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Astron. Astrophys. 333, 629-643 (1998)

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5. Core-envelope decoupling

In this section I present the effects of the choice of the coupling time on the evolutionary tracks of a single star. The results of the model are presented for four different masses: 1, 0.8, 0.6 and 0.5 [FORMULA]. The braking laws are presented for the different models and the different masses in Figs. 8, 9 and 10 where the angular momentum loss is presented as a function of the angular velocity. The braking law computed by Charbonneau (1992) from Weber & Davis model is also plotted. For the three coupling time models presented here, the braking rates are always lower than the braking rate predicted by the Weber-Davis wind model.

[FIGURE] Fig. 8. Braking law for the short coupling time model, [FORMULA] =106 yr. solid line is for 1  [FORMULA], dashed is for 0.8  [FORMULA], dotted for 0.6  [FORMULA] and dash-dotted for 0.5  [FORMULA]. The thick solid line represents Charbonneau's braking law (1992) computed from Weber-Davis model. Values of the parameters are [FORMULA], [FORMULA] and [FORMULA] =35, 15, 10, 2.6  [FORMULA], for 1, 0.8, 0.6 and 0.5 [FORMULA], respectively.

[FIGURE] Fig. 9. Same as Fig. 8 for the intermediate coupling time model, [FORMULA] =2 [FORMULA] yr, and with [FORMULA], [FORMULA] and [FORMULA] =30, 15, 10, 2.7  [FORMULA].

[FIGURE] Fig. 10. Same as Fig. 8 for the long coupling time model, [FORMULA] =5 [FORMULA], and with [FORMULA], [FORMULA] and [FORMULA] =6, 5, 4, 1.2  [FORMULA].

5.1. Choice of the coupling time [FORMULA]

Evolutionary tracks are presented for 3 different coupling times: [FORMULA]  yr, which I will call "short", [FORMULA]  yr, will be called "intermediate", and [FORMULA]  yr, will be called "long". Choices of the coupling times are dictated by both theoretical and observational reasons. The short coupling time is short enough compared to the evolutionary time-scales (contraction, nuclear), so that the star can almost be considered as rotating as a solid-body. The long coupling time (corresponding to almost the age of the Hyades cluster) is far longer than other time scales involved in PMS evolution, so that at the arrival on the MS, stars in clusters like Alpha Per and the Pleiades can be considered as totally decoupled. Finally, I chose an intermediate coupling time which corresponds to the maximal coupling-time required to have a 1  [FORMULA] star in quasi solid-body rotation at the age of the Sun. Each model was calibrated so that 1  [FORMULA] tracks fit the solar value at the age of the Sun, and are therefore submitted to different braking laws. For a given model, braking law parameters [FORMULA], [FORMULA] and [FORMULA] were chosen as the best fit of the observations for 1  [FORMULA] over the complete evolutionary time interval considered here: from 1 Myr to the age of the Sun. And for other masses we scale the value of the saturation parameter [FORMULA]. In all models, the lower the mass, the lower the saturation value (see Figs. 8, 9 and 10).

Each model is discussed with three different disk lifetimes: [FORMULA], [FORMULA] and [FORMULA]  yr, for 1 and 0.5  [FORMULA], and [FORMULA], [FORMULA] and [FORMULA]  yr, for 0.8 and 0.6  [FORMULA]. The first disk lifetime represents a star which would lose its disk almost on the birth-line. Thus, for a given set of parameters (initial period, braking law) it can be considered as an upper limit to the velocities during the evolution. Four different masses are also represented : 1 [FORMULA] in Fig. 11, 0.8 [FORMULA] in Fig. 13, 0.6 [FORMULA] in Fig. 14, and 0.5 [FORMULA] in Fig. 15. Finally, models for 0.5, 0.6, 0.8 and 1  [FORMULA] are presented with an initial period of 8 days, which is the mean observed period during Classical T Tauri phase. For 1  [FORMULA] models evolutionary tracks are also plotted using initial periods of 4 and 16 days, which correspond to minimum and maximum periods found among T Tauri stars (Fig. 12).

