Schatzmans braking mechanism

The relevance of magnetic braking for stars having deep surface convection zones was first recognized by Schatzman (1962). Very briefly, it is assumed that these stars generate episodic mass ejections that act as an expanding plasma in a large-scale magnetic field. As material is ejected from the activity zones, the magnetic field can enforce approximate corotation until the gas has moved out to distances much larger than the star's radius. Beyond this region, because the magnetic stresses become less and less important, the outflowing material can thus leave the star, with each mass element carrying away its angular momentum. As we shall see below, if the gas is kept corotating with the star, a quite small amount of mass loss yields proportionally a much greater loss of angular momentum than matter retaining the angular momentum of the star's surface. Given the efficiency of this mechanism for extracting angular momentum from stars with outer convection zones, the break in the main-sequence rotational velocities can be explained as follows. Since high-mass stars spend relatively little time in the convective phase, magnetic braking is therefore virtually inoperative for these stars. Hence, they suffer very little loss of angular momentum during their pre-main-sequence contraction. In contrast, low-mass stars have a more important convective phase since they retain an outer convection zone all the way to the main sequence. Magnetic braking can thus operate during their entire pre-main-sequence contraction and during their much longer stay on the main sequence. Since the rapid drop in rotational velocity is seen at approximately the point where main-sequence stars develop subphotospheric convection zones, it follows that angular momentum loss preferably occurs in the low-mass stars, thus causing the observed rotational discontinuity in the F-star region. As was shown by Wolff and Simon (1997), recent data strongly suggest that this sharp decrease in mean equatorial velocity along the main sequence, from about 1.6M0 down to about 1.3M0, has already been imposed during the pre-main-sequence phase of stellar evolution. For masses less than about 1.3M0, however, their analysis indicates that further loss of angular momentum occurs rapidly during main-sequence evolution so that, by the age of the Hyades (~ 600 Myr), mean equatorial velocities for stars in the spectral range F8 V-K5 V are remarkably uniform at any given mass and decline from about 11 km s-1 at F8 V to about 4 km s-1 at K5 V (see Figure 1.8).

The strongest support for this rotation-activity connection comes from Wilson's (1966) finding that there is a sudden appearance of Ca II emission in the F-star region along the main sequence, whereas it is never observed among the more massive stars. Obviously, the close agreement between the onset of large rotational velocities and the termination of chromospheres is very suggestive of Schatzman's braking mechanism. More recently, Cameron and Robinson (1989) have found another piece of evidence in support of angular momentum loss via discrete mass ejection. They have obtained time series of high-resolution spectra of the Ha profile in the active, rapidly rotating G8-K0 dwarf AB Doradus. Their spectra show transient absorption features that move through the Ha emission profile on rapid time scales. These features strongly suggest the existence of cool, dense clouds embedded in and corotating with the hot extended corona out to several stellar radii from the rotation axis. Their calculations indicate that angular momentum loss could account for rotational braking on a time scale of no more than 100 Myr. If so, these observations might provide an important clue as to how low-mass stars lose the bulk of their angular momentum upon their arrival on the main sequence.

Another mechanism by which stars with convective envelopes can dispose of a considerable fraction of their initial angular momentum is provided by stellar winds. Following Mestel (1968), it is subphotospheric convection that is again the essential feature of the mechanism. Waves generated in the outer convection zone are dissipated above the photosphere, thus supplying the heat responsible for the formation of a chromosphere and a corona. When the coronal temperatures are too low to generate a thermal wind, however, large centrifugal forces acting on the corotating material can generate an outwardly moving flow (i.e., a centrifugal wind). In both cases, the wind motion accelerates outward from very low values at the bottom of the corona to supersonic values far away from the star's surface. Detailed studies have shown that the angular momentum loss rate is equivalent to that carried by a wind kept strictly corotating with the star out to a radius rA in the circumstellar envelope (e.g., Mestel 1968). By definition, the corotating radius rA is the mean radius of the Alfven surface defined by ha in n

where the indices "A" indicate that the wind speed v, the poloidal field strength H, and the density p are evaluated at r = rA .In the simple model developed by Weber and Davis (1967), where the magnetic field in the thermally driven wind is approximately radial in the corotating frame of reference, the effective corotation prescription gives the following expression for the angular momentum loss rate:

where R is the star's radius and O is the angular velocity of rotation. The importance of the large-scale magnetic field can be seen on the following example. From solar-wind data, one finds that rA « 30R0 for the Sun; hence, by virtue of Eq. (7.2), the rather weak solar magnetic field increases the angular momentum loss by three orders of magnitude over its value calculated without magnetic field. Now, from Eq. (7.1) and the definition of the mass flux at r = rA, dM 2

