The challenge for any theoretical modeling of the rotational evolution of low-mass stars is to provide a convincing scenario that agrees with all of these observational facts. To be specific, the distribution of rotation rates that results from evolutionary calculations must account for the following constraints:

1. the apparent bimodal period distribution of the T Tauri stars,

2. the simultaneous presence of very slow rotators and a tail of very fast rotators in the vicinity of the zero-age main sequence,

3. the rapid spin down of the fastest rotators,

4. the apparent longer spin-down time for the lower mass stars,

5. the fact that all low-mass stars end up as very slow rotators after several 100 Myr, spinning down according to a t-1/2 law as they slowly evolve on the main sequence.

It is to these questions that we now turn.

One of the puzzling aspects of star formation is the substantial discrepancy between estimates of the angular momentum in the cores of molecular clouds and in the youngest optically visible low-mass stars. To be specific, the typical specific angular momentum J/M in a molecular cloud is of the order of 1021 cm2 s-1, while in a typical T Tauri star near the birthline one finds J/M ^ 5 x 1017 cm2 s-1. Since angular momentum is approximately conserved during the near free-fall collapse of aprotostellar cloud, it has often been suggested that spin angular momentum can be converted into orbital angular momentum through fragmentation. In fact, the formation of wide binaries or multiple systems, as well as the formation of circumstellar disks around the fragments, greatly minimizes the angular momentum problem. However, whereas wide binaries are most likely reservoirs for the angular momentum of a collapsing molecular cloud, the existence of close binaries with orbital angular momentum two orders of magnitude smaller remains an open question. Moreover, detailed numerical calculations by Durisen et al. (1984) indicate that each individual fragment should be rotating near break up at the end of its gravitational collapse. This is not borne out by the observations. The presence of a circumstellar disk around some T Tauri stars is also a serious problem since they tend to accrete mass and hence angular momentum from the disk. Following Hartmann and Stauffer (1989), we find that typical accretion rates of 10-7M0 yr-1 are sufficient to spin up a 1M0 star to about half of the breakup velocity in a time comparable to the age of the youngest T Tauri stars, about 1 Myr. Yet, observations reveal that most of the T Tauri stars surrounded by accretion disks are rotating relatively slowly, with v sin i < 20 km s-1. Since this value is one order of magnitude lower than the breakup velocity of a classical T Tauri star, it appears that the processes that control the rotation rate of such a star were probably operative during the early phases of its collapse.

As was pointed out in Section 7.3. T Tauri stars surrounded by accretion disks not only have rotational velocities much smaller than the breakup velocity but have systematically longer rotation periods than stars of similar mass and age that do not exhibit accretion disks. These observations strongly suggest that the accretion disk is acting to counter the spin-up torque expected both from pre-main-sequence contraction and from the deposition of high angular momentum material from the disk onto the star. Broadly speaking, two distinct angular momentum regulation mechanisms have been proposed, both of them relying on the interaction between the magnetosphere of a rotating star and a circumstellar accretion disk.

Konigl (1991) has invoked the theory of Ghosh and Lamb (1979) for accreting magnetic neutron stars and white dwarfs to explain the slow rotation rate of T Tauri stars as the result of magnetic coupling to a truncated disk. In this model the poloidal magnetic field of the star has a closed global structure, modeled as an aligned dipole. Material in the disk that spirals slowly inward moves along the closed field lines and is channeled onto the star at high latitudes. That is to say, the dipolar field disrupts the inner parts of the circumstellar disk and the central star becomes effectively coupled to the disk several radii out. This possibility was investigated by Konigl, who found that a kilogauss field could disrupt the disk at a distance of a few stellar radii from the center and that the spin-down torque transmitted by the field lines that thread the disk beyond the corotation radius could indeed balance the spin-up torque applied by accreting material. More recently, Cameron and Campbell (1993) have shown that a T Tauri star could evolve into a state of rotational equilibrium within the duration of the Hayashi phase, despite the rapid contraction of the star. The resulting rotation rates of their models, which have magnetic fields of a few hundred gauss and an accretion rate of a few 10-8M0 yr-1, are also found to be consistent with the observed rotation rates of classical T Tauri stars.

