## 3. Further assumptions and simplificationsAs 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 and for the initial hydrogen and heavy element relative mass abundances , 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
into the radiative envelope we have applied the
inner boundary conditions for Eq. (4 ) at a radius
, a little above the convective core border
, which allowed us to avoid a singularity when
calculating Relation (10 ) results from the following assumptions: ( The outer boundary conditions applied near the stellar surface were These are direct consequences from the above assumption ( As the initial condition the uniform rotation law was used, where a value of the quantity consistent with other assumptions was calculated as explained below. First, for a specified value of the surface angular velocity an asymptotic solution of Eq. (4) was found by solving the stationary 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
To simplify calculations we have assumed that the radiative
diffusivity The above two simplifications allow us to put and and instead of (5 ) to make use of the approximate formula 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 where is the ordinary (i.e. molecular plus radiative) viscosity and the critical Reynolds number. If condition (14) is not fulfilled then turbulent motions responsible for the turbulent diffusion transport will dissipate through viscous friction. must be fulfilled to justify that effects of heat transfer by turbulence are ignored. is our simplifying assumption which together with the assumption of
a zero This is Zahn's basic assumption which makes it possible to formulate the whole problem as one-dimensional. © European Southern Observatory (ESO) 1999 Online publication: November 26, 1998 |