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Astron. Astrophys. 334, 210-220 (1998)

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3. Numerical simulations

3.1. Input physics

The stellar evolution calculations presented here are obtained with an implicit hydrodynamic stellar evolution code (cf. Langer et al. 1988). Convection according to the Ledoux criterion and semiconvection are treated according to Langer et al. (1983), using a mixing length parameter [FORMULA] and semiconvective mixing parameter of [FORMULA] (Langer 1991a). Opacities are taken from Alexander (1994) for the low temperature regime, and from Iglesias et al. (1996) for higher temperatures. The effects of rotation on the stellar structure as well as the rotationally induced mixing processes are included as in Pinsonneault et al. (1989), with uncertain mixing efficiencies adjusted so as to obtain a slight chemical enrichment at the surface of massive main sequence stars (cf. Fliegner et al. 1996).

The mass loss rate was parameterized according to Nieuwenhuijzen & de Jager (1990), but modified for a rotationally induced enhancement of mass loss as the star approaches the [FORMULA] -Limit (cf. Langer 1997), i.e.


(cf. Friend & Abbott 1986) where


Here, [FORMULA] is the Rosseland opacity. [FORMULA] is not only evaluated at the stellar surface, but its maximum value in the radiative part of the optical depth range [FORMULA] (cf. Lamers 1993; Langer 1997) is used.

The quantitative result for the [FORMULA] -dependence of the mass loss rate obtained by Friend & Abbott (1986) was questioned by Owocki et al. (1996), who performed hydrodynamic simulations of winds of rotating hot stars including the effect of non-radial radiation forces and gravity-darkening in the way described by von Zeipel (1924). However, the only crucial ingredient for the model calculations, which is confirmed by Owocki et al. (1997), is the fact that the mass loss rate increases strongly as the star approaches the [FORMULA] -limit, so that the star cannot exceed critical rotation, but rather loses more mass and angular momentum (cf. also Langer 1998).

To quantify the angular momentum loss due to stellar winds, [FORMULA], we assume that, at any given time, the mean specific angular momentum of the wind material leaving the star equals the specific angular momentum averaged over the rigidly rotating, spherical stellar surface, which we designate as [FORMULA].

The transport of angular momentum inside the star is modeled as a diffusive process according to


where D is the diffusion coefficient for angular momentum transport due to convection and rotationally induced instabilities (cf. Endal & Sofia 1979; Pinsonneault et al. 1991), and the last term on the right hand side accounts for advection. The diffusion equation is solved for the whole stellar interior. In stable layers, the diffusion coefficient is zero. We specify boundary conditions at the stellar surface and the stellar center which guarantee angular momentum conservation to numerical precision.

Our prescription of angular momentum transport yields rigid rotation when and wherever the time scale of angular momentum transport is short compared to the local evolution time scale, no matter whether rotationally induced mixing or convective mixing processes are responsible for the transport. Chaboyer & Zahn (1992) found that meridional circulations may lead to advection terms in the angular momentum transport equation, which may have some influence on the evolution of the angular momentum distribution in the stellar envelope during the main sequence phase (cf. Talon et al. 1997). However, the omission of these terms has no consequences for the spin-up process describe here, since it is dominated by the much faster angular momentum transport due to convection. Since the time scale of convection is generally much smaller than the evolution time, convective regions are mostly rigidly rotating in our models.

The effect of instabilities other than convection on the transport of matter and angular momentum are of no relevance for the conclusions obtained in the present paper. E.g., mean molecular weight gradients have no effect on the spin-up process described below since it occurs in a retreating convection zone (see below).

Calculations performed with a version of the stellar evolution code KEPLER (Weaver et al. 1978), which was modified to include angular momentum, confirm the spin-up effect obtained with our code which is described in Sect. 3.2.

3.2. Results

We have computed stellar model sequences for different initial masses and compositions (cf. Heger et al. 1997), but here we focus on the calculation of a [FORMULA] star of solar metallicity. This simulation is started on the pre-main sequence with a fully convective, rigidly rotating, chemically homogeneous model consisting of [FORMULA] helium and [FORMULA] hydrogen by mass and a distribution of metals with relative ratios according to Grevesse & Noels (1993). Its initial angular momentum of [FORMULA] leads to an equatorial rotation velocity of [FORMULA] on the main sequence, which is typical for these stars (cf. Fukuda 1982; Halbedel 1996).

During the main sequence evolution the star loses [FORMULA] of its initial angular momentum and [FORMULA] of its envelope due to stellar winds. After the end of central hydrogen burning, the star settles on the red supergiant branch after several [FORMULA] and develops a convective envelope of [FORMULA]. During this phase it experiences noticeable mass loss ([FORMULA]), but the angular momentum loss per unit mass lost [FORMULA] is lower than on the main sequence (cf. Sect. 2.2).

Shortly after core helium ignition, the bottom of the convective envelope starts to slowly move up in mass. About [FORMULA] yr later (at a central helium mass fraction of [FORMULA]), the convective envelope mass decreases more rapidly. After another [FORMULA] it reaches a value of [FORMULA]. Up to this time, the star has lost [FORMULA] as a red supergiant, and [FORMULA] of the angular momentum left at the end of the main sequence.

At its largest extent in mass, the convective envelope contains a total angular momentum of [FORMULA], of which [FORMULA] are contained in the lower [FORMULA]. After those layers dropped out of the convective envelope, they have kept only [FORMULA] whereas [FORMULA] remain in the convective envelope; [FORMULA] have been lost due to mass loss. Thus, on average the specific angular momentum of the lower part of the convective envelope is decreased by up to a factor of [FORMULA], while that of the remaining convective envelope increased on average by a factor of 1.5, despite the angular momentum loss from the surface (cf. Fig. 4, Eq. (3)). For comparison, the helium core contains never more than [FORMULA].

