5.2 The solar spin in time

Stellar magnetic activity is fundamentally determined by the stellar rotation period, as I will further discuss in, e.g., Section 5.5.1. Because the magnetized stellar winds transport angular momentum away from the star, a MS solar analog spins down with age. The rate of change of angular momentum is related to the mass loss rate, the spin rate, and the Alfvén radius, RA. One finds
( )m ˙Ω-∝ M˙- RA-- , (6 ) Ω M R ⊙

where m is between 0 and 1 depending on the magnetic field geometry (Weber and Davis Jr, 1967Mestel, 1984Jump To The Next Citation PointStȩpień, 1988Jump To The Next Citation Point). One further has to couple the average surface magnetic field strength, B 0, with Ω. This relation is essentially determined by the magnetic dynamo but can reasonably be parameterized as p B0 ∝ Ω with p probably being 1 or 2 (Mestel, 1984Jump To The Next Citation Point). The approximately observed −1∕2 Ω ∝ t law (“Skumanich law”, Skumanich 1972Jump To The Next Citation Point, see below) is recovered for a thermal wind with p = 1 (Mestel, 1984), i.e., a linear dependence between average surface magnetic field strength and rotation rate. For another detailed study of this problem, see Stȩpień (1988).

Empirically, for solar analogs one finds

0.57 P = 0.21t6 [d] (7 )

(Dorren et al., 1994Jump To The Next Citation Point), where t6 is the age in Myr after arrival on the ZAMS. Equivalently, for the rotational (equatorial) velocity and for the rotation rate (for constant R),

Ω ∝ v ∝ t−60.6±0.1[d] (8 )

(Ayres, 1997Jump To The Next Citation Point) – see Figure 11View Image. These equations imply a decrease in rotation period from ZAMS age to the end of the MS life of a solar analog by a factor of about 20.

View Image

Figure 11: Relation between rotational velocity, v, and age for solar analogs. The diamond-shaped areas show the large scatter of v in young clusters, before rotational convergence has been attained (from Ayres, 1997Jump To The Next Citation Point, reprinted with permission).

At ages of approximately 100 Myr or younger, the stellar rotation period is not a function of age but of the PMS history such as the history of circumstellar-disk dispersal (e.g., Soderblom et al. 1993Jump To The Next Citation Point, see Figure 11View Image). Once the inner disk disappears, the lack of magnetic locking via star-disk magnetic fields and the contraction of the star toward the MS will spin-up the star and thus determine the initial rotation period on the ZAMS. For example, the rotation periods of G-type stars in the Pleiades and the α Per clusters still scatter considerably, ranging from less than a day (the so-called ultra-fast rotators) to many days (Soderblom et al., 1993Jump To The Next Citation PointStauffer et al., 1994Randich et al., 1996Jump To The Next Citation Point), while they (and therefore the stellar X-ray luminosities, see Section 5.5) have converged to a nearly unique value at the age of the UMa Moving Group (300 Myr) or the Hyades (600 – 700 Myr; see Soderblom et al. 1993Jump To The Next Citation PointStern et al. 1995).

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