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

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1. Introduction

White dwarfs are observed to rotate with typical periods of a day. The main sequence progenitors of these stars are also rotating, and it is generally assumed that the rotation of white dwarfs is a remnant of this main sequence rotation. Arguments involving conservation of angular momentum can be used to make this plausible (e.g. Perinotto 1990, Pijpers 1993). A problem with this picture is, however, that progenitors of WD have gone through a giant stage in which at least the envelope rotated very slowly. Thus, it is necessary to assume that the cores of giants remain decoupled rotationally from their envelopes during the entire evolution from main sequence turnoff till the formation of the WD. Since little is known with certainty about the processes that might redistribute angular momentum inside stars, this assumption can not be easily rejected.

There are, however, observational and theoretical reasons to doubt this picture. A strong observational argument is the internal rotation of the Sun. The most recent helioseismological measurements (Elsworth et al. 1995, Kosovichev et al. 1997, Corbard et al. 1997) show that the rotation below the convection zone is esentially uniform, with measured degrees of differential rotation well below the 30% seen at the surface. The known hydrodynamic angular momentum transport processes, even with rather optimistic estimates of their efficiency, leave the Sun with a much too rapidly rotating interior (Spruit et al. 1983). A new hydrodynamic mechanism recently studied in some detail is friction by internal gravity waves excited by the convection zone (Press 1981, Spruit 1987, Zahn 1990, Schatzman 1993). Realistic calculations of this process appear to be difficult, but estimates indicate that it can be more effective than the other hydrodynamic processes (Zahn et al. 1997).

Magnetic fields, on the other hand, have long been known to be very efficient at transporting angular momentum. The torques exerted by magnetic fields become significant already at very low field strengths. For the Sun, for example, a field of less than 10G can provide sufficient torque to maintain the observed uniformity of rotation. A number of mechanisms can provide such weak fields, for example a fossil field (remnant of the star formation process) or a dynamo-like process operating on (a small remnant of) the differential rotation of the core.

In this paper, I develop the consequences of assuming that the cores of giants do, in fact, corotate approximately with their envelopes. After discussing the observational evidence on WD rotation rates I develop theoretical arguments for the existence of effective coupling between the core and the envelope. This predicts very slowly rotating cores in the giant progenitor of a single WD. The rotation of single white dwarfs must then be explained by other processes.

The same applies to neutron stars born in red giants. The observed pulsar rotation periods of the order of a second are much shorter (by a factor [FORMULA] or so) than expected if they formed in approximately uniformly rotating giants, and with our assumption of strong coupling of the core another mechanism also has to be found to explain the rotation of pulsars. The processes differ somewhat for white dwarfs and neutron stars. The arguments for the neutron star case are developed in a companion paper (Spruit & Phinney, 1998). There, we show that the kicks with which neutron stars are born (as inferred from their transverse velocities) also impart angular momentum at amounts sufficient to explain the rotation of most pulsars.

To explain the typical rotation rates periods of single white dwarfs (of which only about 30 have measured rotation rates), I show in Sect.  5that small asymmetries in the mass loss processes during the last phases of evolution on the AGB are sufficient to explain the observed rotation rates. These asymmetries act as a random forcing through which angular momentum accumulates in the envelope. A balance results between this random forcing and the loss of angular momentum by the wind. The evolution of the angular momentum as the mass loss proceeds turns out to be mathematically the same as that of the velocity of a particle experiencing Brownian motion in a gas, and can be described by a Fokker-Planck equation. Solutions of this equation (Sect.  5.1) show that probability distribution of the angular momentum is close to a Maxwellian. The mean angular momentum decreases as the square root of the envelope mass remaining. Current observational evidence relating to the asymmetries needed in this picture is discussed in Sect.  6.

1.1. Rotation speeds of AGB cores

Starting with a rapidly rotating main sequence star, and assuming uniform rotation during the expansion to the giant stage, we can estimate the rotation periods to be expected for white dwarfs evolving from single stars. An early type main sequence (MS) star, rotating near its maximum speed (of the order 400 km/s), and expanding without angular momentum loss onto the asymptotic giant branch (AGB), has a rotation period [FORMULA] yr for [FORMULA] AU (except for a modest difference in gyration radius neglected here). Most early type MS stars rotate significantly slower, so that periods of the order 30-100 yr would be expected for the AGB descendants of earlytype stars. Some of the observed white dwarfs must have descended from solar type stars (F-G), which have periods of the order 30d at the end of their main sequence life. The AGB progenitors of these WD would rotate 100 times slower, with periods of the order of a thousand years.

If the small amount of envelope mass is ignored which settles back onto the core during post-AGB evolution (more about this in Sect.  4.1), these rotation periods would also be inherited by the WDs formed. While there are a few magnetic white dwarfs with inferred rotation periods of at least a century, most WDs for which periods are known rotate much faster. We evidently need another mechanism to explain the rotation of typical single WDs. Before entering the discussion of possible mechanisms, I briefly review the observational evidence on WD rotation.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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