4 The Rotation–Magnetic Field–Activity Relation

The generation of stellar magnetic fields is the result of complex mechanisms acting in the moving plasma of the stellar interior. The variety of magnetic field-related phenomena observed on the Sun should be explainable by a theory of the solar dynamo, but this dynamo continues to pose serious challenges to both observers and theoreticians. Although our knowledge about the solar magnetic properties and its time-dependence is rich and growing with the growing fleet of instrumentation observing the magnetic Sun, this single star can only exhibit magnetic features according to its own properties, and the investigation of the Sun alone will not lead to a full understanding of stellar dynamos in general. It is, therefore, of large interest for a deep understanding of stellar dynamos and, in particular, of the solar dynamo to understand the dependence of magnetic field generation on stellar properties.

The driving force of dynamos operating in the Sun and low-mass stars is the interplay between convective plasma motion, density and temperature stratification, and stellar rotation. Differential rotation and shear play particularly important roles in the most favored versions of sun-like dynamos. For overviews on the solar dynamo and theoretical backgrounds, I refer to the many reviews on this topic, for example Ossendrijver (2003) and Charbonneau (2010). One often-mentioned expectation from stellar dynamo models is the relation between the magnetic field strength and the rotation of a star. The relation is expected because the efficiency of a dynamo can be described by the “dynamo number”, D, that is related to the so-called α-effect that itself depends on rotation. The exact functional dependence between dynamo efficiency and rotation is difficult to assess, but one can argue that a dynamo can only exist if the rotational influence on convection, expressed as the Coriolis number Co, exceeds a certain value in order to create the differential rotation that is required for the dynamo process (see, e.g., Durney and Latour, 1978). The Coriolis number is proportional to the Rossby number that is often used in work on stellar activity, Ro = Prot∕τconv, with Prot the rotational period and τconv the convective overturn time.

The definition of the convective overturn time is not exactly well defined particularly in very low-mass stars (e.g., Gilliland, 1986; Kim et al., 1996). It has therefore been attempted to derive “empirical” turnover times assuming a relation between magnetic activity and stellar rotation. A first and very successful approach was presented by Noyes et al. (1984Jump To The Next Citation Point) who connected observations of stellar chromospheric activity and rotation. Investigating a rich sample of X-ray, coronal, activity measurements, Pizzolato et al. (2003Jump To The Next Citation Point) were able to show a well-defined rotation-activity relation connecting normalized X-ray luminosity and Rossby number. The latter is a mass-dependent function chosen to minimize the scatter in the rotation-activity relation (Figure 18View Image). Empirical convective overturn times in low-mass stars were derived by Kiraga and Stȩpień (2007), and Barnes and Kim (2010) cover a wide range of masses.

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Figure 18: Left panel: Rotation-activity relation showing the normalized X-ray luminosity as a function of Rossby number. Right panel: Empirical turnover time chosen to minimize the scatter in the rotation-activity relation (from Pizzolato et al., 2003Jump To The Next Citation Point, reprinted with permission ESO).

It has been argued that the construction of a mass-dependent empirical convective overturn time in order to minimize scatter in the rotation-activity relation is in principle nothing else than a compensation for a dependence of activity on stellar luminosity (Basri, 1986; Pizzolato et al., 2003). In other words, while normalized activity seems to scale with Rossby number, total (unnormalized) flux seems to scale with rotation period. It is argued that the reason for this is the approximate scaling of τconv with (Lbol)−0.5. Nevertheless, a tight dependence between Rossby number and normalized activity is clearly observed in sun-like and early- to mid-M type stars. The result is that magnetic activity is rising with decreasing Rossby number as long as Ro ≥ 0.1. At Ro ≈ 0.1, activity saturates and does not grow further with decreasing Rossby number. This behavior is interpreted as increasing dynamo efficiency with faster rotation in the regime where rotation is not yet dominating convection (Ro ≥ 0.1). This is what may be expected from the dynamo models introduced above. At fast rotation (Ro ≈ 0.1) the dynamo reaches a level of saturation that cannot be exceeded even if the star is spinning much faster.

