Magnetic field

5.2 Magnetic field

Direct measurements of magnetic fields on the surfaces of cool stars are very difficult. The net circular polarisation of the Sun as a star would be hardly observable with the present observational tools, as the mean magnetic field of the Sun peaks at maximum 8 G.

The ZDI technique (Section 4.3) is an excellent tool for detecting surface magnetic fields. Donati et al. (1999 ) combined Stokes $V$ of more than thousand lines and obtained a time sequence of mean $V$ profiles which shows clear rotational modulation, as magnetic structures are carried across the visible hemisphere of the star. This technique is useful for rapidly rotating stars as the rotation disentangles surface magnetic structures with opposite polarities, reducing polarisation cancellation (see Figure 3 ). On slower rotators the ZDI technique can only reveal large-scale unipolar magnetic fields. It has been intensively used for studying three stars: young dwarfs AB Dor and LQ Hya and RS CVn star HR1099 (Donati et al., 2003b , and references therein). An example of the spot and magnetic field distribution obtained with the ZDI technique is shown in Figure 4 . The common feature of reconstructions for the three stars is that the magnetic field distribution does not coincide with the darkest spots in the temperature images. The radial field component reveals large mid-latitude regions of mixed polarity, while the azimuthal component appears as almost axisymmetric rings of opposite polarities at higher and lower latitudes. Donati et al. (2003b ) interpreted such a field distribution as an indication of large-scale poloidal and toroidal field components on the stellar surface and the underlying dynamo processes distributed throughout the entire convection zone. It was, however, argued by Solanki (2002 ) that the rings of the azimuthal field may represent large penumbral regions with a predominantly horizontal magnetic field.

Most of our current knowledge about magnetic fields on cool stars and in starspots is, however, based on Zeeman broadening measurements, which reveal the distribution of magnetic field strengths with little dependence on the unknown field geometry (Robinson Jr, 1980 ; Saar, 1988 ; Valenti and Johns-Krull, 2001 ). Zeeman broadening is best measured for slowly rotating stars, in contrast to ZDI. Reliable measurements require Zeeman splitting larger than, or comparable to, line widths in the absence of the field. Zeeman splitting is proportional to the field strength $B$ and effective Landé factor $geff$ as

It is also proportional to the square of the wavelength. For this reason, successful measurements of magnetic field strengths can be carried out using red or IR lines with large $geff$ . Also, only stars with strong magnetic fields and large field areas can be studied with this technique.

In order to derive magnetic field strength $B$ and filling factor $f$ , observed line profiles are fitted by the sum of synthetic spectra for magnetic and non-magnetic regions (Saar, 1994 ; Valenti and Johns-Krull, 2001 ):

Such an analysis implicitly assumes that (i) the field is concentrated into flux tubes surrounded by field free regions; (ii) flux tubes are oriented radially in the photosphere; (iii) magnetic regions are distributed uniformly over the surface; (iv) magnetic regions are characterised by a single field strength, and (v) the temperature structure is the same for magnetic and non-magnetic atmosphere (Valenti and Johns-Krull, 2001

). In this approach, the field strength being determined from the splitting is usually well constrained, but the filling factor depends on the unknown temperature structure of magnetic regions. In the case when the used spectral line becomes stronger in starspots, the filling factor can be overestimated, and vice versa. Rüedi et al. (1997 ) concluded that magnetic field strength and filling factor cannot be determined separately for moderately active stars with optical spectra of spectral resolution less than 100 000 and S/N $≤$ 250.

Figure 8: Magnetic field measurements for active dwarfs (circles) and giants (squares) versus the photosphere temperature. Big circles indicate the sunspot umbra ( $B$ = 3 kG) and penumbra ( $B$ = 1.5 kG). The thick solid line is a linear fit to the data, excluding the sunspot umbra (based on data in Table 6).

Figure 9: Magnetic field measurements for active dwarfs (circles) and giants (squares) versus the filling factor. Big circles indicate the sunspot umbra ( $B$ = 3 kG) and penumbra ( $B$ = 1.5 kG). The thick solid line is a linear fit to the data, excluding the sunspot umbra (based on data in Table 6).

Magnetic field measurements for active dwarfs and giants are collected in Table 6 and are plotted in Figures 8 and 9 versus the photosphere temperature and filling factor, respectively. These plots indicate a tendency for cooler dwarfs to have stronger magnetic fields and larger areas covered by them. It is interesting that there is a clear contradiction between spot filling factors measured from light curves and magnetic field filling factors measured from spectral lines (see Figure 10 ). This contradiction suggests that the two filling factors refer to different activity signatures, such as spot umbra and penumbra, or even faculae. The latter, being brighter and possessing relatively strong magnetic fields, would indeed be better seen in atomic lines. This is also supported by results obtained with the ZDI technique, which reveals stronger magnetic fields for intermediate brightness regions (e.g., Donati and Collier Cameron, 1997 ). Thus, it appears that umbral magnetic fields have not been measured as yet.

Figure 10: Filling factors of spots (open symbols) and magnetic fields (filled symbols) on the surfaces of active dwarfs (circles) and giants (squares) versus the photosphere temperature. The thick solid line is a polynomial fit to the spot filling factors. The dashed line is a fit to the magnetic field filling factor, excluding the Sun. A big circle emphasises the sunspot umbra ( $f ∼ 1%$ ) (based on data in Tables 5 and 6).

In order to detect magnetic fields in the starspot umbra it is necessary to employ spectral lines that are very weak outside spots. Zeeman sensitive molecular lines are the lines of choice for this purpose (Berdyugina (2002 ); see Figure 5 ).