[FIGURE] Fig. 11. Angular surface velocity evolutionary tracks for a single star of 1 [FORMULA] are represented for an initial period of 8 d and 3 different disk lifetimes: 0.5, 10 and 30 Myr. Dotted line is for [FORMULA] = [FORMULA]  yr, dashed line is for [FORMULA]  =  [FORMULA]  yr and solid line for [FORMULA] = [FORMULA]  yr. Observations are presented with different symbols: filled dots during PMS are CTTS, empty dots are WTTS. Empty triangles are PTTS, stars are IC2602 and IC2391. Filled dots on the ZAMS are Alpha Per , empty dots are the Pleiades. Crosses are M 34. Filled triangles are the Hyades, and the Sun is a dotted circle.

[FIGURE] Fig. 12. Angular surface velocity evolutionary tracks for a single star of 1 [FORMULA] , for the coupling time scales presented in Fig. 11, and 2 different initial period. Upper tracks are initial period of 4d and lower tracks are 16d

[FIGURE] Fig. 13. Same as Fig. 11 for M=0.8  [FORMULA] .

[FIGURE] Fig. 14. Same as Fig. 11 for M=0.6  [FORMULA]

[FIGURE] Fig. 15. Same as Fig. 11 for M=0.5  [FORMULA]

5.2. Short coupling time

In Figs. 11, 13, 14 and 15, the [FORMULA] = [FORMULA]  yr model is represented by dotted lines. This model is very close to a solid-body model, as angular momentum transport occurs upon a time-scale far shorter than evolutionary time-scales. For the 1  [FORMULA] model, solar calibration leads to [FORMULA], and the best fit of the observations is obtained for [FORMULA] and [FORMULA] = 35  [FORMULA]. With this model it is very easy to account for ultra fast rotators during PMS and ZAMS phases. The largest velocity is more than 70  [FORMULA] ([FORMULA] 140 km [FORMULA]), and the more rapid rotator in the 0.9-1.1  [FORMULA] mass range in Alpha Per has 100  [FORMULA] ([FORMULA] 200 km [FORMULA]  ). Then, from ZAMS ages the stars are rapidly braked. At the age of the Pleiades the largest velocity is however larger than the velocity observed (40  [FORMULA] vs 20  [FORMULA]). This suggests that the spin-down is not strong enough to reproduce the upper limit of velocities both in the Alpha Per and the Pleiades clusters. The rapid braking of rapid rotators requires a strong braking rate for high velocities, which requires a high value of the saturation rate ([FORMULA] = 35  [FORMULA]). A stronger braking law would lead lo lower velocities at the age of Alpha Per . It is however not clear that the age of the Pleiades is 80 Myr, as it is presented here. It has been claimed that the Pleiades were older (about 100 Myr) and some authors even find that the Pleiades cannot be younger than 130 Myr (Basri et al. 1997). If the Pleiades cluster is 100 Myr or more, the short coupling time model leads to a better agreement with the observations of maximum velocities: the track with an initial period of 8 days has an angular velocity of 20  [FORMULA] at 130 Myr.

At the age of the Hyades, all stars have converged to low angular velocity rates between 2 and 3 times the solar value, in accordance with the observations. And braking extends up to the age of the Sun. Observed braking of rapid rotators is thus roughly reproduced from the ZAMS up to the age of the Sun.

Because the only process that keeps the stars from spinning up is the disk-locking, the slow rotators at ZAMS ages and during the early MS are more difficult to account for. The angular momentum losses by magnetic wind occur over a longer time-scale, and are therefore inefficient during the PMS phase. It is then necessary to suppose that the disk-regulating is effective up to at most 30 Myr (Bouvier et al. 1997b for a discussion). A 10 Myr disk lifetime leads to a velocity of 20 km [FORMULA] at the Pleiades'age (we take the Pleiades age to be 80 Myr, Fig. 11), and a 30 Myr disk leads to v = 9-10 km [FORMULA]. Of course a longer period leads to a lower velocity at ZAMS ages and a star that would keep its disk 30 Myr, with an initial period of 16d, arrives on the ZAMS at 4 km [FORMULA].

