk beyond the corotation radius, and tends to spin the central star down. The assumption that the stars accreting material from their surrounding disk are in a rotational equilibrium state, is supported by the observations of the rotation rates of both CTTS and WTTS. The latter tend to rotate faster than the former (Bouvier et al. 1993, Edwards et al. 1993, Choi & Herbst 1996). Computations from Königl (1991), Cameron & Campbell (1993), or Armitage & Clarke (1995) show that the star quickly reaches a constant angular velocity. This requires the existence of a stellar magnetic field of a few hundred Gauss (500 to 1000 Gauss typically), and accretion rates from 10-8 to 10-7 [FORMULA] [FORMULA]. In this model it is supposed that as long as the star is accreting, its rotational period remains constant. From the moment the disk disappears, called disk lifetime, the star freely evolves.

H3 wind braking: The description used here is the description of angular momentum loss as described by Schatzman (1962), and parameterized by Kawaler (1988): angular momentum loss is a function of angular velocity, mass, mass loss and star radius.

[EQUATION]

The exponent n characterizes the field geometry, and a is the power of the linear dynamo relation [FORMULA]. I follow the suggestion by Kawaler and use [FORMULA], corresponding to an "intermediate" field geometry. As discussed by Charbonneau (1992), a braking-law with a fixed value of the exponent in the velocity term is unable to reproduce the standard model of angular momentum loss from Weber and Davis (1967). He pointed out that the WD model for slow rotators is well fitted with an exponent of 3 and the WD model for rapid rotators with an exponent of 2. Keppens et al. (1995) computed an angular momentum loss law from the WD solar wind model and also found that at low rotation rates the law is consistent with [FORMULA] while for fast rotators it scales [FORMULA] and [FORMULA] in the saturated regime. Barnes and Sofia (1996) also investigated the effect of different braking laws in order to reproduce the ZAMS ultra-fast rotators. They found that ultra-fast rotators could not be reproduced with a Skumanich-type braking law and thus required a change of the exponent at high velocities. They found that a saturated braking law with [FORMULA] led to sufficiently high velocities. They also supposed a change of the magnetic configuration, from a more dipolar form during pre-main sequence to a solar-type form on the main sequence, that would lead to a PMS braking scaling [FORMULA].

Observational constraints for slow rotators comes from Skumanich's relation (1972): the velocity decrease of the MS slow rotators is a power-law of the time [FORMULA], where v represents rotational velocity. This relation leads to the braking law [FORMULA]. For rapid rotators, observations from Mayor and Mermilliod (1991) in young clusters lead to [FORMULA].

In this paper, a Kawaler-type description of the braking law is used, with a three-part parameterization. Slow rotators follow the Skumanich regime, while intermediate rotators follow Mayor-Mermilliod regime. This is consistent with Barnes and Sofia assumption of a change of the magnetic configuration somewhere on the ZAMS, as intermediate rotators are mainly found during PMS. In addition it is supposed that saturation of the braking law occurs for high velocities, corresponding to a saturation of the dynamo-generated surface field for high velocities: [FORMULA]. These braking laws write:

[EQUATION]

Skumanich law: [FORMULA] [FORMULA] [FORMULA], with [FORMULA] = [FORMULA]

[EQUATION]

Mayor-Mermilliod law : [FORMULA] [FORMULA] [FORMULA] and [FORMULA] [FORMULA] [FORMULA]

[EQUATION]

Saturation: [FORMULA] [FORMULA] [FORMULA]

Three parameters are thus required to make a full description of the braking law: [FORMULA], [FORMULA] and [FORMULA]. In Sect. 4I discuss how we constrain these parameters from the observations.

3.3. Equations of evolution

Using the above assumptions, equations of evolution for angular velocity write for the envelope:

if t [FORMULA]: [FORMULA]

where [FORMULA] is the initial velocity of the star.

if t [FORMULA]:

[EQUATION]

and for the core:

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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