Single stars

As was noted in Section 1.1, two basic methods have been used to measure rotational velocities of single stars. One of them consists of extracting rotational broadening from a spectral line profile, from which one infers the projected equatorial velocity v sin i along the line of sight. The other one consists of determining the modulation frequency of a star's light due to the rotation of surface inhomogeneities (such as spots or plages) across its surface. If observable, this modulation frequency is a direct estimate of the star's rotation period Prot, which is free of projection effects. Hence, given a radius R for the star, this period can be transformed into a true equatorial velocity v (= & R = 2n R/Prot).

The spectrographic method has proven useful in determining the projected velocities for stars of spectral type O, B, A, and F. In fact, v sin i measurements can only be used in a statistical way because the inclination angle i is generally unknown. Evidence for random orientation of rotation axes is found in the lack of correlation between the measured values of v sin i and the galactic coordinates of the stars. For randomly oriented rotation axes, one can thus convert the average projected equatorial velocity (v sin i) for a group of stars to an average equatorial velocity (v), taking into account that the average value (sin i) is equal to n/4. Numerous statistical studies have been made over the period 1930-1970. The main results pertaining to stellar rotation have been assembled

Fig. 1.5. Mean projected equatorial velocities for a number of different classes of stars as compared with normal main-sequence stars. Source: Slettebak, A., in Stellar Rotation (Slettebak, A., ed.), p. 5, New York: Gordon and Breach, 1970. (By permission. Copyright 1970 by Gordon and Breach Publishers.)

Fig. 1.5. Mean projected equatorial velocities for a number of different classes of stars as compared with normal main-sequence stars. Source: Slettebak, A., in Stellar Rotation (Slettebak, A., ed.), p. 5, New York: Gordon and Breach, 1970. (By permission. Copyright 1970 by Gordon and Breach Publishers.)

by Slettebak and are summarized in Figure 1.5. In this figure the mean observed rotational velocities for single, normal, main-sequence stars are compared with the mean observed v sin i s for giant and supergiant stars, Be stars, peculiar A-type and metallic-line stars, and Population II objects.

The distribution of rotational velocities along the main sequence is quite remarkable: Rotation increases from very low values in the F-type stars to some maximum in the B-type stars. However, a different picture emerges when one considers the mean rotation periods rather than the mean equatorial velocities. This is illustrated in Table 1.1 which lists typical values of the masses, radii, equatorial velocities, angular velocities, and rotation periods. Note that the periods reach a minimum value of about 0.56 day near spectral type A5, and they increase rather steeply on both sides so that the G0- and O5-type stars have approximately the same rotation period. The large observed values (v) for the upper main-sequence stars are thus entirely due to the large radii of these stars.

The open circles in Figure 1.5 represent mean rotational velocities for stars belonging to the luminosity classes III and IV; they are connected by a broad cross-hatched band, thus suggesting uncertainties in the mean rotational velocities for the giant stars. According to Slettebak, the very low point at spectral type A0 can probably be interpreted in terms of selection effects. In any case, the broad band indicates that the early-type giants rotate more slowly than the main-sequence stars of corresponding spectral types, whereas for the late A- and F-types the giants rotate more rapidly than their main-sequence counterparts.

Table 1.1. Average rotational velocities of main-sequence stars.

Spectrum

M

R

v

Q

Prot

(class V)

(M0)

(R0)

(km s-1)

(10-5 s-1)

(days)

O5

39.5

17.2

190

1.5

4.85

B0

17.0

7.6

200

3.8

1.91

B5

7.0

4.0

210

7.6

0.96

A0

3.6

2.6

190

10.0

0.73

A5

2.2

1.7

160

13.0

0.56

F0

1.75

1.3

95

10.0

0.73

F5

1.4

1.2

25

3.0

2.42

G0

1.05

1.04

12

1.6

4.55

Source: McNally, D., The Observatory, 85, 166, 1965.

Source: McNally, D., The Observatory, 85, 166, 1965.

This behavior can be interpreted as an evolutionary effect. As we know, the rapidly rotating B- and A-type main-sequence stars evolve to luminosity classes III and IV in later spectral types. But then, the drop in rotation as the star's radius increases is compensated by the steeper drop in rotation along the main sequence, so that the evolving star still has a larger equatorial velocity than its main-sequence counterpart. As we shall see in Section 6.5, the drop in rotation for the giants takes place between spectral types GO III and G3 III; the drop for subgiants occurs a little earlier, at spectral types F6 IV to F8 IV.

Supergiants and Population II stars are shown schematically near the bottom of Figure 1.5. The supergiants of all spectral types do not show conspicuous rotations. They show no sudden decrease in rotation either, although rotational velocities up to 90 km s-1 are observed for spectral types earlier than F9. The apparent rotation velocities of Population II stars are also small, with v sin i values smaller than 30 km s-1. Note also that the mean rotational velocities of the peculiar A-type stars and metallic-line stars are considerably smaller than the means for normal stars of corresponding spectral types. Finally, going to the other extreme, we note that the Be stars rotate most rapidly, and individual rotational velocities of 500 km s-1 have been observed by Slettebak. These stars are shown separately on Figure 1.5, with arrows indicating that their mean rotational velocities are in reality larger than shown. (As we shall see in Sections 6.3.2 and 6.3.4, however, there are no early-type stars with rotation rates anywhere near the critical rate at which centrifugal force balances gravity at the equator.) As a rule, the white dwarfs rotate rather slowly, with typical v sin i values of order 20 km s-1, and none of them rotates faster than 60 km s-1.

