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Astron. Astrophys. 334, 1000-1006 (1998)

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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 [FORMULA] 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 [FORMULA] (in polar coordinates), located at the mass level [FORMULA] in the star, is conserved in a contraction or expansion, and therefore the lagrangian derivative of its angular momentum is [FORMULA]. Using the relation between lagrangian and eulerian expressions, and applying the equation of continuity, we obtain the equivalent form

[EQUATION]

Assuming that angular momentum is transported only through advection (by the velocity field [FORMULA]) and through turbulent diffusion (of viscosity [FORMULA]), this time derivative is governed by

[EQUATION]

We have explicited the velocity [FORMULA] in its radial and meridional components [FORMULA] et [FORMULA]. But from now on we shall distinguish between the radial expansion or contraction of the star [FORMULA] and the components [FORMULA] and [FORMULA] of the meridional circulation,

[EQUATION]

Assuming, as in Zahn (1992), that the rotation depends little on latitude, due to strong horizontal diffusion, we write

[EQUATION]

with [FORMULA], the horizontal average being taken over the angular momentum:

[EQUATION]

With such a "shellular" rotation law, the meridional circulation is of quadrupolar type, as it would be for uniform rotation. One has then

[EQUATION]

Multiplying equation (2.2) by [FORMULA] and integrating over [FORMULA], we obtain for the radial diffusion of angular momentum

[EQUATION]

Note that the change in radius [FORMULA] 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 [FORMULA] in hydrostatic models (in hydrodynamic ones it may even be shorter). The characteristic time associated to the transport of [FORMULA] by the circulation is (cf. Zahn 1992)

[EQUATION]

where [FORMULA] is the gravity at the surface. Thus, we clearly have in general [FORMULA] (except possibly in very external zones). This inequality means that the term in [FORMULA], which is due to the secular contraction expansion of the star, can be dominant with respect to [FORMULA] ; 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 [FORMULA] through [FORMULA]:

[EQUATION]

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).

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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