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

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4. Effects of varying the parameters

The evolution of a single star is submitted to 6 parameters related to the initial velocity, the disk lifetime, the wind braking law and the coupling time. In this section I present the effect of each parameter, independently of the value of the coupling time-scale. The effect of varying the coupling time is discussed in the next section.

4.1. Disk lifetime

The effect of the disk lifetime [FORMULA] for a 1  [FORMULA] star is presented on Fig. 3. The track for [FORMULA] =0.5 Myr corresponds to a star that looses its disk on the birth-line and, for a given initial velocity, sets an upper limit on velocities during the PMS and MS evolution. The 10 Myr and 30 Myr lifetimes set a lower limit on velocities and are used to compare the predictions of the different models for ZAMS slow rotators. The longer the disk-lifetime, the slower the rotator on the ZAMS. It is obvious for the surface velocity, but it is less obvious for the core velocity. The core appears with the same angular velocity than the envelope. If the disk still remains, the core immediately shows an acceleration, while the envelope keeps a constant angular velocity.

[FIGURE] Fig. 3. Evolutionary tracks of a 1  [FORMULA] star for different disk lifetimes: 0.5, 1, 5, 10 and 30 Myr. Surface velocities are represented by solid lines and core velocities by dashed lines. Model is presented for a coupling time of 2 [FORMULA] yr, [FORMULA] = 30 [FORMULA], [FORMULA] = 2.7 [FORMULA] and [FORMULA] = 4.2 [FORMULA].

The disk-regulation nevertheless keeps the core from spinning up too quickly. From Eqs.  2, Eq.  1writes:

[EQUATION]


The angular momentum exchanges is a function of the difference of rotation between the core and the envelope, and as long as the star is coupled to the disk, the envelope keeps a very low rotation rate, leading to a large value of the quantity [FORMULA], and thus a slower spin up of the core.

Therefore the disk-lifetime has an indirect consequence on differential rotation. On the other hand, the disk lifetime has no influence on MS velocities, past a few 109  yr, nor on the moment when convergence between the core and the envelope is reached.

4.2. Initial velocity

Classical T Tauri observations indicate that at an age of a few [FORMULA] yr, stars have rotational periods in the 4-16 days range with a peak around 8 days (see Sect. 2). This initial spread in velocities will remain at later ages. Fig  4 shows the evolution of a 1 [FORMULA] star that would appear in the T Tauri phase with an initial period of 4, 8 and 16 days. I present here velocities for radiative and convective zones in the case where the star would keep its accretion disk during 106 yr and for the following values of the model parameters: [FORMULA] = 2 [FORMULA] yr, [FORMULA] = 2.7 [FORMULA], [FORMULA] = 4.2 [FORMULA], and [FORMULA] = 30 [FORMULA].

[FIGURE] Fig. 4. Evolutionary tracks for the envelope (thick lines) and the core (dotted lines) velocities of a 1  [FORMULA] star for 3 different initial periods: [FORMULA] d (upper track), 8d (middle track) and 16d (lower track). See text for values of the parameters.

A difference in the initial velocity remains all over the PMS and early MS evolution. A star that would have an initial velocity twice as large as another , will rotate twice as fast during the entire PMS phase. On the other hand, initial velocity has no influence upon the final velocity when MS braking is achieved, in other words, the MS star does not keep memory of its initial velocity. Initial velocity has no influence either on differential rotation: the core velocity varies in the same way the surface velocity does, and the time when convergence is attained does not change.

Effects of different initial velocities can be distinguished from the effects of different disk lifetimes for slow rotators only. More precisely, if the disk disappears before the core develops, evolutionary tracks look exactly the same in both cases. Difference occurs when the disk survives after the core has developed because the envelope keeps a constant period while the core spins up. In the case of different initial periods, even for slow rotators the core and the envelope both spin up.

4.3. The braking law

As described in the previous section, 3 parameters are used to describe the magnetic braking law: [FORMULA], the constant for slow rotators, [FORMULA], for intermediate rotators, and [FORMULA], the rotational value of the saturation. The value of the rotation rate at which braking goes from the Skumanich regime to the Mayor-Mermilliod regime is defined by : [FORMULA]   [FORMULA].

Fig. 5 shows the influence of the constant of the Skumanich's law for slow rotators, [FORMULA], for the core and the envelope. Three different values are presented: 7.5  [FORMULA], 3  [FORMULA] and 1.2  [FORMULA]. The values of the other parameters are [FORMULA], corresponding to [FORMULA]  = 19.5, 4.9, and 1.22  [FORMULA], and [FORMULA]  = 30  [FORMULA]. The effect of [FORMULA] is concentrated on slow rotators on the MS for both the envelope and the core velocities. The value of [FORMULA], by influencing the value of [FORMULA], influences the moment the tracks enter the Skumanich's regime on the main sequence. Thus, the lower [FORMULA] the weaker the braking law on the main sequence, and the earlier the star enters the Skumanich's regime. Both effects contribute to a weaker braking of the slow rotators.

