Zeeman signatures in atomic lines due to starspots are expected to be extremely small, with typical relative amplitudes of 0.1%. Detecting them requires measurements of polarisation with noise level in Stokes as low as 10–4, while the current instrumentation allows for the best relative noise level of 10–3. Semel (1989) and Semel and Li (1996) proposed a multi-line approach for increasing the signal-to-noise (S/N) ratio of the measured polarisation, which has resulted in first detection of the circular polarisation signal in a cool star (Donati et al., 1997). A combination of Stokes profiles using a multi-line technique called Least Squares Deconvolution (LSD) is based on the weak field approximation, i.e., one assumes that the magnetic splitting of spectral lines is smaller than their local Doppler broadening. In this case the Stokes signal is proportional to the derivative of the intensity profile , i.e.,et al., 1997). It was also used for temperature mapping using Stokes observations of faint stars with short rotational periods for which high signal-to-noise spectra cannot be obtained through longer exposure times.
Applying an inversion technique, similar to those used for Doppler imaging (see Section 4.2), to all four Stokes parameters, one can recover the distribution of the temperature and magnetic field vector over the stellar surface. Three numerical codes based on the Maximum Entropy method (Brown et al., 1991; Hussain et al., 2000) and the Tikhonov regularisation (Piskunov and Kochukhov, 2002) have presently been developed. In practice, however, measuring the full Stokes vector for cool stars is difficult, because magnetic signatures in Stokes and are considerably smaller than in Stokes . Obtained Zeeman–Doppler images of cool stars are based on measurements of only Stokes and are certainly not unique and provide limited information for the interpretation. A lack of information on different components of the magnetic vector can be overcome by assuming a certain relation between the components. For instance, Hussain et al. (2001) prescribed the field to be potential and reconstruct its distribution from circularly polarized line profiles. Piskunov and Kochukhov (2002) suggested a special type of regularisation based on spherical harmonic expansion. In this case the field distribution is forced to take the form of such an expansion which is acceptable only for stars with clearly dominating multipole structures like, e.g., Ap stars.
In the inversion procedure three components of the magnetic field vector are generally represented by radial, azimuthal, and meridional fields. To some extent they contribute to the line of sight component observed in Stokes at different rotational phases and different Doppler shifts. For instance, the radial field will dominate the Stokes near the centre of the stellar disk, while the azimuthal field will be most noticeable in the circular polarisation near the stellar limb. This allows to recover some parts of the magnetic field components from Stokes observations. An example of such restoration is shown in Figure 4. When interpreting the results of Zeeman–Doppler images obtained only from Stokes and , one has to take into account that the magnetic field distribution is underdetermined for each component and that there might be a cross-talk between different components (Donati and Brown, 1997).
© Max Planck Society and the author(s)