Alternatively, Shu et al. (1994) have proposed a model in which shielding currents in the surface layers of the disk are invoked to prevent penetration of the stellar field lines everywhere except near the corotation radius Rcorot, where the Keplerian angular velocity of the disk matches the angular velocity of the star. Exterior to Rcorot, matter diffuses onto field lines that bow outward, resulting in a magnetocentrifugally driven wind with a mass loss rate proportional to the disk accretion rate MD. Matter interior to Rcorot diffuses onto field lines that bow inward and is funneled onto the star's surface. It is found that this flow actually results in a trailing-spiral configuration for the magnetic field and that it transfers angular momentum from the star to the disk as long as the corotation radius remains significantly greater than the star's equatorial radius R. As was shown by Ostriker and Shu (1995), for an aligned stellar dipole of strength m = HR3 the corotation radius is given by where H is the field strength at the equator, G is the constant of gravitation, and M is the star's mass. Parenthetically note that the Ghosh-Lamb theory yields a similar relation, except for the value of the numerical constant, which is not exactly known but should be of order unity. Making use of Kepler's third law, one obtains the star's rotation period as

Letting Pmt = 8 days, M = 0.5Mo, and Md = 10-1MQ yr-1 in Eq. (7.10), one finds that m = 7.32 x 1036 in cgs units. This value corresponds to H ^ 800 gauss and

Rcorot ^

4.4R for a star with R = 3R0. Thus, given reasonable values for the stellar parameters, appropriately slow rotation rates are obtained for the classical T Tauri stars.

The hypothesis of disk-regulated angular momentum provides an attractive framework for understanding the rotational evolution of low-mass pre-main-sequence stars. No commonly accepted model exists at the present time, however, since the fine details of the disk-star interaction are still to be modeled quantitatively. Nonetheless, ample evidence now exists that an accretion disk may play a fundamental role in regulating the rotation rate of a classical T Tauri star, holding its angular velocity almost fixed during Hayashi track evolution. This locking results in net transfer of specific angular momentum from the central star to the disk, so that the total angular momentum of the star steadily decreases in time until its regulating accretion disk is fully dissipated. If

so, then the observed bimodal period distribution for T Tauri stars clearly indicates that the fast rotators are stars that, for one reason or another, are not strongly locked to an accretion disk during Hayashi track evolution. Hence, because they remain free to spin up in response to changes in moment of inertia as they contract, they also cover a wider range of rotation periods than their disk-locked counterparts. As was noted by Choi and Herbst (1996), the gap in the histogram of T Tauri stars is evidence of the rapid evolution through which a star passes on its way to another mechanical equilibrium, once released from its disk induced rotational lock.

In Section 7.3 we have seen that the rotation distribution among main-sequence dwarfs of spectral type G and later in very young open clusters consists of a narrow peak at v sin i = 10 km s-1 or less and an extended tail of rapid rotators, with v sin i > 100 km s-1 (see Figure 7.5). As was noted by Cameron, Campbell, and Quaintrell (1995), the presence of fast and slow rotators in the same cluster suggests that this peak-and-tail distribution is already established when the cluster stars reach the zero-age main sequence. In order to check the validity of that assumption, they have thus expanded the work of Cameron and Campbell (1993) to determine how disk braking might affect the histogram of rotation periods for low-mass stars on the zero-age main sequence. For disk masses of a few hundredths of a solar mass or more, and dynamo-generated field strengths of a few hundred gauss, their numerical calculations indicate that the net (magnetic plus accretion) torque is sufficient to pull the star's rotation into quasi-static equilibrium before the end of the Hayashi phase, with the resulting rotation rate being one order of magnitude lower than the breakup rate. Thence, by the time this equilibrium breaks down due to the dwindling accretion rate, the star's rotation is effectively independent of both the disk mass and the initial angular momentum of the star. For lower disk masses, however, such an equilibrium is never established so that the star can retain a greater fraction of its initial angular momentum. The histogram of rotation rates that results from a reasonable choice for the distribution of disk masses has the form of a low-velocity peak and an extended high-velocity tail. The slow rotators are the stars that evolved into rotational equilibrium as classical T Tauri stars and gave away most of their initial angular momenta to their former disks; the stars in the tail are those with lower initial disk masses, in which rotational equilibrium was never established during the Hayashi phase. If this is the case, then their model provides a natural explanation for the histograms depicted in Figure 7.5.

Since the early 1990s, much theoretical effort has been expended in trying to understand the rotational history of a low-mass star, both before and during the main-sequence phase. Notably, MacGregor and Brenner (1991) have developed a particularly simple description of the transport of angular momentum within the interior of a solartype star. In this section I shall briefly describe their model, its use in conjunction with a suitable parameterization for the angular momentum loss resulting from magnetized stellar winds, and some of the numerical results obtained by Keppens, MacGregor, and Charbonneau (1995).

Broadly speaking, their approach to constructing an evolutionary sequence is to simplify matters by separating computation of the rotational evolution from that of the internal, structural evolution. Accordingly, we shall assume that the effects of rotation on internal structure are small, so that an evolutionary track for a spherical star of the same mass can be used to calculate the star's radius R, the radius of the convection zone base Rconv, the mass of the radiative core Mcore, and the moments of inertia, Icore, and Iconv, of the core and envelope. We shall further assume that the radiative core and the convective envelope each rotate rigidly, although not necessarily at the same rate. If ^core and ^conv are the angular velocities of the core and envelope, then the angular momenta of these regions are Jcore = Icore^core and Jconv = Iconv^conv. With these assumptions, the equations governing the time evolution of these angular momenta can be derived by considering the processes by means of which angular momentum is redistributed and lost.