Eq. (7.2) can be rewritten in the form dJ 2 O , 0,2

Since the conservation of magnetic flux implies that HA r2A = H0 R2 in the case of a purely radial field, Eq. (7.4) becomes dJ O (H0 R2)2, (7.5)

where H0 is the average surface magnetic field. If a linear relationship of the form H0 a Q is assumed for the dynamo-generated magnetic field, with J a MR2Q Eq. (7.5) yields d Q ,

After integrating Eq. (7.6), one obtains

which is identical to Skumanich's empirical law (see Eq. [1.7]). This is a most fortunate coincidence since it implies that a simple formulation of angular momentum loss via magnetically channeled stellar winds is adequate to describe the rotational evolution of solar-type stars on the main sequence. As we shall see in Section 7.4.2, however, such a formulation does not describe adequately the spin-down of the very rapidly rotating low-mass stars in young open clusters. In fact, there is now clear indication that the angular momentum loss-rate saturates for surface rotational velocities in excess of 10-20 km s-1.

To the best of my knowledge, there is as yet no complete theory that explains the existence of a dynamo saturation in the most rapid rotators. However, there is increasingly convincing observational evidence to support the idea that the dynamo activity of a late-type star scales with its rotation rate. Dynamo saturation was originally inferred by Vilhu (1984) from the observation that the chromospheric and coronal emission fluxes depend only weakly on rotation at high angular velocities. More recently, Patten and Simon (1996) have undertaken a program to measure photometric rotation periods and X-ray luminosities for late-type stars in the young open cluster IC 2391 (age ~ 30 Myr). In Figure 7.1 we plot the X-ray luminosity LX against the rotation period Prot for solar-type

Fig. 7.1. The X-ray luminosity LX as a function of rotation period Prot for solar-type stars in the IC 2391 (filled triangles), a Persei (filled squares), Pleiades (filled diamonds), and Hyades (open circles) clusters. Source: Patten, B. M., and Simon, T., Astrophys. J. Suppl., 106,489, 1996.

_7 1 ■ I ■ ■ ■ 1 I ■ 1 ■ ■ I ' 1 ■ 1

Fig. 7.2. The normalized X-ray luminosity RX = LX /L boi as a function of the Rossby number Nr . Plotted are data for IC 2391 (filled triangles), a Persei (filled squares), the Pleiades (filled diamonds), the Hyades (open circles), and field main-sequence stars (open squares). Source: Patten, B. M., and Simon, T., Astrophys. J. Suppl, 106,489, 1996.

_7 1 ■ I ■ ■ ■ 1 I ■ 1 ■ ■ I ' 1 ■ 1

Fig. 7.2. The normalized X-ray luminosity RX = LX /L boi as a function of the Rossby number Nr . Plotted are data for IC 2391 (filled triangles), a Persei (filled squares), the Pleiades (filled diamonds), the Hyades (open circles), and field main-sequence stars (open squares). Source: Patten, B. M., and Simon, T., Astrophys. J. Suppl, 106,489, 1996.

stars in IC 2391 and, for comparison, for older stars from a Persei (age ~ 50 Myr), the Pleiades (age ~ 70 Myr), and the Hyades (age ~ 600 Myr). One readily sees that there is an overall decline in the median rotation rate and X-ray luminosity with age. Note also that the older cluster stars trace out a definite correlation between LX and Prot, whereas those in IC 2391 show at best a weak correlation between these two parameters. Following current practice, in Figure 7.2 we present an alternative representation of this activity-rotation plot, which greatly reduces the scatter when stars of different masses are combined together as in Figure 7.1. In Figure 7.2 we depict again the whole sample of stars, ranging in spectral type from late-F to M. The ordinate is the normalized X-ray luminosity, RX = LX/Lbol, where Lbol is the bolometric luminosity; the abscissa is the Rossby number,

rconv where rconv is the turnover time of turbulent convective motions in the outer convection zone. (Compare with Eq. [2.30], which is the standard definition of the Rossby number.) Note the clearly defined discontinuity near log10 NR = -0.5 and the saturation plateau at smaller values of the Rossby number. The existence of this plateau is often ascribed to a change in the nature of the stellar dynamo for the most rapid rotators.

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