4. Mass and angular momentum loss on the AGB
A large fraction of the star's mass is lost in the last phases of evolution on the AGB (e.g. Habing, 1990). Most is ejected in the form of a superwind () lasting on the order of yr (e.g. Vassiliadis & Wood, 1993). It is believed to be driven by pulsational instability and radiation pressure on dust (Fleischer at al. 1992, Sedlmayr & Carsten 1995, Höfner & Dorfi, 1997), or possibly by sound waves (e.g. Pijpers & Hearn, 1989). The mass loss is probably not steady because the stellar pulsation is an important part of the driving. Also, dust formation in the expanding flow is an unstable process (Höfner & Dorfi, 1997). Thus the envelope is probably ejected in the form of a (large) number of light shells. The mass loss is also believed to be modulated on longer time scales by the thermal pulses of the AGB star.
A small fraction of the star's envelope (on the order ) settles back onto the core after the superwind ceases. Most of the angular momentum is lost together with the mass of the envelope, but because of the large size of the envelope, even the small amount of mass remaining might conceivably contain enough angular momentum to form a significantly rotating white dwarf. Thus we need to look in some detail at the angular momentum balance of the mass losing AGB envelope. First, I show that if the superwind is axially symmetric and has the specific angular momentum of the stellar photosphere from which it is ejected, the angular momentum remaining after envelope ejection is far too small to produce a significantly rotating white dwarf.
4.1. Axially symmetric mass loss
where is the mass loss rate, , the photospheric radius and rotation rate of the envelope. The factor 2/3 is due to the variation of specific angular momentum over the surface. Since the envelope is convective, it is a good approximation to assume that it rotates uniformly. Because of the very large radius of the envelope, the core contributes very little to the star's moment of inertia, even if the envelope mass is quite small. By angular momentum conservation the star's angular momentum varies as
In stars, , so that the angular momentum of the star decreases more rapidly than its mass. This is because the specific angular momentum of the mass leaving the star is higher than the average specific angular momentum of the star (by a factor ). Since the envelope mass varies strongly, the gyration radius can not be taken as constant. The total moment of inertia of the star can be written as the sum of core and envelope contributions:
where is the distance to to rotation axis, and
If the envelope contains most of the stellar mass, is approximately that of a polytrope of index 1.5, . For the estimates below I assume this value. For a degenerate core of mass , is of the order 0.19. The gyration radius of the star as a whole is then
For , , the first term is negligible for envelope masses larger than , so that
where and are the initial angular momentum and envelope mass. The steep dependence on implies that only a small fraction of is retained. An upper limit on the final rotation rate is obtained by assuming the AGB star to rotate critically, . The rotation rate of the post-AGB core then becomes
where indices and denote the post-AGB core and the AGB star, respectively. If at the end of the superwind phase an envelope mass of not more than is left, we get a final rotation period of at least a year.
The effect depends rather critically on the index m in (15). If the wind corotates with the star out to some radius , for example because of an atmospheric magnetic field, the specific angular momentum in the wind is increased by the factor , and the index m would become
Magnetic fields are known to exist in Mira envelopes from the circular polarization of the SiO masers (Barvainis et al. 1987, Kemball & Diamond 1997). The values of the field derived are uncertain since they depend on the degree of saturation of the masers (Nedoluha & Watson 1994). A strength of a few tenths of a Gauss, however, would already cause significant additional angular momentum loss by the wind.
The conclusion is that even a maximally rotating AGB star, with its huge amount of angular momentum, will produce only a nearly non-rotating white dwarf if mass loss is axisymmetric. This results from the fact that almost all the envelope is lost, combined with the higher than average specific angular momentum taken away by the mass lost. Physically, as mass is lost from the photosphere, the envelope expands, causing spindown by angular momentum conservation.
Let me summarize the assumptions made in arriving at this, perhaps surprising, conclusion. The first is that core of the AGB star corotates approximately with the envelope when the phase of rapid mass loss sets in. The others are the rather minimal assumptions that the (convective) envelope rotates approximately uniformly, and that the mass lost in the wind carries at least the specific angular momentum of the photosphere of the star.
© European Southern Observatory (ESO) 1998
Online publication: April 20, 1998