In the Pleiades a large proportion of stars (about 50 %) have [FORMULA] lower that 10 km [FORMULA]. The solid-body model then requires that a large fraction of stars have long-lived disks. But the fraction of long disk lifetimes can be substantially reduced if we take into account the effect of the [FORMULA] distributions. The contamination by [FORMULA] factor is very important in the Pleiades cluster as all the stars are slow or moderate rotators. The mean value of [FORMULA] is [FORMULA], so that stars with true velocities between 10 and 13 km [FORMULA] will on average have [FORMULA] to the range 8-10 km [FORMULA]. [FORMULA] contamination is a statistical effect, and there probably is some stars with very low velocities, but the true fraction of slow rotators is lower than the observed fraction. Recent [FORMULA] measurements in the Pleiades cluster (Queloz et al. 1997a, 1997b) suggest that the fraction of true velocities lower than 10 km [FORMULA] is [FORMULA] 35 % for solar-type stars, thus lower than the fraction of [FORMULA].

Bouvier et al. (1997b) suggested that the initial distribution of period at T Tauri ages is well fitted with a Gaussian curve with a mean period of 8 d. In their model, with a disk lifetime of 10 Myr, a star with an initial period of 8 d reaches the age of the Pleiades with a velocity of 10 km [FORMULA]. It thus requires that 35 % of the stars keep their disk longer than 10 Myr in order to reproduce the 35 % of slow rotators. In their paper Bouvier et al. summarize the different PMS disks observations and surviving disk fractions estimations at different ages. At the moment, and without taking into account PMS stars dispersed in star forming regions recently discovered by ROSAT, the fraction of stars still surrounded by a disk at an age of 10 Myr is 10 to 30 % (Strom et al. 1995, Lawson et al. 1996). It thus seems difficult to explain that 35 % of the stars have rotational velocities lower than 10 km [FORMULA] at the age of the Pleiades. In the present model (Fig. 11), stars which decouple from their disk at an age of 10 Myr reach the age of the Pleiades with a velocity of 20 km [FORMULA]. Such a difference with Bouvier et al. model can be explained by a difference in the braking law. In that case, it is even more difficult to explain the large fraction of slow rotators on the ZAMS.

By the time the stars arrive on the ZAMS, at the Alpha Per cluster age, the slow rotators are submitted to a weak braking with the short coupling time model. It is very difficult indeed to slow down slow rotators if we consider solid-body rotation, because the braking applies on the entire star. The consequence is that between the age of the young clusters IC2391 and IC2602 (30 Myr) and the age of the Pleiades, the slow rotators - below 10 km [FORMULA] - should keep roughly the same velocity. In other words, the proportion of very slow rotators in these clusters must be roughly the same. This point will be discussed in Sect.  6.

0.8  [FORMULA] and 0.6  [FORMULA] models require, respectively, [FORMULA] =15 and 10  [FORMULA] to account for fast rotators in young clusters (Fig. 13 and 14). For both 0.8 and 0.6  [FORMULA], rapid rotators and their braking during the main sequence, are well fitted. As for 1  [FORMULA] stars a large fraction of stars are slow rotators.

At later ages on the MS, the short coupling time models find that the lower the mass, the lower the velocity. The braking law was chosen so that the 1  [FORMULA] model fits the solar value. For 0.8  [FORMULA] model, velocities at the age of the Sun are between 0.8 and 0.9 the solar value. For 0.6  [FORMULA], evolutionary tracks stop before reaching solar ages, but from the position of the tracks at the Hyades age, 0.6  [FORMULA] model should reach lower values that the 0.8  [FORMULA] model . This point is thus in agreement with the observations in the Hyades cluster.

