Comparisons with Rotation Velocities in Clusters

The Pleiades and Hyades are the best studied clusters and they offer us a picture of the evolution of rotation at ages of 110 and 600 Myr, respectively (Fig. 21.5). Observations of rotation are available down to 0.1 M0. The Pleiades represent the best case of rotational velocities of stars close to the ZAMS (a star with a mass of 0.7 M0 is just reaching the ZAMS in this cluster). The following features can be noticed [550] in the Pleiades: a large spread of rotation velocities over the whole mass range, a big group of very slow rotators, a group of fast rotators (about 1/3 break-up), a bimodal distribution of velocities between 0.6 and about 1M0 and the M dwarfs with M < 0-3 M0 have less spread in rotation.

For the Hyades, the rotation velocities are much reduced compared to those in the Pleiades, except for the M dwarfs where the reduction is moderate. Other clusters in

Fig. 21.5 Distributions of the rotational velocities in the Pleiades and Hyades as a function of Teff. The heavy shaded areas indicate where most stars are lying; the lighter shaded area shows the location of a significant fraction of Pleiades stars. The gray areas indicate regions where stars may be present. The scale at the top of the left diagram is an indicative mass scale. Adapted from J. R. Stauffer [550]

Fig. 21.5 Distributions of the rotational velocities in the Pleiades and Hyades as a function of Teff. The heavy shaded areas indicate where most stars are lying; the lighter shaded area shows the location of a significant fraction of Pleiades stars. The gray areas indicate regions where stars may be present. The scale at the top of the left diagram is an indicative mass scale. Adapted from J. R. Stauffer [550]

the same range of ages confirm the picture provided by the Hyades and Pleiades. The interpretation of the observations (Fig. 21.5) is consistent with the various effects considered above and in Sect. 13.2.

- Disk locking: this process is necessary to account for the very slow rotation of a large fraction of solar-type and lower mass stars as in the Pleiades. Magnetic braking alone does not produce enough loss of angular momentum during the pre-MS phase to account for the high number of very slow rotators. Moreover, if it would do so, it would kill stellar rotation during the MS evolution phase, which is not the case.

- Magnetic braking: the general decrease of rotation with ages from the Pleiades to the Hyades is interpreted as resulting from the magnetic braking due to stellar winds.

- Saturation effect: as seen in Sect. 13.2.1, the saturation effect is supported by X-ray observations of magnetic activity. It allows us to account for the survival of some relatively fast rotators on the ZAMS, like in the Pleiades.

- Mass dependence: the limited decrease of rotation between the M dwarfs of the Pleiades and of the Hyades is likely due to the fact that most M stars have a rotational velocity above the saturation velocity, which is very low for such masses (cf. 13.37 and subsequent remarks).

- Core-envelope partial coupling: there are evidences that the angular momentum of the radiative core of slow rotators contribute to sustain the rotation of convective envelopes over a timescale of 100-200 Myr [482].

The role of binarity with respect to rotation is uncertain. The formation of binaries in the pre-stellar phase may absorb a lot of angular momentum initially present (Table 21.1) and may thus contribute to reduce rotation. However, the presence of a companion around a forming star may also destroy or truncate the circumstellar disk and thus reduce the role of disk locking, thus favoring higher rotation in binaries. However in some cases, binarity may also extend the disk lifetime and thus favor slow rotators [14]. No clear theoretical picture is emerging about the effect of binarity on the initial rotation velocities. Observations show rather similar distributions of rotation velocities in single and double stars [61].

Figure 21.6 shows quantitative comparisons [36] between cluster observations and models of 0.6 and 1.0 M0 with and without disk locking. In all models, magnetic braking has been included with the saturation effect and mass dependence. Differential rotation is present, with some mixing processes treated as a diffusion. To make a long story short, models starting from the T Tauri stage with rotation periods of 4 and 16 days, respectively, corresponding to the lowest and highest periods observed are considered. The various observational constraints at different ages (in particular the occurrence of groups of low and fast rotations in the Pleiades) can only be satisfied if the models with small periods (fast rotation) have disks and the

Fig. 21.6 Evolution of the rotation periods as a function of time for 0.6 and 1.OM0 models. In each diagram, the upper continuous line applies to a starting period of 16 days, a disk is present with a lifetime of 107 yr (continuous line) and with 3 x 106 yr (broken line); the lower curve applies to a starting period of 4 days, no disk is present. The lower and upper boxes represent the observed period for the fast and slow rotators in a Per and the Pleiades. The gray bar at 106 yr represents the observed range of periods in T Tauri stars. The gray bars at 2.5 x 108 and 6 x 108 yr represent data in NGC 3532 and the Hyades for the corresponding masses. Adapted from S. Barnes, S. Sofia & M. Pinsonneault [36]

Fig. 21.6 Evolution of the rotation periods as a function of time for 0.6 and 1.OM0 models. In each diagram, the upper continuous line applies to a starting period of 16 days, a disk is present with a lifetime of 107 yr (continuous line) and with 3 x 106 yr (broken line); the lower curve applies to a starting period of 4 days, no disk is present. The lower and upper boxes represent the observed period for the fast and slow rotators in a Per and the Pleiades. The gray bar at 106 yr represents the observed range of periods in T Tauri stars. The gray bars at 2.5 x 108 and 6 x 108 yr represent data in NGC 3532 and the Hyades for the corresponding masses. Adapted from S. Barnes, S. Sofia & M. Pinsonneault [36]

models with high periods (low rotation) have no disks. The last part of the evolution of the period with age of the fast rotators in Fig. 21.6 obeys the Skumanich relationship, which as mentioned in Sect. 13.2.1 tends to reduce the velocity dispersion at high ages. As a last remark, the evolution of the fast sequence is better explained if the moment of inertia of only the outer zone is participating, while the whole star participates in the spin-down of the slow sequence. This may suggest that the degree of coupling depends on rotation.

The models with disk locking and a disk lifetime of 3 x 106 yr (dashed line) and 107 yr (upper continuous curve) in Fig. 21.6 are able to reproduce the slowest rotators in the young clusters. As mentioned above, these models have differential rotation. For solid body rotation, the disk lifetimes obtained from such comparisons would be of the order of 10-20 Myr. This is much longer than the order of million(s) years suggested by IR evidences of disks in pre-MS stars.

We remark that it may be a little embarrassing to need different processes with various parameters to account for the evolution of rotation velocities in solar-type and lower mass stars. However, this may just be the result of the richness of the physical processes acting in the evolution of these stars. Asteroseismological observations of pre-MS stars may provide decisive constraints.

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