6. Discussion and conclusions
The main assumption of the model presented here is a decoupling between the stellar core and the envelope, and the existence of angular momentum transfer, controlled by a coupling time-scale . I computed evolutionary tracks for 3 different coupling time-scales and compared the results to the observations. If the coupling time-scale is short compared to the contraction time scale (a few Myr vs a few 10 Myr), then the star will nearly rotate as a solid-body. If it is long, then decoupling will have non negligible effects on rotational evolution. I first summarize the effects of the models presented here.
With a long coupling time model (i.e. 500 Myr) it is very easy to keep a low surface velocity, as wind braking only applies on the envelope. This model leads to quite a good agreement with the slow rotators fraction at ZAMS ages, even if only short disk lifetimes are considered ( 10 Myr). Rapid braking of rapid rotators between the age of Alpha Per and the Pleiades is also in good agreement with the observations. However, it is more difficult to explain the existence of these rapid rotators on the ZAMS. The existence of rapid rotators requires a weak braking law, inconsistent with the subsequent rapid braking. Furthermore due to the anguular momentum core resurfacing, braking nearly stops on the MS, in contradiction with the observations. With a coupling time of 500 Myr, the angular momentum transfer from the core to the envelope really becomes effective at the age of the M 34/M 7 clusters, and is far from being finished by the age of the Sun. The internal rotational profile of the Sun thus cannot be reproduced.
Results for an intermediate coupling time (20 Myr) are worse for both rapid and slow rotators. In young clusters such as Alpha Per and the Pleiades, the problem is the same as in the long coupling time model: rapid rotators require a weak braking law to arise, but rapid spin-down requires a strong braking law. As the coupling time is in the order of the age of these clusters, angular momentum transfer begins to be effective. This leads to an increase of the velocities of very slow rotators, and braking of rapid rotators occurs only on a very short time scale. In older clusters such as M 34 and the Hyades, the transfer is in progress, so that braking is slow, and the model predicts too large velocities. At the age of the Sun, however, the transfer is finished and the star is a quasi-solid body, even for the most rapid rotators.
The main problem of the short coupling time model (1 Myr) is the existence of a large number of slow rotators in young clusters. The only process that keeps the star from spinning up is the disk-locking. It is then necessary to suppose that a fraction of the disks can survive as long as 40 Myr. A short coupling time can easily reproduce rapid rotators on the ZAMS and rapid decrease of these rotators from the age of Alpha Per for 1 stars (and from the age of the Pleiades for lower masses) to the age of the Sun. This model is also in good agreement with the observed solar rotational profile.
As pointed out in Sect. 3.2, most of the theories of angular momentum transfer predict that the coupling time is a function of angular velocity (core, envelope or both). In a way, the effect of the difference between the core and envelope rotational velocities is taken into account in the angular momentum exchanges quantity . In spite of this, rapid rotators are easier to account for with solid-body rotation, over the all mass range, while slow rotators are well fitted with a strongly decoupled model. For the latter, the coupling time scale should be at least of order of the age of the Pleiades (100 Myr) so that angular momentum transfer is only effective on the main sequence, thus allowing ZAMS slow rotators. On the other hand, the coupling time scale should not be much longer so that there is no important differential rotation left at the age of the Sun.
Fig. 16 presents the evolutionary tracks for 0.8 and 1 models with an initial period of 8 days, and with two different coupling times: I use a short coupling time to fit rapid rotators, and a model with a coupling time of 100 Myr to fit slow rotators. As discussed in previous section and in this one, a solid-body model is able to reproduce the braking of rapid rotators all over the main sequence phase for both masses. On the other hand, the model with =100 Myr can easily explain existence of slow rotators in the 0.9-1.1 mass range with disk-lifetimes of 10 Myr at most. The tracks presented in Fig. 16 start with an initial period of 8 days and can fit velocities as low as 6 km at the age of the Pleiades, i.e an important fraction of the stars in the Pleiades. A T Tauri distribution with periods up to 16 days can explain the existence of the Pleiades very slow rotators which have velocities lower than 6 km .
But ZAMS slow rotators still pose severe problems to the models. Slow rotators in the mass range 0.6-0.9 are more difficult to account for because they are more numerous, and also because a 0.8 model leads to a higher velocity at the Pleiades age, than a 1 model (using the same initial period and disk lifetime).
In their paper, Krishnamurthi et al. (1997) claimed that there is a significant decrease of the fraction of slow rotators between the age of Alpha Per and the age of the Pleiades: there are no stars in the velocity range 0-7.5 km in Alpha Per , while 10% of the Pleiades stars have velocities lower than 7.5 km . But recent observations in these clusters -presented on the figures of rotational evolution- show that 1) stars in the mass range 0.9-1.1 with velocities lower than 6 km are not braked, 2) stars with lower than 10 km are only slightly braked, 3) stars in the mass range 0.6-0.9 are significantly braked at all velocities (Queloz et al. 1997a, 1997b, Allain et al. 1997).
It seems to be different at early ages. The mean rotation in the young clusters IC2391 and IC2602 (30 Myr) is higher than in Alpha Per . But for older clusters, there is no evidence that the mean rotation in M 34 is larger than in the Pleiades. Conversely, Hyades' mean rotation is definitely lower than in the Pleiades.
How slow rotators evolve during the ZAMS and early-MS phases is a crucial issue to constrain the models. From Fig. 16, results for =100 Myr are in good agreement with the 1 observations of very slow rotators but this model seems predicts a too important braking for rotators with velocities between 10 and 20 km . On the other hand, this model finds a braking for slow rotators in the mass range 0.6-0.9 consistent with the observations, while a solid-body model cannot brake these stars.
From the results presented on this paper, I conclude that rapid rotators can be assimilated to solid bodies, and that slow rotators are submitted to an important differential rotation. The parametric description used here is however too basic to explain velocity effects, especially for moderate rotators, and a more physical description, handling angular momentum transfer in the radiative core, is required. Masses effects - probably through the depth of the convective zone- are also very important in angular momentum transfer. More observations are clearly needed to constrain the models both at different ages and different masses.
© European Southern Observatory (ESO) 1998
Online publication: April 20, 1998