7.1 A scaling law for saturated planetary and stellar dynamos

The surface magnetic field of a star is controlled by its rotation rate, but magnetism saturates when rotation is faster than a critical velocity, perhaps associated to the rate where Ro ≈ 0.1. The value of the field strength at this saturation level, however, may vary between objects and depend on additional parameters. Christensen et al. (2009Jump To The Next Citation Point) suggested a scaling law based on energy-flux consistent with the maximum magnetic fields found in rapidly rotating low-mass stars, and some planetary fields like Earth and Jupiter (Figure 25View Image). In this picture, the available heat flux in the convection zone is converted into magnetic energy. A few assumptions are necessary to explain fields of other planets, but the general idea is that a single scaling relation may hold in objects of vastly different dimensions like planets, brown dwarfs, and stars (for a deeper discussion of flux scaling relations, see Christensen, 2010).
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Figure 25: Magnetic energy density in the dynamo vs. a function of density and bolometric flux (both in units of J m–3) according to Christensen et al. (2009Jump To The Next Citation Point). The scale on the right shows r.m.s. field strength at the dynamo surface. Blue: T Tauri stars; red: old M dwarfs. Black lines show the rescaled fit from Christensen et al. (2009Jump To The Next Citation Point) with 3σ uncertainties (solid and dashed lines, respectively). The stellar field is enlarged in the inset. Brown and grey ellipses indicate predicted locations of a brown dwarf with 1500 K surface temperature and an extrasolar planet with seven Jupiter masses, respectively (from Christensen et al., 2009).

In order to test a relation like this one, magnetic fields must be measured in stars in the saturated dynamo regime, i.e., at fast rotation. So far, this was possible only in pre-MS stars and M dwarfs, and fields on the order of a few kG were found here. The two species have available a comparable amount of heat flux so that the scaling relation yields comparable results for both groups, which is consistent with the empirical results from the field measurements. In order to further test the applicability of the scaling relation, it would be necessary to observe the saturation level in stars of very different nature, brown dwarfs, and finally exoplanets. The scaling law provides a prediction for magnetic fields in brown dwarfs; according to the relation, magnetic fields on the order of several hundred Gauss up to kG-strength should exist on rapidly rotating, evolved brown dwarfs.

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