During pre-main-sequence contraction, angular momentum is reapportioned between the core and the envelope as a consequence of the gradual conversion of the stellar interior from a nearly fully convective state to one in which most of the mass is contained within the radiative core. Thus, if dMcore/dt denotes the rate of growth of the core mass, angular momentum exchange will occur at the rate jdMcore/dt, where

is the specific angular momentum of material in the thin spherical shell about the radius r = Rconv(t) that is undergoing assimilation at the core at time t.

We now assume that the torque exerted by the magnetically controlled wind extracts angular momentum only from the surface convection zone. The resulting deceleration of the convective envelope causes a shear to develop at the core-envelope interface. In a real star, this would lead to the creation of interfacial stresses that would act to redistribute angular momentum between the two regions. In the MacGregor-Brenner heuristic model, one simulates this transport process by assuming that an amount of angular momentum

^core + ^conv is transferred from the core to the envelope in a specified time Tc. Note that an instantaneous exchange of angular momentum A J would equilibrate ^core and ^conv, thereby restoring an angular momentum distribution that satisfies the essential stability condition defined in Eq. (3.98).

In the absence of magnetic coupling with an accretion disk, the combination of the foregoing effects can be written down in the form dJcore A J , . ^^core

dt Tc dt for the core, and dJconv A J _ ^^^core Jconv /t i

for the surface convection zone. In these equations, tc is the prescribed core-envelope coupling time and Tw is the e-folding time for wind-induced angular momentum loss from the convective envelope. (The time scale tw needs to be calculated from a reasonable model for the steady-state expansion of the stellar corona.) Once Rconv, dMcore/dt, 7core,

Fig. 7.6. The evolution of the rotation rate (in units of = 3 x 10-6 s-1) of the core, ^core, and the convective envelope, ^conv (thicker lines) for a single star. Panel A: For a 1 M0 star, with initial equatorial velocity i>eq = 15 km s-1 and coupling time scale Tc = 20 Myr, for three different dynamo prescriptions. The solid lines are for a linear dynamo; the dashed lines for a dynamo saturated at ^conv > 5Q0; and the dash-dotted lines for a dynamo saturated at ^conv > 1O^0. Panel B: The rotational histories fora 1M0 star having ueq = 15kms-1anda linear dynamo for Tc = 5 Myr (dashed lines), 20 Myr (solid lines), and 50 Myr (dash-dotted lines). Panel C: A 1M0 star, with Tc = 20 Myr and a linear dynamo, for ueq = 5 km s-1 (dashedlines), ueq = 15 km s-1 (solid lines), and ueq = 25 km s-1 (dash-dotted lines). Panel D: For a star of mass 0.8M0 (dashed lines), 0.9M0 (dash-dotted lines), and 1.0M0 (solid lines), with Tc = 20 Myr, a linear dynamo, and i>eq = 15 cm s-1. Source: Keppens, R., MacGregor, K. B., and Charbonneau, P., Astron. Astrophys., 294,469, 1995.

Fig. 7.6. The evolution of the rotation rate (in units of = 3 x 10-6 s-1) of the core, ^core, and the convective envelope, ^conv (thicker lines) for a single star. Panel A: For a 1 M0 star, with initial equatorial velocity i>eq = 15 km s-1 and coupling time scale Tc = 20 Myr, for three different dynamo prescriptions. The solid lines are for a linear dynamo; the dashed lines for a dynamo saturated at ^conv > 5Q0; and the dash-dotted lines for a dynamo saturated at ^conv > 1O^0. Panel B: The rotational histories fora 1M0 star having ueq = 15kms-1anda linear dynamo for Tc = 5 Myr (dashed lines), 20 Myr (solid lines), and 50 Myr (dash-dotted lines). Panel C: A 1M0 star, with Tc = 20 Myr and a linear dynamo, for ueq = 5 km s-1 (dashedlines), ueq = 15 km s-1 (solid lines), and ueq = 25 km s-1 (dash-dotted lines). Panel D: For a star of mass 0.8M0 (dashed lines), 0.9M0 (dash-dotted lines), and 1.0M0 (solid lines), with Tc = 20 Myr, a linear dynamo, and i>eq = 15 cm s-1. Source: Keppens, R., MacGregor, K. B., and Charbonneau, P., Astron. Astrophys., 294,469, 1995.

and Iconv are known along an evolutionary track, Eqs. (7.13) and (7.14) can be integrated to yield the rotational evolution of the core and envelope of a low-mass star.

