2. Transport of the angular momentum
The equation describing the transport of angular momentum is necessary in order to follow-up the evolution of the angular velocity in stars. In slightly different forms, this equation has been derived for example by Jeans (1929), Tassoul (1978), Chaboyer and Zahn (1992). Written in eulerian coordinates, the equation generally applies to a stationary star. This means that contraction or expansion occurring at any level in an evolving star are not accounted for, although these contribute most, in general, to the changes of the internal rotation. In the following we derive the equation applicable to evolving stars.
A mass element (in polar coordinates), located at the mass level in the star, is conserved in a contraction or expansion, and therefore the lagrangian derivative of its angular momentum is . Using the relation between lagrangian and eulerian expressions, and applying the equation of continuity, we obtain the equivalent form
We have explicited the velocity in its radial and meridional components et . But from now on we shall distinguish between the radial expansion or contraction of the star and the components and of the meridional circulation,
Assuming, as in Zahn (1992), that the rotation depends little on latitude, due to strong horizontal diffusion, we write
with , the horizontal average being taken over the angular momentum:
With such a "shellular" rotation law, the meridional circulation is of quadrupolar type, as it would be for uniform rotation. One has then
Multiplying equation (2.2) by and integrating over , we obtain for the radial diffusion of angular momentum
Note that the change in radius of the given mass shell is now included in the eulerian formulation of this conservation equation. The characteristic timescale for the change of r is in hydrostatic models (in hydrodynamic ones it may even be shorter). The characteristic time associated to the transport of by the circulation is (cf. Zahn 1992)
where is the gravity at the surface. Thus, we clearly have in general (except possibly in very external zones). This inequality means that the term in , which is due to the secular contraction expansion of the star, can be dominant with respect to ; it must therefore be included in evolutionary models.
Another possibility is to use the lagrangian formulation, and to consider r as the coordinate linked to through :
In that case the effects of expansion or contraction are automatically included; such a lagrangian treatment was applied for instance by Talon et al. (1997).
© European Southern Observatory (ESO) 1998
Online publication: June 2, 1998