Massive main-sequence (MS) stars, i.e. stars with producing He at the expense of H in their convective cores, are observed as early B- or O-type stars. During the last two decades observational data have accumulated (see, for instance, a review paper of Lyubimkov (1996)) indicating that some additional mixing (in the sense that it is not included in the standard stellar structure and evolution calculations) is active in these stars. The most convincing arguments in favour of the existence of such mixing in massive MS stars are measurements of increased N and He abundances (sometimes accompanied by a deficit of C) in atmospheres of the OB-stars. Lyubimkov (1984, 1989) was the first to point out that atmospheres of a majority of (presumably) single early B-type MS stars were enriched in N and even in He and that those enrichments seemed to correlate with the stellar age and mass. A more recent re-analysis carried out by Gies & Lambert (1992) confirmed Lyubimkov's results on N.
Considerable He overabundances (up to three times the solar He abundance) were also found in luminous OB-stars (most of them are still on the MS) by the Munich group (Herrero et al. 1992). These stars are on average more massive than the early B-stars studied by Lyubimkov. The He abundance anomalies in the OB-stars were shown to be accompanied by mass discrepancies. It turned out that masses of stars derived from comparison of their position in the HR diagram with theoretical evolutionary tracks (the so-called "evolutionary masses") were systematically higher than "spectroscopic masses" estimated from stellar spectra and, independently, from the radiation-driven stellar wind theory.
Langer (1992), Weiss (1994) and Denissenkov (1994) proposed that both, helium and mass, discrepancies might be attributed to additional mixing (of as yet unknown nature) operating in the radiative envelopes of the OB-stars. Stellar material enriched in He becomes more transparent for radiation and this results in an increase of star's luminosity (, where µ is the mean molecular weight (Kippenhahn & Weigert 1994)). As a consequence, the evolutionary mass can be overestimated. It should be noted that in his next publication Herrero (1994) reduced the previousely announced mass discrepancies, however, without changing values of the He overabundances.
A massive MS star has a convective core surrounded by a radiative envelope. It is important to recall that almost no He is produced outside this convective core (see Fig. 3a below). Therefore, any mechanism of additional mixing must provide it with the ability to penetrate the core and, of course, to operate fast enough for material from deep layers to reach the atmosphere during the OB-stars MS life-times.
The MS OB-stars are known to be very fast rotators, therefore, a natural idea is to connect additional mixing in their radiative envelopes with rotation. Several years ago Zahn (1992) elaborated an original scheme describing how the mixing might be initiated and sustained in a radiative zone of a single non-magnetic rotating star. His only assumption has been that turbulence induced by various instabilities associated with star's differential rotation is highly anisotropic. The resulting turbulent viscosity has a horizontal component strongly dominating over a vertical one. Among the instabilities induced by rotation Zahn has distinguished shear instability as possessing the shortest development time-scale, horizontal and vertical shear flows being naturally produced by the classical Eddington-Sweet meridional circulation (Eddington 1925; Vogt 1925; Sweet 1950) when it is redistributing angular momentum.
The basic assumption of horizontally dominating turbulence ensures that the star settles in a state of "shellular" rotation with the angular velocity depending only on the distance from the center r. Zahn (1992) derived an expression for appropriate for shellular rotation. He also obtained an equation governing the transport of angular momentum by meridional circulation competing with turbulent diffusion, the latter being generated by shear instabilities. The stronger turbulence in the horizontal direction has also to be taken into account when one considers a redistribution of chemical elements by the meridional circulation. Chaboyer & Zahn (1992) have shown that it results in a reduction of the advective flow which carries a tracer with the meridional circulation, an effect called "horizontal erosion" by them.
Since 1992 several important modifications have been introduced to Zahn's original scheme: (i ) transport by turbulent diffusion arising from the horizontal shear instability is no longer considered because shear flows on the level surfaces are thought to be effectively hindered by the strong turbulent viscosity in the horizontal direction (Zahn 1997); (ii ) a new expression for the vertical component of the turbulent viscosity which takes into account the horizontal erosion and radiative leakage has been proposed by Talon & Zahn (1997); (iii ) more recently, Maeder & Zahn (1998) developed a modified scheme which allows for the evolution of a star, i.e. for changing profiles of a star's structural parameters, and offers a solution of the problem of simultaneous treatment of meridional circulation and semiconvection.
Zahn's scheme gives a self-consistent solution of the problem of rotationally induced mixing in stellar radiative zones in the sense that the rotation profile is no longer chosen arbitrarily (for instance, const, as was often assumed before) but instead is let to adjust itself with time as the angular momentum gets redistributed inside the star. A similar algorithm was used earlier by Endal & Sofia (1976), Pinsonneault et al. (1989) and, recently, by Fliegner et al. (1996), but they incorrectly described the angular momentum transport by meridional circulation as a purely diffusive process (Zahn 1992).
Recently, Zahn's scheme has been applied to study stellar evolution with rotation of 9, 20, 40 and 60 MS stars by Meynet & Maeder (1997) and to calculate the He and CNO abundance profiles built up by meridional circulation and turbulent diffusion in a 9 star by Talon et al. (1997). However, in those works approaching the steady-state rotation during which the function evolves from the assumed initial uniform rotation to an asymptotic distribution was not discussed. The authors had simply used the asymptotic solutions of Zahn's angular momentum transport equation as representative ones for and only allowed them to change slowly with time due to the evolution of the star's internal structure on the MS.
In this paper we discuss the massive MS stars' approach to the asymptotic steady-state rotation as well as some related problems. Because relaxation times required for the stars in question to reach the state of asymptotic rotation are found to be short compared to the MS life-times and, consequently, changes of the H (and He) abundance taking place during these times can be neglected, we use ZAMS models and do not follow the evolution of the stars. In order to find out how the results obtained depend on stellar mass and rotation rate we performed calculations for two values of M, 10 and 30 , and for the 10 model two different values of the surface rotational velocity were considered.
© European Southern Observatory (ESO) 1999
Online publication: November 26, 1998