At this point of evolution, the transition to the blue sets in in our model. Within the next [FORMULA] another [FORMULA] drop out of the convective envelope which then comprises only about [FORMULA] but has still the full radial extent of a red supergiant. The ensuing evolution from the Hayashi line to the blue supergiant stage takes about [FORMULA] (cf. Fig. 3). During this time, the angular momentum transport to the outermost layers of the star continues, tapping the angular momentum of those layers which drop out of the convective envelope.

[FIGURE] Fig. 3. Track of the [FORMULA] star in the HR diagram during its blue loop. The two arrows indicate the direction of the evolution. The thick drawn part of the track corresponds to the time span shown in Figs. 6 and 7. The star spends [FORMULA] between two neighboring tick marks. The thick dot indicates where the star obtains its maximum mass and angular momentum loss rates; see the dotted line in Fig. 6.

[FIGURE] Fig. 4. Evolution of the internal specific angular momentum profile in the hydrogen-rich envelope of the [FORMULA] sequence during the first part of core helium burning which is spent as a red supergiant. Dashed lines show specific angular momentum versus mass coordinate in the convective part of the envelope for different times, from [FORMULA] to [FORMULA] before the red [FORMULA] blue transition, with those reaching down to lower mass coordinates corresponding to earlier times and extending to higher mass coordinates, because the stellar mass decreases with time. The solid line shows the angular momentum profile for those mass zones which have dropped out of the convective envelope, which remains frozen in later on. The dotted line marks the angular momentum profile in the inner stellar region which is never part of the convective envelope.

The contraction of the star by a factor of [FORMULA] (cf. Fig. 5) reduces the specific moment of inertia at the surface by [FORMULA] and [FORMULA] would be increased by the same factor if j were conserved locally. This would imply an increase of the equatorial rotational velocity by a factor of [FORMULA] (cf. Fig. 7, "decoupled"). The true increase in the rotational velocity is one order of magnitude larger, due to the spin-up effect described in Sect. 2, and it would be even larger if the star would not arrive at the [FORMULA] -limit (see Fig. 7) and lose mass and angular momentum at an enhanced rate.

[FIGURE] Fig. 5. Evolution of the radii of different mass shells as a function of time for a period including the transition from the red to the blue supergiant stage of the [FORMULA] model. The zero-point on the x-axis corresponds to the thick dot in Fig. 3 and corresponds roughly to the time of the red [FORMULA] blue transition. Except for the uppermost solid line, which corresponds to the surface of the star, the lines trace Lagrangian mass coordinates. The mass difference between the lines is [FORMULA]. Shading indicates convective regions.

For local angular momentum conservation [FORMULA] scales as [FORMULA]. A value of [FORMULA] is found on the red supergiant branch in our calculation, and [FORMULA] for the blue supergiant stage. This by itself would increase [FORMULA] by a factor of [FORMULA] during the red [FORMULA] blue transition. Actually, [FORMULA] changes from [FORMULA] at the red supergiant branch at the beginning of central helium burning to [FORMULA] - to critical rotation - at the red [FORMULA] blue transition, i.e. by a factor [FORMULA], despite significant mass and angular momentum loss.

At an effective temperature of [FORMULA] the Eddington-factor [FORMULA] at the surface (as defined above) rises from a few times [FORMULA] to [FORMULA], mainly due to an increase in the opacity; the luminosity remains about constant. Around [FORMULA] of the angular momentum of the star are then concentrated in the upper [FORMULA]. Since [FORMULA] becomes close to 1, the mass loss rate rises to values as high as several [FORMULA] (cf. Fig. 6) in order to keep the star below critical rotation, with the result that these layer are lost within a few [FORMULA]. This is by far the most dramatic loss of angular momentum, i.e. the highest value of [FORMULA], the star ever experiences. The specific angular momentum loss [FORMULA] reaches a peak value of [FORMULA] (cf. Fig. 6).

[FIGURE] Fig. 6. Evolution of characteristic stellar properties as function of time, during the first part of the blue loop of our [FORMULA] model (cf. Fig. 3). The time zero-point is defined as in Fig. 5 and is marked by the dotted line. Displayed are: the angular momentum loss rate [FORMULA] (A), the total angular momentum J (B), the mass loss rate [FORMULA] (C), the specific angular momentum loss rate [FORMULA] (D), and the effective temperature [FORMULA] (E).

After arriving at the blue supergiant stage, the star still experiences a high angular momentum loss rate for some time, since the major part of the star's angular momentum is still concentrated in the vicinity of the surface. Interestingly, the star now even spins faster than when rigid rotation of the whole star were assumed (cf. Fig. 7), because within the blue supergiant's radiative envelope the angular momentum is not transported downward efficiently. However, due to mass loss, this deviation does not persist long.

[FIGURE] Fig. 7. Equatorial rotation velocity as function of time (solid line) compared to the Keplerian (dashed line) and the critical (dotted line) rotation rate; the latter two are different by the factor [FORMULA]. During the red supergiant phase it is [FORMULA] and the two lines coincide, while during the blue supergiant phase [FORMULA] rises to 0.4. The dash-dotted line shows the evolution of the surface rotation rate if there were no angular momentum transport in the convective envelope, and the dash-triple-dotted line shows how the surface rotation rate would evolve if the whole star would maintain rigid rotation.

Due to the long duration of the blue supergiant phase, several [FORMULA] in comparison to the [FORMULA] the red [FORMULA] blue transition takes, the total angular momentum J is reduced by another factor [FORMULA] although both, the mass and the angular momentum loss rate, become smaller with time. In total, after the blue loop the star has [FORMULA] times less angular momentum than before, and thus red supergiants which underwent a blue loop rotate significantly slower than those which did not.

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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998