4.3 A posteriori knowledge about detectability of magnetic fields

I have argued in Section 3.1 and 4 that the detection of magnetic fields from optical lines is extremely difficult, and that our picture of the non-saturated part of the rotation-magnetic field relation may be biased by influences of activity on line profiles other than the Zeeman effect. In other words, the relation between rotation period and measured fields may be driven by the presence of starspots that are generated when stars rotate more rapidly, and they influence the line profiles in a way that may mimic the presence of Zeeman splitting. It is currently very difficult to assess the influence of this effect. The rotation-magnetic field relation would probably look similar to the one shown in Figure 19View Image, but its absolute values may differ significantly. One consequence of this might be the mismatch between the solar average field and the value predicted according to its rotation period.
View Image

Figure 21: Estimated average magnetic fields as a function of equatorial rotational velocity. Equatorial velocities are calculated for stars with measured rotation periods in Noyes et al. (1984Jump To The Next Citation Point) and Donahue et al. (1996Jump To The Next Citation Point). Average fields are estimated from the relation given in the text.

As an interesting exercise, we can take the rotation-magnetic field relation in the unsaturated part of Figure 19View Image and estimate magnetic field strengths for a sample of stars with measured rotation periods. The relation in Figure 19View Image can be approximated by Bf  = 70 Ro–1.5. We can then take empirical Rossby numbers (using measured rotation periods) from the work of Noyes et al. (1984) and Donahue et al. (1996), convert rotation period into approximate surface rotation velocity for each star, and estimate the average magnetic field strength from the relation between Bf and Ro. The result of this exercise is shown in Figure 21View Image. We can conclude that according to the relation in Figure 19View Image, sun-like stars with surface rotation velocities of 5 km s–1 generate average magnetic fields of approximately Bf  = 200 G. Kilo-Gauss field strengths are generated in stars that rotate as rapidly as veq = 15 km s–1. A rough approximation to the relation shown in Figure 21View Image is Bf ≈ 50veq, with Bf in Gauss and veq in km s–1. We can insert this relation into Equation (4View Equation) to achieve a very rough estimate of the ratio between rotational broadening and Zeeman splitting in sun-like stars. The resulting ratio is

ΔvZeeman- v ≈ 0.07 λ0g, (13 ) eq
with λ0 in µm. Thus, at optical wavelengths, the approximate Zeeman shift according to the rotation-magnetic field relation in the non-saturated dynamo regime is usually well below 10% of rotational broadening. Given the limitations and systematic uncertainties of detailed spectral synthesis, this is a very challenging problem for Zeeman observations. So far, conclusive investigations of Zeeman splitting at infrared wavelengths are lacking, but there is certainly a great need to verify the rotation-magnetic field relation at longer wavelengths λ0.

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