[FIGURE] Fig. 5. Evolutionary tracks of a 1  [FORMULA] star for 3 different values of the parameter [FORMULA]: 7.5 [FORMULA] upper track, 3 [FORMULA] is middle track,and 1.2 [FORMULA] is lower track. Tracks for the envelope are thick lines, and for the core are thin lines. The correspondent values of [FORMULA] are 19.5, 4.9 and 1.2 [FORMULA].

The other parameters are the constant of the braking law which applies for moderate rotators [FORMULA], and the value of the rotation rate at which saturation occurs [FORMULA]. To investigate the effect of [FORMULA] I take extreme values: 5.5, 30 and 60  [FORMULA]. For the lower saturation rate the braking goes from the Skumanich regime to the saturated regime with no intermediate Mayor-Mermilliod regime. For the higher rotation rate the stars are most of the time in the Mayor-Mermilliod regime. Fig. 6 presents 3 tracks corresponding to three different values of the disk lifetime: 0.5, 10 and 30 Myr. The values of the other parameters are [FORMULA]   [FORMULA] yr, [FORMULA], [FORMULA] ([FORMULA]  = 5.4  [FORMULA]). The effect of the parameter [FORMULA] is concentrated on rapid rotators for high saturation values and affects slow rotators for low [FORMULA] only. As for [FORMULA] the effect is essentially sensitive between 10 and 100 Myr, i.e from ZAMS ages. The lower the saturation rate the weaker the braking law and the higher the rotation on the ZAMS. For [FORMULA]  = 60  [FORMULA], the maximum velocity is only 50  [FORMULA], so the saturation value is never reached. For long disk lifetimes [FORMULA]  = 30 and 60  [FORMULA] tracks are superimposed: the saturation value is never reached, and hence the braking law is the same for both tracks. In addition of changing the maximal value reached on the ZAMS, a difference in [FORMULA] changes the moment when this maximum is reached: the lower [FORMULA] the later the braking occurs. Furthermore, the effect of a weaker braking law is also seen during early MS, where it is more difficult to brake ZAMS ultra fast rotators (UFR's).

[FIGURE] Fig. 6. Evolutionary tracks of a 1  [FORMULA] star for 3 different values of the parameter [FORMULA]: 5.5  [FORMULA] is solid line , 30  [FORMULA] is dashed line, and 60  [FORMULA] is dotted. See text for values of the parameters.

Early pre-main sequence evolution (before 10 Myr) is independent of the value of [FORMULA] as the evolution during this phase is completely dominated by the contraction effects. Finally, the final velocity, at 1010 yr, is not dependent on [FORMULA].

Now, to investigate the effect of [FORMULA], [FORMULA] is set to 300  [FORMULA], so that the saturation value is never reached, and the star is most of the time in the Mayor-Mermilliod regime. The evolutionary tracks are presented for 3 values of [FORMULA]: 1.05  [FORMULA], 4.2  [FORMULA] and 1.68  [FORMULA] ([FORMULA]  = 1.36, 5.42 and 23.05  [FORMULA] respectively, see Fig. 7). [FORMULA] is set to 2 [FORMULA] and [FORMULA]. The value of [FORMULA] affects tracks for both rapid and slow rotators. The lower [FORMULA], the higher the velocity on the ZAMS, and the later the braking. Differences in the tracks remain during early phases of the main sequence and progressively disappear. The rotation rate at the age of the Sun still slightly depends on the value of [FORMULA].

[FIGURE] Fig. 7. Surface rotation evolutionary tracks of a 1  [FORMULA] star for 3 different values of the parameter [FORMULA]: [FORMULA] =1.05 [FORMULA] ([FORMULA] = 1.4  [FORMULA]) is solid line , 4.2 [FORMULA] ([FORMULA] = 5.4  [FORMULA]) is dashed line for , and 1.68 [FORMULA] ([FORMULA] = 23.0  [FORMULA]) dotted. See text for values of the parameters.

The values of the braking law parameters can be constrained by the observations: as [FORMULA] and [FORMULA] poorly affect the tracks for slow rotators on the main sequence [FORMULA] is completely determined by the solar calibration of the model. And a combination of the two parameters [FORMULA] and [FORMULA] is determined to fit both rapid rotators on the ZAMS, and their rapid braking on the early MS.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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