To put the relation between stellar age and axial rotation on a firm quantitative basis, several authors have obtained projected equatorial velocities for stars belonging to open clusters and associations. Detailed statistical analyses have been made by Bernacca and Perinotto (1974) and Fukuda (1982). In Figure 1.6, which is derived from data presented by Fukuda, we compare the average rotational velocity loci for field and cluster stars. As was done in Figure 1.5, the data have been grouped to smooth out irregularities in the distributions of (v sin i) along the main sequence (see also Section 6.3). Figure 1.6 shows that field and cluster stars of spectral type O, B, and A have mean projected rotational

Fig. 1.6. Mean projected equatorial velocities for early-type field and cluster stars. Note that the open-cluster F dwarfs rotate more rapidly than their older, field counterparts. Source: Stauffer, J. R., and Hartmann, L. W., Publ. Astron. Soc. Pacific, 98, 1233, 1986. (Courtesy of the Astronomical Society of the Pacific.)

Fig. 1.6. Mean projected equatorial velocities for early-type field and cluster stars. Note that the open-cluster F dwarfs rotate more rapidly than their older, field counterparts. Source: Stauffer, J. R., and Hartmann, L. W., Publ. Astron. Soc. Pacific, 98, 1233, 1986. (Courtesy of the Astronomical Society of the Pacific.)

velocities in the range 150-200 km s-1. Within each spectral type, the mean rotational velocities of the field stars earlier than spectral type F0 are almost the same as those in clusters. Later than spectral type F0, however, the rotational velocities steeply decrease with increasing spectral type, dropping to below 20 km s-1 at spectral type G0. Note also that the F-type cluster stars, which are generally younger than the field stars, rotate more rapidly than their field counterparts. This result confirms Kraft's (1967) original finding that the mean rotational velocities of late-Fandearly-G stars decline with advancing age. This correlation between rotation and age was quantified shortly afterward by Skumanich (1972), who pointed out that the surface angular velocity of a solar-type star decays as the inverse square root of its age. To a good degree of approximation, we thus let

which is known as Skumanich's law. (Other mathematical relations between rotation and age have been suggested, however.) As we shall see in Section 7.2, such a spin-down process is consistent with the idea that magnetically controlled stellar winds and/or episodic mass ejections from stars with outer convection layers continuously decelerate these stars as they slowly evolve on the main sequence.

An inspection of Figure 1.5 shows that appreciable rotational velocities are common among the normal O-, B-, and A-type stars along the main sequence, whereas they

Fig. 1.7. Rotational velocity distribution for a Persei members. Source: Stauffer, J. R., Hartmann, L. W., and Jones, B. F., Astrophys. J., 346, 160, 1989.

virtually disappear near spectral type F5. Several photometric and spectroscopic studies made during the 1980s have confirmed that late-type, old field dwarfs with few exceptions are slow rotators, with true equatorial velocities less than 10 km s-1 in most stars. Fortunately, because continuous mass loss or discrete mass ejections cause spin-down of stars having convective envelopes, this sharp drop in rotational velocities along the main sequence is considerably reduced in younger stellar groups. Hence, clues to the rotational evolution of low-mass stars may be gained from the study of stars belonging to open clusters. This is illustrated in Figures 1.7 and 1.8, which depict, respectively, the rotational velocity distributions for lower main-sequence stars in the a Persei cluster (age ~ 50 Myr) and in the Hyades (age ~ 600 Myr). Figure 1.7 shows that the young a Persei cluster has a large number of very slowly rotating stars and a significant number of stars with projected equatorial velocities greater than 100 km s-1. This is in contrast to the older Hyades, where G and K dwarfs are slow rotators, with the mean equatorial velocity appearing to decrease at least until spectral type K5. There is one prominent exception in Figure 1.8, however, a K8 dwarf that is the earliest known member of a population of relatively rapidly rotating late K- and M-type Hyades stars. These are genuine evolutionary effects that will be discussed in Section 7.4.2.

Other essential clues to the initial angular momentum distribution in solar-type stars can be obtained from the rotational velocity properties of low-mass, pre-main-sequence stars. These stars are commonly divided into two groups: the classical T Tauri stars, which have evidence of active accretion, and the weak-line T Tauri stars, which do not. Several photometric monitoring surveys have successfully determined rotation periods for a large number of these stars. It appears likely that most of the weak-line stars rotate faster than the classical T Tauri stars. Moreover, as was originally found by Attridge and Herbst

1

| l . l

1 1 1

1 1 1

T

1 1

1

1

-

-

- •

-

•• • •

_

• •

• •

_

-

-

F7

GO G5

K0

K5

1 . . .

1 , ,

i

i

■ 1

B-V COLOR

Fig. 1.8. Rotational velocity distribution for 23 Hyades stars. Source: Radick, R. R., Thompson, D. T., Lockwood, G. W., Duncan, D. K., and Baggett, W. E., Astrophys. J., 321, 459, 1987.

(1992), the frequency distribution of rotation periods for the T Tauri stars in the Orion Nebula cluster is distinctly bimodal. Figure 1.9 illustrates the frequency distribution of known rotation periods for these stars, combining the data for the Trapezium cluster, the Orion Nebula cluster, and other T associations. This combined distribution is clearly bimodal, with a sparsely populated tail of extremely slow rotators. The implications of this bimodality will be further discussed in Section 7.4.1.

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