## 3. Description of the model## 3.1. Evolutionary modelsThe evolutionary models for 1, 0.8, 0.6 and 0.5 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).
## 3.2. HypothesesThe model is based on 3 assumptions: internal differential rotation, disk-locking and solar-type wind braking.
The exchanges considered here have been described by MacGregor (1991) who suggested that a quantity of angular momentum is extracted from the (fast) rotating core and is transfered to the (slower) envelope, and is defined by: Where and are the moment of inertia for the radiative and convective zones respectively: The quantity is transfered over a time-scale , called coupling time, introduced as a free parameter of the model. If is short, the transfer of would be almost instantaneous and would equilibrate and , 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 ,
contained in a thin shell at a radius and with
a velocity becomes radiative, (see
Fig. 2)and the amount of angular momentum which is transfered
from the envelope to the core during the time interval
The exponent 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 , where 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: . These braking laws write:
Three parameters are thus required to make a full description of the braking law: , and . In Sect. 4I discuss how we constrain these parameters from the observations. ## 3.3. Equations of evolutionUsing the above assumptions, equations of evolution for angular velocity write for the envelope: if t : where is the initial velocity of the
star. if t : and for the core: © European Southern Observatory (ESO) 1998 Online publication: April 20, 1998 |