The mechanism to spin-up stellar envelopes presented above enhances the specific angular momentum in the surface layers of the star by some factor (cf. Sect. 3.2); a question to be considered separately is the origin of the angular momentum, and also in which cases critical rotation is reached. For single stars, the available angular momentum seems to be limited to that of the ZAMS star, reduced by the angular momentum lost during its evolution. A way to supply a considerable amount of angular momentum to the star would be the capture of a companion star or planet (Soker 1996; Podsiadlowski 1998). This might occur when the star first becomes a red supergiant. The resulting configuration would have a much higher angular momentum than a normal single star, but still, due to the high moment of inertia of the supergiant, it could initially be far away from critical rotation. Due to the spin-up mechanism described here, such a star might then approach critical rotation even before the transition into a blue supergiant.
The mechanism to spin-up stellar envelopes (Sect. 2), which was applied to a post-red supergiant star in Sect. 3, may operate in all evolutionary phases during which stars evolve from the Hayashi-line towards the hotter part of the HR diagram. These comprise the transition of pre-main sequence stars form their fully convective stage to the main sequence, the transition of low and intermediate mass stars from the Asymptotic Giant Branch (AGB) to central stars of planetary nebulae, the blue loops of stars in the initial mass range , and the transition of massive red supergiants into Wolf-Rayet (WR) stars. For all four phases, observational evidence for axisymmetric circumstellar matter exists.
A high value of during the evolution off the Hayashi line can be expected to have notable influence not only on the mass and angular momentum loss rate as in the case studied in Sect. 3, but also on the geometry of the wind flow. At present, theoretical predictions of the latitudinal dependence of stellar wind properties for rapidly rotating stars are ambiguous. Bjorkman & Cassinelli (1993) have proposed that angular momentum conservation of particles in a pressure free wind which is driven by purely radial forces leads, for sufficiently large values of , to high ratios of equatorial to polar wind density, i.e. to disks or disk-like configurations. Owocki et al. (1996) found that this might no longer be the case when the non-radial forces occurring in hot star winds are accounted for. For cooler stars, the prospects of disk formation may thus be better (cf. also Ignace et al., 1996). To discuss the geometry of the winds of our models is beyond the scope of this paper; however, we want to point out that the maximum values of and occur at about (cf. Sect. 3.2 and Fig. 6), so that an equator-to-pole wind density ratio larger than one may still be justified.
4.1. Blue loops during or after core helium burning
An interesting example where the spin-up mechanism described in this paper should almost certainly have played a rôle is the progenitor of SN 1987A. In fact, the SN 1987A progenitor is the only star of which we are reasonably sure that it performed a blue loop, in this case after core helium exhaustion (cf. Arnett et al. 1989)
The structures observed around SN 1987A seem to be rotationally symmetric with a common symmetry axis, which may suggest that rotation has played a major rôle in their formation. The inner of the three rings is currently explained by the interaction of the blue supergiant wind with the wind of the red supergiant precursor (cf., Chevalier 1996). However, this interaction would result in a spherical shell in case of spherically symmetric winds. To understand the ring structure of the inner interaction region, and maybe also the two outer rings, significant rotation appears therefore to be required (cf. Martin & Arnett 1995), which may be provided by our spin-up mechanism. We may note that for this mechanism to work it is insignificant what actually triggered the blue loop; i.e. it would work as well for single star scenarios with a final blue loop (cf. Langer 1991b; Meyer 1997; Woosley et al. 1998) or for binary merger scenarios which predict a final red blue transition of the merger star (cf. Podsiadlowski 1998).
We do not expect to find ring nebulae frequently around blue supergiants (cf. Brandner et al. 1997), since the time scale on which our model (cf. Sect. 3) shows very high surface rotation rates is rather small compared to the typical life time of a blue supergiant (cf. Langer 1991a), and because they are quickly dissolved by the blue supergiant wind. SN 1987A is an exception, since here the transition to the blue happened only short time ago (i.e. the supernova exploded only shortly after the transition). However, a certain type of B supergiants, the B[e] stars, show emission line features which might be due to a circumstellar disk (cf. Zickgraf et al. 1996). The location of the less luminous subgroup of B[e] stars in the HR diagram (Gummersbach et al. 1995) is in fact consistent with a blue loop scenario for their evolution. I.e., their disks might be produced by the spin-up mechanism described here (cf. also Langer & Heger 1998).
