4.3 Zeeman-Doppler Imaging

As an extension of the temperature and abundance mapping of the stellar surface, a magnetic Zeeman-Doppler imaging (ZDI) method was introduced by Semel (1989

) and further developed by Donati et al. (1989 ), Semel et al. (1993 ), Brown et al. (1991

), and Donati and Brown (1997

). The technique is based on the analysis of high-resolution spectropolarimetric data and allows for disentangling magnetic field distribution on the stellar surface due to different Doppler shifts of Zeeman-split local line profiles in the spectrum of a rotating star (Figure 3

). In the absence of rotation, the net circular polarisation signal in spectral lines would be zero due to mutual cancellations of contributions of regions of opposite field polarity.

Figure 3:

Observed circular polarisation in the spectrum (panel b) is a sum of opposite polarity Stokes $V$ profiles due to spots located at different Doppler shifts (panel a). From Semel (1989

). See also explanatory animations by P. Petit at

http://webast.ast.obs-mip.fr/people/petit/zdi.html

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 $V$ 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 $V$ 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 $V$ signal is proportional to the derivative of the intensity profile $I(v)$ , i.e.,

where $gi$ is the effective Landé factor and $ci$ is the wavelength of the $i$ -th spectral line. It is assumed further that the local line profiles are self-similar and scale in depth and width with the central depth $d i$ and wavelength, i.e.,

where $Z(v)$ is a so-called mean Zeeman signature, which is constant for all lines. The LSD Stokes $V$ profile can be obtained as a sum over many individual lines:

Blends and splitting patterns should be treated explicitly for each line. The gain factor in the S/N ratio can be as large as 30 when using more than 2000 line profiles. The LSD technique allowed for detection of magnetic fields on various types of cool stars, from pre-main-sequence stars to evolved giants (Donati et al., 1997 ). It was also used for temperature mapping using Stokes $I$ 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 $Q$ and $U$ are considerably smaller than in Stokes $V$ . Obtained Zeeman-Doppler images of cool stars are based on measurements of only Stokes $I$ and $V$ 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 $V$ at different rotational phases and different Doppler shifts. For instance, the radial field will dominate the Stokes $V$ 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 $V$ observations. An example of such restoration is shown in Figure 4 . When interpreting the results of Zeeman-Doppler images obtained only from Stokes $I$ and $V$ , 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 ).

Figure 4:

Zeeman-Doppler images of the young active K0 dwarf AB Dor. From Donati and Collier Cameron (1997