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Astron. Astrophys. 341, 181-189 (1999)

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3. Further assumptions and simplifications

As said in the introduction, we have used ZAMS model stars and have not followed their evolution because our objective has been to calculate stars' approaching the steady-state rotation which was found to happen rather quickly compared to the MS life-times. Two ZAMS models of [FORMULA] and [FORMULA] for the initial hydrogen and heavy element relative mass abundances [FORMULA], [FORMULA] have been prepared. The stellar evolution code we used for this is an updated version of the same one already used by Denissenkov & Weiss (1996). The modifications, which include the employment of the latest OPAL opacity (Rogers & Iglesias 1992; Iglesias & Rogers 1996) and equation of state (Rogers et al. 1992) tables, have been described in Schlattl et al. (1997). With these models all necessary stellar structure parameters entering Eqs. (1 -8) and independent of the rotation state were calculated once and considered to be fixed furtheron.

We did not modify the equation of hydrostatic equilibrium to take into account effects of rotation on the stars' structure because according to Meynet & Maeder (1997) these are very small.

Assuming that convection penetrates up to a level [FORMULA] into the radiative envelope we have applied the inner boundary conditions for Eq. (4 ) at a radius [FORMULA], a little above the convective core border [FORMULA], which allowed us to avoid a singularity when calculating U with formula (1). We have used the following inner boundary conditions:

[EQUATION]

where

[EQUATION]

Relation (10 ) results from the following assumptions: (i ) the convective core rotates as a rigid body due to a large turbulent viscosity of convective origin; (ii ) the star conserves its angular momentum, i.e. mass loss is ignored (we discuss this point in more detail in the conclusion).

The outer boundary conditions applied near the stellar surface were

[EQUATION]

These are direct consequences from the above assumption (ii ).

As the initial condition the uniform rotation law [FORMULA] was used, where a value of the quantity [FORMULA] consistent with other assumptions was calculated as explained below.

First, for a specified value of the surface angular velocity [FORMULA] an asymptotic solution of Eq. (4) was found by solving the stationary equation

[EQUATION]

which is deduced from (4 ) by assuming that the partial derivative with respect to time equals zero (cf. Zahn 1992; Urpin et al. 1996; Talon et al. 1997). The boundary problem for the third order ODE (12) (the boundary conditions being now reduced to the first row from (9 ) and (11) supplemented with the requirement that [FORMULA] at the surface) was solved by a shooting method. After that the appropriate value of [FORMULA] was chosen to satisfy the angular momentum conservation law.

To simplify calculations we have assumed that the radiative diffusivity K is much greater than the turbulent diffusion coefficient [FORMULA]. Another simplification was ignoring energy and µ-gradient producing nuclear reactions in the radiative envelope. Let us emphasize again that we did not follow stellar evolution and therefore our "radiative envelope" was actually a zone stretching from a position of the convective core border at its maximum extent up to the stellar surface. The standard stellar evolution calculations show that during star's MS life the convective core shrinks and leaves behind itself a zone of variable chemical composition which becomes later a semiconvective zone in massive MS stars. We do not discuss here the rather difficult problem of how the additional mixing succeeds to overcome the µ-gradient barrier (cf. Maeder & Zahn 1998) but instead address the problem of whether the mixing is fast enough to bring products of nuclear reactions from the radius [FORMULA] to the surface while the star is on the MS. In this respect our second simplification is permissible.

The above two simplifications allow us to put [FORMULA] and [FORMULA] and instead of (5 ) to make use of the approximate formula

[EQUATION]

Solutions of the full nonstationary problem (Eqs. (1-4 ), (6), (8 -11) and (13 )) were obtained by a Newton relaxation method.

Before coming to a discussion of results of the calculations we are going to write down consistency criteria ensuring that Zahn's scheme does work. These must be checked a posteriori .

[EQUATION]

where [FORMULA] is the ordinary (i.e. molecular plus radiative) viscosity and [FORMULA] the critical Reynolds number. If condition (14) is not fulfilled then turbulent motions responsible for the turbulent diffusion transport will dissipate through viscous friction.

The second inequality

[EQUATION]

must be fulfilled to justify that effects of heat transfer by turbulence are ignored.

[EQUATION]

is our simplifying assumption which together with the assumption of a zero µ-gradient transforms (5 ) into (13).

[EQUATION]

This is Zahn's basic assumption which makes it possible to formulate the whole problem as one-dimensional.

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© European Southern Observatory (ESO) 1999

Online publication: November 26, 1998
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