The velocity distributions for 0.5  [FORMULA] stars show significant differences with other masses in young clusters like Pleiades and Alpha Per , and for evolved ones like Hyades. In the formers, the distributions are quite flat - there is no peak of the distribution for slow rotators - and for the latter, rapid rotators (with velocities up to 25 km [FORMULA]) still exist. These distributions are well fitted with a short coupling time model with [FORMULA] = 2.6  [FORMULA], which means that braking goes from Skumanich's law to saturated regime, with no intermediate Mayor-Mermilliod regime. Velocity braking occurs at 150-200 million years, and is not achieved at the age of the Hyades. It also means that at the age of the Pleiades, maximum is not attained, and 0.5  [FORMULA] stars have not reached the ZAMS.

With 0.5 and 0.6  [FORMULA] models it is more difficult to account for rapid rotators in young clusters than with 0.8 and 1  [FORMULA] models. A larger initial velocity would however account for rapid rotators.

In conclusion, the main problem for the solid-body model is the existence of numerous slow rotators in young clusters. This might not really be a problem if these slow rotators are [FORMULA] contaminated. Precise determinations of the true slow rotators fraction, both at the Alpha Per cluster and the Pleiades ages are required to answer the question whether solid-body model, with a realistic disk lifetimes distribution, can reproduce the slow rotators observations.

5.3. Intermediate [FORMULA]

In Figs. 11, the intermediate [FORMULA] model is represented by dashed lines. I find the best model to have nearly the same braking law as the short coupling time model: [FORMULA], [FORMULA] and [FORMULA] = 30  [FORMULA]. A weaker braking law would lead to faster rotators on the ZAMS, but would also lead to faster rotators on the MS. The rapid braking phase is very short, and rapid rotators are not significantly braked between Alpha Per and the Pleiades.

What makes a rapid spin-down so difficult to obtain is that with [FORMULA] = 2 107  yr, angular momentum transfer from the core to the envelope begins at ages typically between those of the clusters IC2602/2391 and Alpha Per . This makes braking very inefficient during the early stages of MS as the rate of angular momentum transfered from the core to the envelope compensates the loss of angular momentum from a magnetic wind. This leads to a plateau of the rotation curve between 70 Myr and 250 Myr. On the other hand, with a stronger braking law, it would be impossible for the stars to reach high velocities of 200 km [FORMULA] on the ZAMS. The best braking law is then a compromise between the existence of fast rotators and rapid spin-down of these rotators.

With this model very slow rotators are even more difficult to account for than with the short coupling time model. With a disk lifetime of 30 Myr, the short [FORMULA] model leads to velocities of 10 km [FORMULA] on the ZAMS (Fig. 11), while the intermediate [FORMULA] model leads to a larger value of the rotation rate of 12 km [FORMULA]. This can be explained as follows : after 30 Myr, angular momentum transfer from the core to the envelope for the intermediate coupling time model is in progress, thus leading to higher values of the rotation rate. It is the opposite if we consider the tracks with a disk lifetime of 10 Myr : the short coupling time model leads to higher velocities than the intermediate model. This difference between the two disk lifetimes tracks can be explained by core-envelope decoupling effects: between 10 and 30 Myr, the angular momentum transfer from the core to the envelope is not yet effective and braking is more efficient for the de-coupled model.

With the intermediate coupling time model angular momentum transfer is really effective between a few 10 Myr and a few 100 Myr, so that breaking nearly stops (this effect is especially important for solar-type stars). At later ages, minimum velocities reached for clusters M 34 and Hyades are too large compared to observed velocities. And maximal velocities are as well far too large.

