Axial rotation along the upper main sequence

In Section 1.3 we summarized the mean rotational properties of single stars. It is the purpose of this section to provide further information about the rotation patterns in specific groups of early-type, main-sequence stars.

6.3.1 Rotation in open clusters

Figure 1.6 provides a comparison between the average rotational velocities of cluster and field stars. It is immediately apparent that the (v sin i) values of the, generally younger, cluster stars are similar to those of the field stars, except that for spectral types later than F0 the cluster stars rotate more rapidly than the field stars. A somewhat different picture emerges when one compares the (v sin i) values for members of individual cluster and field stars. That this is indeed the case is illustrated in Figure 6.6, which shows that open clusters and associations often differ in their mean projected rotational velocities.

The question immediately arises whether the (v sin i) values of a given cluster are unusual because of high or low equatorial velocities, v, or because of preferential inclination angles, i, of the rotation axis. Unfortunately, we do not yet know whether the rotation axes are oriented at random in space or whether there exists a preferential direction in some (if not all) clusters. Hereafter we shall assume that alignment of axes does not contribute appreciably to the unusual projected rotational velocities that are observed in some clusters. With regard to the causes of the differences between clusters,

Pleiades

Pleiades

Fig. 6.6. Mean projected equatorial velocities for several open star clusters compared with field main-sequence stars. Adapted from Kraft (1970). Source: Gray, D. F., The Observation and Analysis of Stellar Photospheres, Cambridge: Cambridge University Press, 1992.

Fig. 6.6. Mean projected equatorial velocities for several open star clusters compared with field main-sequence stars. Adapted from Kraft (1970). Source: Gray, D. F., The Observation and Analysis of Stellar Photospheres, Cambridge: Cambridge University Press, 1992.

three likely explanations have been considered, namely, evolutionary expansion effects, the proportion of binaries, and the proportion of peculiar stars.

When a star leaves the zero-age main sequence and expands, its rotational velocity decreases. Since the brightest stars in a cluster evolve faster than the less luminous ones, such an evolutionary effect could possibly explain the low rotational velocities of the brightest stars in, for example, IC 4665 (see Figure 6.6). However, the fact that evolutionary expansion is not the main cause of this "turn-down" effect in clusters is well illustrated by the a Persei cluster, where the evolved stars have larger, rather than smaller, mean rotational velocities than field stars! As we shall see in Section 6.3.5, there are at least two ways in which the initial rotational velocities of stars may be gradually modified: by tidal interaction in closely spaced binaries (e.g., the Am stars) and by magnetic braking in magnetic stars (e.g., the Ap stars). Thus, if some clusters differ in their number of spectroscopic binaries or peculiar stars, we might expect that their (v sin i) values will also depart significantly from the mean rotational velocities of field stars. Detailed studies have shown that clusters with rapidly rotating stars have far fewer binaries and Ap stars than clusters with stars having normal or low rotational velocities (e.g., Levato and Garcia 1984). Hence, we conclude that tidal interaction and magnetic braking are quite effective in reducing rotational velocities, so that a large part of the differences between clusters in their (v sin i) values can be assigned to different frequencies of binaries and Ap stars. But then, as was correctly pointed out by Abt (1970), we have succeeded only in shifting the problem from trying to explain the various mean rotation rates in clusters to trying to explain these frequency differences.

6.3.2 The angular momentum diagram

As was shown in Section 6.2.2, there is as yet no clear expectation for the angular momentum distribution within an early-type star. At this writing, however, the most reasonable guess seems to be uniform rotation or mild differential rotation. Using the assumption that these stars rotate as solid bodies, we shall now derive an important relation between total angular momentum and mass along the main sequence. To the best of my knowledge, McNally (1965) was the first to obtain that relation.

The total angular momentum of a uniformly rotating body is given by the product of I, its moment of inertia, and its angular velocity of rotation, Since the observations give the mean equatorial velocity for each mass interval, we divide this quantity by the mean radius R to obtain the mean angular velocity. Thus, for randomly oriented rotation axes, the mean value of the total angular momentum is given by the simple relation

where all quantities are functions of stellar mass. The usual mass-spectral type relation can be used to obtain the <v sin i > values as functions of mass. Theoretical models provide us with the functions R(M) and I(M) for selected mass intervals.

Updating Kraft's (1970) analysis, Kawaler (1987) has re-derived the mean angular momentum < J(M)> along the main sequence using current stellar models and rotational velocities. In Figure 6.7 the circles represent a sample of normal single stars, whereas data indicated by crosses include Am and Be stars in the sample. For comparison, also shown is the line < J(M)> that corresponds to rotation at breakup velocity vcrit, that is, where

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