4.2. Red supergiant Wolf-Rayet star transition
The transition of massive mass losing red supergiants into Wolf-Rayet stars is the massive star analogue of the AGB post-AGB star transition. It occurs when the mass of the hydrogen-rich envelope is reduced below a critical value (cf., e.g., Schaller et al. 1992) and should not be confused with the blue loops discussed in Sect. 4.1. As in the post-AGB case (Sect. 4.4), the spin-up mechanism can be expected to work. Unlike in the case discussed in Sect. 3.2, the major part of the envelope is lost due to stellar winds before the red supergiant Wolf-Rayet star transition, with the consequence of considerable angular momentum loss. Therefore, it may be more difficult for the star to reach critical rotation during the contraction.
However, there are signs of asphericity in the ring nebula NGC 6888 around the Galactic Wolf-Rayet star WR 136 which has been interpreted as swept-up red supergiant wind shell by García-Segura & Mac Low (1995) and García-Segura et al. (1996). Also, Oudmaijer et al. (1994) report on bi-polar outflows from IRC+10420, a massive star just undergoing the red-supergiant Wolf-Rayet star transition. IRC+10420 is currently an F type star, i.e. it has an effective temperature at which we expect the maximum effect of the spin-up mechanism discussed here (cf. Sect. 3.2). Therefore, in the absence of a binary companion, the spin-up mechanism may yield the most promising explanation for the bipolar flows.
4.3. Pre-main sequence evolution
As mentioned in Sect. 3.2, we started our sequence from a fully convective pre-main sequence configuration. We expected the spin-up mechanism found for the post-red supergiant stage to be also present in the contraction phase towards the main sequence. An analysis of this evolutionary phase showed in fact its presence, although the efficiency of the spin-up was found to be somewhat smaller. During that part of the pre-main sequence contraction phase where the convective region retreats from the center of the star to its surface, the star would have increased its equatorial rotational velocity by a factor of if angular momentum were conserved locally, but it was spun-up by a factor of . I.e., the spin-up mechanism described in Sect. 2 resulted in an additional increase of the rotational velocity of a factor of . This was not enough to bring the star to critical rotation in this phase. However, if the initial rotation rate would have been larger, a phase of critical rotation during the pre-main sequence stage would well be possible.
While a fully convective pre-main sequence stage may not be realistic for massive stars and is just assumed in our case for mathematical convenience, pre-main sequence stars of low and intermediate mass are supposed to evolve through a fully convective stage (Palla & Stahler 1991). During the transition from this stage to the main sequence, the spin-up mechanism described in this paper might operate. Pre-main sequence stars in the corresponding phase, i.e. past the fully convective stage but prior to core hydrogen ignition, are often found to have disks. They correspond to the T Tauri stars at low mass (e.g., Koerner & Sargent 1995) and to central stars of Herbig Haro objects at intermediate mass (e.g., Marti et al. 1993). However, the disks are usually interpreted as remnants of the accretion process which built up the star. Since pre-main sequence stars are often found to be rapid rotators (cf. Walker 1990), we may speculate here about a possible contribution to the disk due to decretion from the star reaching critical rotation due to spin-up (cf. Krishnamurtihi et al. 1997).
4.4. Post-AGB evolution
Low and intermediate mass stars leave the AGB when the mass of their hydrogen-rich envelope decreases below a certain value. When this happens, the envelope deflates, the stellar radius decreases, and the energy transport in the envelope changes from convective to radiative. The spin-up mechanism described in Sect. 2 can be expected to operate in this situation, as in the red supergiant Wolf-Rayet star transition (Sect. 4.2).
Whether or not post-AGB stars can reach critical rotation due to this spin-up is not clear. Certainly, the heavy mass loss during the evolution on the AGB spins the envelope down according to the mechanism sketched in Fig. 1. The ratio of the rotation rate to the critical rotation rate may be further affected by random kicks due to asymmetric mass loss, which may keep the envelope at some level of rotation (cf. Spruit 1998), or by transport of angular momentum from the core to the envelope, which may be efficient during the thermal pulses, and by the evolution of the critical rotation rate, which depends on the Eddington factor and thereby on the opacity coefficient (cf. García-Segura et al. 1998).
Clearly, the spin-up mechanism described in this paper may help to bring post-AGB stars - or more specific: stars which just left the AGB, i.e. central stars of proto-planetary nebulae - closer to critical rotation. It may play a rôle in explaining axisymmetric flows which are often observed in central stars of proto-planetary nebulae (cf. Kwok 1993) and the shapes of bi-polar planetary nebulae (cf. Schwarz et al. 1993; Stanghellini et al. 1993; García-Segura et al. 1998).
© European Southern Observatory (ESO) 1998
Online publication: May 12, 1998