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Astron. Astrophys. 333, 603-612 (1998)

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6. Discussion

In the standard view, the rotating single WD derive from the rotating cores of giants, which somehow avoided spinning down in the slowly rotating convective envelope. I argue, instead, that rotating cores in giants are an unattractive idea, especially if these cores are magnetic. Unless the magnetic WD acquired their fields after emerging from the envelope, the observed dipole moments are so large that a strong interaction with the slowly rotating convective envelope would be very hard to avoid.

I recall the classical demonstration (e.g. in Mestel 1953, 1961) that rather weak magnetic fields (magnetic energy a small fraction of the rotational energy) can already transmit enough torques to maintain corotation between core and envelope. Such a weak field could be inherited from the star formation process. In order to prevent these torques from acting, any magnetic field in the core would have to be very weak or very accurately shielded from the convective envelope. In addition, a differentially rotating, initially nonmagnetic core is unstable to the growth of a small scale dynamo magnetic field, initiated by a magnetic shear instability (Balbus & Hawley 1992). The conditions for existence of this instability in stars were studied in detail already by Acheson (1978) who showed, in particular, that thermal diffusion allows it to operate under a much wider range of conditions than in the adiabatic case.

The very weak differential rotation in the core of the Sun (e.g. Kosovichev et al.1997), for which no good explanation has been put forward except magnetic torques, is strong evidence for the operation of magnetic effects. While the arguments given here do not constitute a proof, I feel they are sufficiently compelling that approximately uniform rotation is a reasonable hypothesis, and is at least as plausible as the traditional assumption, which implies a core rotating [FORMULA] - [FORMULA] times faster than the envelope for the entire duration of the RGB and AGB.

I have explored the consequences of the assumption of approximately uniform rotation for AGBs stars in the process of shedding their envelopes. If this mass loss is strictly axisymmetric, the remaining core rotates very slowly (period more than 10 years). This is just the consequence of angular momentum conservation: the wind takes away almost the entire envelope, but the specific angular momentum it carries away is that of the stellar photosphere, which is larger than the average specific angular momentum of the envelope.

On the other hand, only small nonaxisymmetries in the mass loss process suffice to give the star enough `kick' to explain the angular momentum of single white dwarfs. Such kicks could be associated with mass loss events at the pulsation period of the star or dust-formation episodes in the atmosphere. I have illustrated this with a calculation of the evolution of the probability distribution of the star's angular momentum under the combined action of many small nonaxisymmetric kicks and the angular momentum loss in the wind. The degree of asymmetry required is found to be of the order [FORMULA].

Present theories for AGB mass loss are not detailed enough to calculate such asymmetries, but observational indications for asymmetries exist. Interferometric images of red supergiants ([FORMULA]  Sco, [FORMULA]  Ori and [FORMULA]   Her: Tuthill et al. 1997), speckle reconstructions ([FORMULA]  Ori: Kluckers et al. 1997) and HST imaging ([FORMULA]  Ori: Gilliland & Dupree, 1996) show pronounced `hot spots' on their surfaces. Assuming that such nonuniform photospheric conditions persist during the superwind phase, one would expect them to also affect the dust formation that is thought to be essential for the driving of the wind. The required asymmetry is obtained if a few (5 say) such spots are present, and the wind locally generated above these spots is slightly non-radial by a few tenths of a degree. That the mass flow is indeed asymmetric already close to the stellar photosphere is shown by speckle imaging (IRC 10216: Osterbart et al. 1997), and especially by mm-wave interferometric images of the SiO maser emission. These show a highly clumpy and time dependent structure (Diamond et al. 1994, Humphreys et al. 1996, Pijpers et al. 1994). This maser emission occurs at a few stellar radii, which is also the region where the backreaction of the wind on the star (`kick') takes place. Though the SiO maser emission is very sensitive to small changes in the local physical conditions, models of the emission (Lockett & Elitzur 1992, Bujarrabal 1994) should give estimates of the degree of inhomogeneity in the physical conditions in the wind.

Measurement of deviations from radial flow in proper motion studies of the masing clumps in the wind should enable direct determination of the asymmetries relevant for the kick process described in this paper.

An issue mentioned here only briefly is the origin of the 5 or so very slowly rotating ([FORMULA] yr) magnetic white dwarfs. A possible explanation is angular momentum loss in a radiation driven, but magnetized, wind during post-AGB evolution. This possibility will be further explored elsewhere.

The coupling between core and envelope proposed here would also imply that the cores of pre-supernovae on the giant branch are so slowly rotating that very slowly ([FORMULA] hr) rotating neutron stars would result even if angular momentum were conserved during core collapse. While these would not show up as pulsars, one would have to argue that none of the observed pulsars were formed in red giants, which feels like an unattractive idea. It turns out, however, that the kicks neutron stars receive at birth and which give them their high observed space motion, are strong enough to impart a significant rotation as well. This idea is developed further in a separate paper (Spruit & Phinney, 1998).

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998