In Figure 7.6 we illustrate the influence of the model parameters on the rotational evolution of a single star. Panels A, B, and C are calculated for a 1M0 star; they depict the effect of varying the dynamo prescription, the coupling time scale tc, and the initial equatorial velocity ueq. Panel D illustrates the rotational evolution of stars of different mass. Obviously, an important feature of these solutions is the convergence of rotation rates after a time of the order of 1 Gyr. It is also apparent that the rotational memory of a solar-type star is effectively lost at the age of the present-day Sun. In fact, all models considered end up rotating at nearly the present-day solar rotation rate (^0 ^ 3 x 10-6 s-1), with essentially no internal differential rotation.

Panel A of Figure 7.6 shows how the phenomenological dynamo prescription influences the rotation evolution of a 1M0, with ueq = 15 km s-1 and Tc = 20 Myr. The solid line corresponds to a linear dynamo, that is, a dynamo for which the strength of the mean coronal magnetic field increases linearly with rotation (see Eqs. [7.5] and [7.6]). The dashed lines and dash-dotted lines correspond to saturated dynamos, in which the mean coronal field saturates when the star rotates faster than, respectively, 5 and 10 times the present-day solar rotation rate. One readily sees that dynamo saturation reduces the angular momentum loss from the stellar wind since a lower field strength at the base of the corona causes less efficient magnetocentrifugal acceleration of the plasma. The angular momentum carried away by the stellar wind is therefore reduced, so that higher rotational velocities are achieved and sustained for a larger time. As was shown by Keppens and coworkers, a linear dynamo produces adequate spin-down early in the evolution but fails to produce sufficiently rapid rotators at the ages of a Persei and the Pleiades. Their analysis also shows that a saturated dynamo can explain the observed large spreads in rotation rates but the level of saturation is constrained by the requirement of achieving spin-down to slow rotation by the Hyades age (see Figure 7.5).

Making use of their parametric model for the rotational evolution of a single star, Kep-pens and coworkers have also investigated how the distribution of rotational velocities for late-type stars in the mass range 0.8-1.OM0 evolves with age. Starting from an initial distribution compiled from observations of rotation among T Tauri stars, they found that reasonable agreement with the observationally inferred rotational evolution of solar-type stars is obtained for: (i) a linear dynamo that saturates beyond 20 times the present-day solar rotation rate, (ii) a coupling time scale tc of the order of 10 Myr, (iii) a mix of stellar masses consisting of roughly equal numbers of 0.8M0 and 1.0M0 stars, and (iv) disk regulation of the surface rotation up to an age of 6 Myr for stars with initial rotation periods larger than 5 days. The first requirement is in agreement with the observed saturation in chromospheric and coronal emission fluxes in the fastest rotators (see Section 7.2). As they noted, however, a number of discrepancies remain. In particular, their calculations fail to produce a sufficiently large proportion of slow rotators (ueq < 10 km s-1) on the zero-age main sequence.

At this juncture it is appropriate to compare these results with some of the model calculations made by Barnes and Sofia (1996). Following closely the method described in Section 5.4.1, these authors have computed the overall redistribution of angular momentum by making use of a simple diffusion equation and some ad hoc prescription for their coefficient of eddy viscosity (see Eq. [5.43]). As usual, the values of that coefficient were obtained by requiring that the present-day Sun rotates at the observed rate. A suitable parameterization was also used to describe the angular momentum loss through the action of a magnetically channeled stellar wind. An important conclusion of their work is that angular momentum loss without saturation is unable to account for the presence of the fastest rotators in young star clusters, regardless of the initial rotation periods. Moreover, calculations of evolutionary models in the mass range 0.6-1.0M0 show that the saturation threshold is different for G, K, and M stars, with lower-mass stars saturating at lower angular velocities. Because lower-mass stars have deeper convective envelopes, this result seems to indicate that turbulent convection contributes significantly to the dynamo-generated magnetic fields of low-mass stars.

Insofar as comparison is possible, these results are quite similar to those obtained by Keppens and coworkers. In particular, both studies indicate that dynamo saturation is required to maintain a considerable spread in rotation rate at least until the age of the a Persei cluster (see Figure 7.5). Both studies also show that the observed spin-down of the slow rotators in the young open clusters is in better agreement with differentially rotating models than with rigidly rotating models. Since these investigations were carried out by means of models that make use of quite distinct parameterizations to treat angular momentum loss and redistribution, there is thus compelling evidence that saturated magnetized stellar winds, structural evolution, and core-envelope decoupling are the main agents determining the rotational history of a low-mass star. As was pointed out in Section 7.4.1, however, the effects of disk regulation during the pre-main-sequence phase should also be taken into account since disk-star magnetic coupling prevents, to some extent, spin-up associated with decreasing moment of inertia during that contraction phase.

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