Astron. Astrophys. 321, 465-476 (1997)

2. The rotating stellar models

2.1. The computational method

Rotation may affect the equations of stellar structure in four ways (cf e.g. Endal & Sofia 1976) :

1. Centrifugal forces reduce the effective gravity at any point not on the axis of rotation.
2. Since the centrifugal force is not, in general, parallel to the force of gravity, equipotential surfaces are no longer spheres.
3. Because the radiative flux varies with the local effective gravity (the von Zeipel effect, 1924), the radiative flux is not constant on an equipotential surface.
4. Rotation may induce some mixing processes.

In the present models, we consider these four effects.

In general, to incorporate into the stellar structure equations, the effects of rotation, the method devised by Kippenhahn & Thomas (1970) is used (see e.g. Endal & Sofia 1976; Pinsonneault et al. 1990; Fliegner & Langer 1995; Chaboyer et al. 1995). The main idea of this method is to replace the spherical stratification which prevails in non rotating stars by a rotationally deformed stratification. The problem can be kept one dimensional in the case where the effective gravity (i.e. the gravity decreased by the effect of the centrifugal force) can be derived from a potential (conservative case). Indeed, in this case the pressure P and the density keep a constant value on an equipotential. The temperature T remains also constant if the chemical composition is homogeneous on equipotentials. Thus in a rotating star, the stellar structure equations almost keep the same form as in a non rotating star if they are written on equipotentials

It is quite straightforward to introduce this procedure into a stellar evolutionary code. Unfortunately it applies only in the case a conservative potential exists i.e. in case a cylindrical symmetry for the angular velocity distribution prevails (Poincaré-Wavre theorem, see Tassoul 1978). As already stated in the introduction above, the theory of Zahn (1992) assumes that turbulence is anisotropic, with a stronger transport in the horizontal directions than in the vertical one. This enforces a rotation rate which, to first approximation, remains constant on isobar ("shellular" rotation). Obviously such a rotation law does not fall into the conservative case and the Kippenhahn & Thomas method cannot be used. In the Appendix, we show however that it is possible to adapt the Kippenhahn & Thomas method to the case of a "shellular" rotation law and we describe in details the way we have implemented the effects of shellular rotation in the stellar structure equations.

2.2. The physical ingredients

The solar initial chemical composition (mass fraction of the heavy elements ), the nuclear reaction rates, the mass loss rates are as in the previous grids of stellar models from the Geneva group (see Schaller et al. 1992). The new OPAL radiative opacities from Iglesias & Rogers (1993) are used, complemented at temperatures below 6000 K with the atomic and molecular opacities by Kurucz (1991).

We considered the Ledoux criterion for convection without semiconvective diffusion. Let us emphasize here that the main conclusion of this paper concerning the inhibiting effect of the molecular weight gradient is not depending on this particular choice (see section 4).

In rotating stars some mixing is expected caused by the meridian circulation (Eddington 1925, 1926; Vogt 1925). This mixing process not only transports the chemical species but also advects angular momentum, the conservation of which induces differential rotation. In the nearly inviscid stellar material, turbulent motion will appear as a result of various instabilities produced by differential rotation. Among these instabilities, the shear instability plays certainly the most important role (Spiegel & Zahn 1970; Zahn 1974, 1975). We have thus the following logical links between these various effects: rotation implies meridian circulation, which in turn implies differential rotation which produces shear instabilities. How the shear instability will interact with the meridian circulation pattern is not an easy problem. The main difficulty comes from the impossibility to describe from first principles the properties of the turbulent motions sustained by the shear. However Zahn (1992), starting from the unique and reasonnable assumption that the turbulence is more vigourous in the horizontal direction than in the vertical one, succeeded in establishing in a coherent way the equations for the transport of the chemical species and of the angular momentum resulting from these hydrodynamical processes. Let us note here that the above conjecture on the anisotropy of the turbulence leads to a "shellular rotation law" which seems to be realised in the interior of the Sun (Tomczyk et al. 1995).

Following Zahn (1992), the transport of the chemical species is treated with a diffusion equation. The diffusion coefficient is composed of two terms: a term accounting for the effects of the vertical turbulence induced by the shear and a term describing the concommitant effects of the circulation and of the horizontal turbulence induced by the shear. Turbulent motions are sustained when the Reynolds number is above a critical value , of the order of 3000 (l is the characteristic size of an eddy, v its velocity and the viscosity). For current inner conditions in massive stars, this requirement is generally fullfilled (Maeder & Meynet 1996). More constraining is the Richardson criterion (see Chandrasekhar 1961) which may completely suppress the effects of shear mixing. Indeed, this criterion states that a sufficient condition for stability (i.e. for no mixing) is that the Richardson number be superior to 1/4. As we shall see below, this Richardson criterion plays an important role in deciding which region of a star can or cannot be mixed by rotational turbulence. Generally this stability criterion is fullfilled in stellar regions where there is a molecular weight gradient (µ-gradient), even quite modest, and thus no mixing occurs. However the Richardson criterion as it is classically given (see e.g. Zahn 1992) needs some modifications in order to take into account:

1) In a more consistent way the radiative losses and the vertical µ-gradients.

2) The coupling between the shears and their effects on the T-gradient.

3) The diffusion both in the radiative and semiconvective zones.

Maeder (1995a) has established the Richardson criterion taking into account the effects indicated on point 1). Maeder & Meynet (1996) have pursued its improvement by considering points 2) and 3). In this paper we shall consider the Richardson criterion and the associated diffusion coefficient as they have been established by these last authors. It can be expressed as

where K is the thermal diffusivity and is the maximum Peclet number (divided by a factor 6) of the turbulent eddies. is related to and to the angular velocity gradient through the expression:

with

We consider here the case of "no wind" described by Zahn (1992, see this reference for the meaning of the other variables and for a thorough discussion of these expressions).

For massive stars and for asymptotic regime (see Zahn, 1992), the advection of angular momentum by the meridional currents are compensated by the diffusion of angular momentum through shear flow. In this case one can consider that the angular momentum in a given shell remains constant with time (see Appendix A.4 for the expression of the specific angular momentum). We shall further discuss the consequences of this simplifying assumption.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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