We know from the Sun that activity is caused by magnetic fields. Together with expectations of the relation between rotation and magnetic field strength from dynamo theory, it is straightforward to conclude that the reason for the observed rotation-activity relation is a rotation-dependence of magnetic field generation, i.e., what we observe is a direct consequence of the magnetic dynamo efficiency. From an observational standpoint, this is not entirely clear because all we have discussed so far is that activity scales with rotation (or Rossby number), but this may also be due to a constant magnetic field translating to observable activity in a fashion that depends on rotation. A direct link between rotation and magnetic field observations was shown by Saar (1996aJump To The Next Citation Point, 2001Jump To The Next Citation Point), the observational basis of this work was discussed in Section 3.1. In sun-like and young stars it is found that magnetic field strengths indeed are a function of rotational period: Bf follows a relation that is proportional to some power of Ro consistent with expectations. In sun-like stars, however, magnetic fields can not be measured at very low Rossby numbers (saturated regime) because spectral line widths are too broad due to the rotational broadening at the corresponding rotation rates. Mainly because of the smaller radii, and perhaps also because of longer convective overturn times, the relation between Rossby number and equatorial velocity favors the detection of Zeeman splitting at low Rossby numbers (in the saturated regime) in low-mass stars (see, e.g., Reiners, 2007); M dwarfs have very small radii (and long overturn times) so that for small Rossby numbers the corresponding surface velocities are relatively low. This allows measuring magnetic fields of M stars well within the saturated regime. For M dwarfs of spectral type M6 and earlier, Reiners et al. (2009aJump To The Next Citation Point) found that average magnetic fields indeed show evidence for saturation at low Rossby numbers. This can be interpreted as evidence that saturation of activity at high rotation rates is a consequence of saturation of the average magnetic field and that B itself is limited (in contrast to a limit of the filling factor f or of the coupling between magnetic fields and non-thermal heating). Unfortunately, it was not yet possible to separate B from f in the measurements of magnetic flux density Bf so that the true range and variation of both field strength B and filling factor f remains unknown. Although there is good evidence for a firm upper limit, there is still room for some variation of B as a function of Ro in the saturated regime. Further observations, and especially information about values of both B and f are highly desired.

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Figure 19: Magnetic fields as a function of Rossby number. Crosses are sun-like stars Saar (1996a, 2001), circles are M-type of spectral class M6 and earlier (see Reiners et al., 2009a). For the latter, no period measurements are available and Rossby numbers are upper limits (they may shift to the left hand side in the figure). The black crosses and circles follow the rotation-activity relation known from activity indicators. Red squares are objects of spectral type M7 – M9 (Reiners and Basri, 2010Jump To The Next Citation Point) that do not seem to follow this trend (τconv = 70 d was assumed for this sample).

The relation between magnetic flux density and Rossby number is shown in Figure 19View Image. Crosses are from sun-like stars and define the rising, unsaturated part of the rotation-activity relation, and circles are M-type stars defining saturation at a few kilo-Gauss average field strength. At least for sun-like stars and early-M stars, the rotation-magnetic field relation seems to be rather well defined. Looking back to the discussion on detectability of magnetic fields, at least in sun-like stars, some cautious doubt may be allowed as to the relation between Rossby number and Bf in the “low-field” regime (Bf  < 1 kG). First, in principle, the detection of magnetic fields in this region and, in particular, from optical data, is extremely difficult and the significance of the data points is difficult to assess (see Section 4.3). Second, assuming that the Sun has a Rossby number somewhere between 2 and 0.5 (Prot ∼ 26 d, τconv ∼ 12 – 50 d), the average magnetic flux for the Sun is on the order of 20 – 100 G, which is significantly above the value detectable in the Sun if it is observed as a star.

One interesting and relatively firm conclusion from M dwarf magnetic field measurements is that the typical upper limit for average magnetic fields is of the order of a few kilo-Gauss, average fields of 10 kG are not observed, and the upper limit does not seem to significantly depend on temperature. This contradicts the prediction of a close correlation between (maximum) magnetic field strength and spectral type introduced by assuming a limiting influence of buoyancy forces on the dynamo efficiency (Durney and Robinson, 1982). However, this conclusion is only valid for M-type main sequence stars because we have no good estimate of maximum field strengths in F – K-type stars, and magnetic fields in pre-main sequence stars may follow different rules.

 4.1 The dynamo at very low masses
 4.2 Magnetism and Hα activity
 4.3 A posteriori knowledge about detectability of magnetic fields

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