2.2 Broad band polarization

Obtaining high resolution spectra of cool stars, in particular of very faint M stars and cooler objects, is challenging because the required signal-to-noise ratios are difficult to reach. It is, therefore, very desirable to develop a method to measure magnetic field properties from low-resolution spectroscopy or even photometry. Leroy (1962) proposed that broad band linear polarization can be caused by differences between saturation of π and σ components (Figure 5View Image). Based on the observation of linear polarization in different filters by Koch and Pfeiffer (1976), Mullan and Bell (1976) present evidence for a magnetic field of 10 kG strength on the bright spotted dwarf BY Dra. This scenario, however, was ruled out by several later measurements. Huovelin and Saar (1991) and Saar and Huovelin (1993) modeled broad band linear polarization in cool stars including polarization from scatter in the stellar atmosphere. They show that broad band polarization probably dominates over Rayleigh and Thomson scattering. Measurements of linear polarization in cool stars were reported, e.g., by Tinbergen and Zwaan (1981) and Alekseev (2003Jump To The Next Citation Point), but the correspondence to magnetic regions is not entirely clear. Alekseev (2003) show linear polarization depending on wavelength with higher degrees of polarization at short wavelengths. This dependence is expected if the signal comes from the magnetic surface of the star, but in some cases the detected polarization strongly exceeds the maximum level expected. Thus, a supplementary source of polarization is suggested, which is proposed to be most likely the remnant of a circumstellar disk. If such a disk is required, however, polarization due to magnetism and polarization from the disk are difficult to disentangle. Other potential sources of broadband linear polarization include light source anisotropy (Al-Malki et al., 1999) and stellar flares (Saar et al., 1994b).
View Image

Figure 9: Stokes V plotted vs. the derivative of Stokes I (weak field approximation). The linear relation shows that the star has a magnetic field, and from the slope of the relation magnetic field strengths on the order of 2 kG are derived for the two configurations (reprinted with permission from Bagnulo et al., 2002Jump To The Next Citation Point,  ESO).

Bagnulo et al. (2002) demonstrated a method they used to successfully measure magnetic fields in hot stars from low spectral resolution. Assuming that the weak-field approximation holds, they plot circular polarization V against the derivative of intensity, dI ∕dλ (see Equation (5View Equation)). If the weak-field approximation holds and the intensity derivative is proportional to Stokes V, the longitudinal magnetic field can be determined from the slope of their relation (Figure 9View Image). The method is fairly straightforward in hot stars with well-separated hydrogen lines that all have very similar Landé factors. The method has not yet been applied successfully to cool stars (understood here as stars of spectral type F and later) and it is not clear whether it would work given the large number of blended lines with very different Landé factors. Kolenberg and Bagnulo (2009) applied the method to RR Lyr stars that are technically similar to cool stars (regarding their outer convection zones) but have spectra very different to later spectral types. Nevertheless, this may be a promising method to determine longitudinal net field strengths in cool stars that are not observable at very high spectral resolution.

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