Fig. 6.7. Mean angular momentum as a function of stellar mass, assuming solid-body rotation at the surface rate. The circles represent the sample of normal single stars ofFukuda (1982); the crosses represent the same sample, but include Am and Be stars. The solid line represents the angular momentum for main-sequence models rotating at breakup velocity. Source: Kawaler, S. D., Publ. Astron. Soc. Pacific, 99, 1322, 1987. (Courtesy of the Astronomical Society of the Pacific.)

surface gravity and centrifugal force are equal with vcrit = (GM/R)1/2. These results are consistent with the (v sin i) values being the same fraction of the critical velocity vcrit for all main-sequence stars more massive than 1.5M0.

For normal single stars earlier than spectral type F0, the relation between mean angular momentum per unit mass, (j) = (J )/M, and stellar mass is well represented by a power-law relation of the form (j) a Ma with a = 1.09. (When the Am and Be stars are included in the sample, however, one finds that a = 1.43.)* The low-mass stars (M < 1.5M0) deviate from this simple power-law relation, as evidenced by their slow rotational velocities in Figure 1.6. As we shall see in Section 7.2, this sharp break at mass 1.5M0 can be attributed to angular momentum loss by magnetically controlled winds or episodic mass ejections from stars with outer convection zones. Accordingly, since the high-mass stars (M > 1.5M0) have no appreciable convective envelopes that could support winds or mass ejections, it is generally believed that these stars have retained most of their initial angular momentum. Hence, it seems likely that the simple power law (J) a Ma+l expresses a fundamental relation between the angular momentum content of an early-type star and its mass, where stars are given, on the average, an amount of angular momentum in proportion to their masses. t

6.3.3 The rotational velocity distributions

Figure 1.6 is a plot of the (v sin i) values against spectral type for single, main-sequence stars. In this section we shall briefly discuss the distribution of v sin i at a given spectral type. Extensive surveys of projected rotational velocities have been assembled by Wolff, Edwards, and Preston (1982). Figure 6.8 illustrates the observed distributions of v sin i for a number of spectral type ranges.

It is immediately apparent that these distributions are all strikingly similar: They peak at low values of v sin i and decrease slowly with increasing rotational velocity, with a maximum of about 350 km s-1 at all spectral types. Note that the early B-type stars are unique only in having a larger percentage of stars with v sin i smaller than 40 km s-1. The decrease in rotational velocity for the late A-type stars is also worth noticing since it indicates that the braking mechanism that spins down the stars of later spectral type is already partially operative in the A-type stars. Note also that these observational results rule out simple Maxwellian distributions for the v sin i s along the upper main sequence.

The similarity of the observed distributions strongly suggests that the same physical mechanisms are involved in determining the rotational velocities of all upper-main-sequence stars. Unfortunately, without a clear understanding of the star-formation process and early stellar evolution, we are still unable to explain why slow rotation (i.e., v sin i < 100 km s-1) is so prevalent among these stars.

6.3.4 Rotation of Be and shell stars

If one excepts remnants such as neutron stars and pulsars, the stars of most rapid rotation are the emission-line B stars (i.e., the Be stars). There is now widespread

* The values originally obtained by McNally and Kraft were a = 0.80 and a = 0.57, respectively. t As was shown by Brosche (1963) and others, the (J) a M2 rule is closely obeyed over the mass range 1018-1048 g, from asteroids up to clusters of galaxies. Explanations have been presented by Wesson (1979) and by Carrasco, Roth, and Serrano (1982).

Fig. 6.8. The observed distribution of projected equatorial velocities as a function of spectral type. Hatched areas show spectroscopic binaries discovered to date among the stars within each group. Source: Wolff, S. C., Edwards, S., and Preston, G. W., Astrophys. J., 252, 322, 1982.

agreement that matter is leaving the Be stars at their equator, with the resultant equatorial disk giving the emission seen in the hydrogen lines. Some Be stars also develop, from time to time, a network of deep and narrow absorption lines and they are then called shell stars. They are also characterized by extremely broad absorption lines, which, when interpreted as due to axial rotation, makes them as a class the most rapidly rotating Be stars. As was pointed out by Slettebak (1979), this suggests that the shell stars are edge-on normal Be stars: The difference in spectra is due to differences in inclination of the rotation axes.

Mean values of the observed v sin i s range between about 200 and 250 km s"1, with the largest v sin i s being in the neighborhood of 400 km s"1. This raises at once the following question: Do the Be and shell stars rotate at their critical velocity at which centrifugal force balances gravity at the equator? The answer to that question is flatly no. Indeed, as can be seen in Figure 6.2, the theoretical breakup velocities are much larger than 400 km s"1 in the mass range 3-15M0, which corresponds to the masses of normal B-type stars and probably to those of Be-type objects as well.