On the other hand, for a coupling time of 2 107  yr, stars at the age of the Sun have all been braked to a few km [FORMULA], and rotate almost like solid-bodies. This coupling time model is therfore able to reproduce the solar rotation profile, while a longer coupling time would lead to differential rotation for solar-type stars at solar age (see Sect.  5.4)

For 0.8 and 0.6  [FORMULA] models [FORMULA] is set to 15 and 10  [FORMULA], respectively (Fig  13 and 14). For slow rotators there is no significant difference between the short and the intermediate models. For 0.6 [FORMULA] models there are much less differences between short [FORMULA] model and the intermediate coupling model than for 1  [FORMULA] tracks. More precisely, the two models find almost the same velocities for highest and slowest rotators in clusters Alpha Per , the Pleiades, M 34 and even Hyades. There are two possible explanations to this. First, for 0.6  [FORMULA] stars, the evolutionary time is longer than for higher masses, and longer than the intermediate coupling time of 2 107  yr presented here. In that case, this model acts nearly as a solid-body model. Opposite to higher masses, this model leads to rotations rates lower than the short coupling time in the last evolutionary track - with [FORMULA] = 30 Myr. As the MS is not reached yet, spin-up from contraction remains the main process that controls the angular momentum evolution, and is more efficient in the case of solid-body rotation. Second, the lower the mass, the smaller the radiative core. The role of the core -a reservoir of angular momentum- is thus less important for lower masses as for higher masses. Therefore, inferred differential velocity is also less important.

For masses lower than 0.6  [FORMULA], [FORMULA] is set to 2.7  [FORMULA]. As for 0.6  [FORMULA] models , evolutionary tracks for this model are not significantly different from the short coupling time model. The slight difference is explained by different braking laws. The plot of the core velocity shows that the intermediate coupling time model acts like a solid model for 0.5 [FORMULA] (see previous section).

With this model, the coupling time is short enough to reproduce the solar rotational profile. But at least for 1  [FORMULA] stars, it is difficult to account for 1) the existence of UFR's on the ZAMS, and, 2) the rapid spin-down of these UFR's at the age of the Pleiades, 3) the important spin-down down to a few km [FORMULA] at the age of the Hyades and 4) slow rotators at the early stages of the main sequence evolution. For low masses of 0.5 and 0.6 [FORMULA], this model behaves almost like a solid-body model, and leads to a good agreement with the observations.

In their model, Keppens et al. claimed that a coupling time of [FORMULA] = 10 Myr is able to reproduce the observed velocity distributions at different ages, from the T Tauri phase up to the age of the Sun. From the work presented here, a model with a coupling time of 20 Myr leads to important differences with the observations. Differences in the assumptions of the models lead to differences in the resulting angular momentum evolution. Keppens et al. used an initial distribution mixing 0.8 and 1.0  [FORMULA] stars. But there are important differences between the 0.8 and 1  [FORMULA] observed velocity distributions, especially in the Pleiades, where solar-mass stars all have velocities lower than 60 km [FORMULA] while the largest observed velocity among 0.8  [FORMULA] is larger than 100 km [FORMULA] (Queloz et al. 1997). So the important braking observed for solar-type stars between the age of Alpha Per and the age of the Pleiades is not reproduced by their model. Furthermore, they artificially increase the number of very slow rotators among solar-type stars, as 0.8  [FORMULA] stars are more easily braked than 1  [FORMULA]. Finally, even with adding 0.8  [FORMULA] stars to their sample, they obviously cannot reproduce the large number of stars with velocities lower than 10 km [FORMULA] on the ZAMS. The maximum disk lifetime they use is 6 Myr, but even if they use a longer disk lifetime, they cannot produce more very slow rotators. From the tracks presented on Fig. 3, the longest disk lifetime (30 Myr) leads to a minimal velocity on the ZAMS of 10 km [FORMULA], while a a disk lifetime of 1 Myr leads to a velocity of 20 km [FORMULA]. It it thus easy to reproduce velocities in the range 10-20 km [FORMULA] on the ZAMS with an intermediate coupling time model. It is indeed the result found by Keppens et al., as at the Pleiades age, 60% of the stars are in this velocity range.