In order to gain further insight into the problem, Porter (1996) has made a detailed statistical study of the projected rotational velocities of these stars. In his discussion the fundamental parameter is not v sin i, however, but the equatorial velocity of the star as a fraction of the breakup velocity, w = v/vcrit, where vcrit is the critical equatorial velocity of the star at breakup rotation. The distribution functions of normal Be stars and shell stars as functions of w sin i are shown in Figure 6.9. One readily sees that the projected equatorial velocities for shell stars are significantly larger than those for normal Be stars. Statistical tests further indicate that shell stars and normal Be stars are simply related by inclination. This, taken along with theoretical shell line profiles generated in edge-on disks, leads to the following conclusions: (i) shell stars are normal Be stars viewed edge-on and (ii) the shell star distribution with i = 90° is a good representation of the distribution of

Fig. 6.9. Distribution functions of normal Be stars (top) and Be-shell stars (bottom) as functions of w sin i. Source: Porter, J. M., Mon. Not. R. Astron. Soc., 280, L31, 1996. (Courtesy of Blackwell Science Ltd.)

Fig. 6.10. Distributions of equatorial rotational velocities for two samples of A5-F0 stars. The right distribution is for 234 normal class V stars plus 23 stars with weak X4481 lines; the left distribution is for 133 Ap+Am stars. Source: Abt, H. A., and Morrell, N. I., Astrophys. J. Suppl, 99, 135, 1995.

Fig. 6.10. Distributions of equatorial rotational velocities for two samples of A5-F0 stars. The right distribution is for 234 normal class V stars plus 23 stars with weak X4481 lines; the left distribution is for 133 Ap+Am stars. Source: Abt, H. A., and Morrell, N. I., Astrophys. J. Suppl, 99, 135, 1995.

the ratios v/ vcrit for all Be stars. In accordance with a previous study by Chen and Huang (1987), it is therefore concluded that the distribution function of all Be stars is sharply peaked at w = 0.7, although there is a tail of the distribution to the smaller ws.

6.3.5 Rotation of Am and Ap stars

The metallic-line (Am) and peculiar A-type (Ap) stars have small projected rotational velocities, v sin i, relative to the means for normal stars of corresponding spectral types. As we shall see in Section 6.4, the abnormal compositions of the Am and Ap stars can be explained very well by microscopic diffusion processes. That is, because these stars are slow rotators, they are most likely to possess quiet radiative envelopes in which gravitational sorting of the chemical elements is possible, thus leaving abnormal atmospheric abundances.

The distributions of equatorial rotational velocities for representative samples of chemically peculiar (Ap and Am) stars and normal A-type stars are shown in Figure 6.10. Thus, after deconvolving the v sin i distribution and assuming random orientation of the rotation axes, one finds that all the rapid rotators have normal spectra while nearly all the slow rotators have Am or Ap spectra. There is a 10% overlap, corresponding to 39 too many normal stars with sharp lines in the sample. According to Abt and Morrell (1995), this overlap is due to their failure to detect all the abnormal stars so that a specific rotational velocity is probably sufficient to determine whether a star will have a normal or abnormal spectrum. This statement has been recently challenged by Budaj (1996, 1997), however.

It has been known for some time that all, or most, Am stars are spectroscopic binaries. A detailed study of the frequency of Am stars among those binaries has been made by Ginestet et al. (1982) and Budaj (1996). Their analyses indicate that the orbital period distribution of the Am stars has a prominent peak in the period range 2-15 days, which is also the region where synchronization is observed. As they showed, this period range coincides with the largest gap in the orbital period distribution of nonpeculiar spectroscopic binaries of spectral types A4-F1, IV and V. To be specific, in the period range 2-100 days, it is found that about 85% of the binaries are Am stars. However, although the Am stars are also observed at larger orbital periods, there is a conspicuous gap in the period range 180-800 days. In Section 8.4.4 we shall explain how tidal interaction in binaries with period smaller than 100-200 days can effectively cause their components to have low rotational velocities and thus become Am stars.

In contrast to the Am stars, however, the slow rotation of the Ap stars does not appear to be due to tidal interaction in close binaries. What is, then, the mechanism responsible for the abnormally low rotation rates of the Ap stars, when compared to normal stars of corresponding temperature and luminosity? Unfortunately, whereas their slow rotation is generally attributed to some kind of magnetic braking, there remains considerable controversy as to whether most of their angular momentum is lost before or during the main-sequence phase. Observations of Ap stars in open clusters and associations of varying ages can answer that question. According to Wolff (1981), measurements of v sin i values strongly suggest that the Si-type Ap stars lose angular momentum after they reach the main sequence, while those of the Sr-Cr-Eu group might do so prior to the main-sequence phase. As was noted by North (1984) and Borraet al. (1985), however, her conclusions are based on line-broadening measurements, which are affected not only by the sin i projection factor but also by the magnetic field strength via Zeeman broadening. This is the reason why they have determined accurate photometric rotation periods of magnetic Ap stars belonging to open clusters and associations. Both studies show that the young cluster stars have essentially the same rotation periods as the older field stars, indicating either that the Ap stars have lost most of their angular momentum before they reach the main sequence or that they are intrinsically slow rotators from their formation on. This result has been recently confirmed by North (1998), who found no evidence for any loss of angular momentum on the main sequence, thus confirming earlier results based on less reliable estimates of surface gravity.

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