5.4. Long [FORMULA]

For this model, solar calibration leads to [FORMULA], and I use [FORMULA] and [FORMULA] = 6 [FORMULA]. A coupling time of 5 [FORMULA]  yr is far longer than PMS evolutionary times of the 1  [FORMULA] stars. The core and the envelope are then completely decoupled up to a few 100 Myr, i.e., far after the star has reached the age of the Pleiades. As only the convective envelope is subject to wind braking it is very easy to keep slow rotators at T Tauri ages slowly rotating up to the age of the Hyades. From the moment the star stops its contraction it is significantly braked.

For rapid rotators, this decoupling leads to a very efficient braking between the age of Alpha Per and the age of the Pleiades, more efficient than in the case of solid-body model. This rapid braking is indeed required to account for the decrease of velocities of the most rapid rotators in both clusters. During the same time scale slow rotators are also braked : a star with a velocity of 5 km [FORMULA] in Alpha Per has a velocity of 3 km [FORMULA] in the Pleiades. In that case, a disk lifetime value of 10 Myr is sufficient to account for velocities as low as 5 km [FORMULA] at the age of Alpha Per and 3 km [FORMULA] in the Pleiades.

While the spin-down is consistent with the observations during the early MS phases, it is less efficient on the MS at later ages because by an age of a few 108  yr angular momentum transfer from the core to the envelope becomes effective. A spread in the velocities then remains during MS. A spread in the velocities is really observed in the old clusters M 34 and the Hyades, but the model cannot reproduce the decrease of the maximum velocity between these two clusters, as the model predicts more a plateau than a braking. Therafter, it is necessary to suppose a strong Skumanich braking law (i.e a large value of [FORMULA]) to fit the braking at later ages down the solar value. But a main problem remains for initial ultra fast rotators as they cannot spin-down enough to reach convergence at the age of the Sun. Furthermore, with this model, a strong differential rotation remains inside the star at the age of the Sun: for the slowest rotators, the radiative core still rotates 4 times as fast than the envelope, and for fastest rotators, the core rotates 10 times as fast. And this is in contradiction with the observations of the angular velocities in the solar interior.

Another problem arises with Alpha Per 's rapid rotators, as the maximum velocity achieved is only 100 km [FORMULA]. A larger initial velocity would lead to a larger velocity at ZAMS ages. But such rapid rotators would also lead to far too large velocities during MS phase (see Fig. 12).

Model of 0.8 and 0.6  [FORMULA] require, respectively, [FORMULA] =5 and 4  [FORMULA] (Fig  13 and 14). In young clusters, the same holds for 0.8 and 0.6 [FORMULA] models as for the 1 [FORMULA] model: it is difficult to fit rapid rotators, but the fits of slow rotators are quite good. Conversely, for the Hyades cluster, 0.6 and 0.8 [FORMULA] models lead to a spread larger than observed, while the 1 [FORMULA] model spread fits the data well.

For 0.5  [FORMULA], [FORMULA] is set to 1.2 [FORMULA] to account for ZAMS rapid rotators. Evolutionary tracks look quite the same as in other coupling time models for the same mass to the age of the Pleiades. The differences to other coupling time models occur after the ZAMS, where braking is much more rapid. Therefore, at the age of the Hyades, this model cannot account for the fastest rotators (50  [FORMULA]). It also finds a lower limit (1  [FORMULA]) which is too low, whereas Hyades slow rotators have rotation rate between 4 and 8  [FORMULA]. The long coupling-time is still too long compared to the evolutionary time, and as for the 0.8 or 0.6  [FORMULA] models in M 34, it leads to too slow rotators during the MS phase. This model would eventually require a weaker braking law to reproduce the 0.5 [FORMULA] Hyades members.

The main successes of this model is to easily explain 1) the rapid decrease of rapid rotators between Alpha Per and the Pleiades, and 2) the existence of a large number of very slow rotators from T Tauri ages to the age of the Sun in the 0.9-1.1  [FORMULA] mass range without requiring long disk lifetimes. But this model cannot reproduce 1) ZAMS rapid rotators, 2) braking of rapid rotators during the main sequence, 3) large velocities of low-mass stars in the Hyades cluster and 3) the rotational profile of the Sun.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998