4.2 Doppler Imaging

During the last two decades the Doppler imaging technique has been extensively employed for studying starspots on active stars. The main idea of the technique is to use high-resolution spectral line profiles of rapidly rotating stars for mapping the stellar surface. This idea was first formulated by Deutsch (1958 ), while the first inversion technique with minimisation was developed by Goncharskii et al. (1977

). It was first used for mapping chemical peculiarities on the surface of Ap stars. Modelling of photometric variations of late-type active stars has revealed that cool starspots are often quite large, covering up to 20% of the stellar surface. Such starspots should have resulted in noticeable line profile variations which were first observed in spectra of the RS CVn-type star HR 1099, and from which the first Doppler image of a spotted star was obtained by Vogt and Penrod (1983 ).

The Doppler imaging technique aims to restore starspot distribution information which is contained in time varying line profiles of rotating stars. If the star rotates rapidly enough, so that the rotation broadening of a line profile is significantly larger than the local line profile at a single point on the stellar surface, then a cool spot on the stellar surface will result in a “bump” in the profile (Figure 2 ). This “bump” moves across the profile, as the star rotates, with the velocity amplitude depending on the spot latitude. Inversion of a time series of the stellar line profiles results in a map, or image, of the stellar surface.

Figure 2:

Spectral line profiles for a model fast rotating star with no spots (dashed line) and with a spot moving across the disk as the star rotates (solid line). See also animation at

http://www.astro.phys.ethz.ch/staff/berdyugina/private/StellarActivity/StellarAct.html

An assumption on the nature of starspots is the main part of the model calculations. Cool spots will modify the flux integrated over stellar disk. Therefore, the intensity of radiation $I(T (s),c, m)$ emitted by the stellar surface from the point $s$ in the direction $m$ at the wavelength $c$ is defined by the local temperature $T (s)$ , which is assumed to be the effective temperature of the model atmosphere used for the calculation of the local line profile at the point $s$ . Integration over the stellar disk, given rotational phase $f$ and set of wavelengths, results in the residual flux $rc(f)$ which contains information on the temperature distribution on the stellar surface $T(s)$ and, therefore, is to be compared with the observed residual flux $rocbs(f)$ . Thus, the integration determines the direct transformation for a subsequent inversion. A comparison of the residual fluxes determines the discrepancy function $D(T )$ . By minimising $D(T )$ , one can obtain a unique solution with minimum variance. Such a solution is not however feasible due to noise in data. Smoothing the noise results in a multitude of different stable solutions. Searching for the unique and stable solution is a so-called ill-posed inverse problem, and there are different approaches and methods developed for solving it.

A common way of choosing the unique solution with a given level of goodness of the fit is to invoke some additional constraints $R(T )$ , which usually determine special properties of the solution. Therefore, the original ill-posed minimisation problem is replaced by another, which has a unique solution:

$P(T ) = D(T ) + /\ .R(T ), (2)$

where $/\$ is a Lagrange multiplier and $R(T )$ is a regularisation functional, which makes the solution unique. The value of $/\$ should be selected so that if $T (s)$ minimises $P(T )$ , then the rms deviation of the fit of the profiles is of the order of the noise in the observations.

Methods developed with such assumptions differ by the definition of $R(T )$ . The Tikhonov regularisation method (TRM), applied by Goncharskii et al. (1977 ) and Piskunov et al. (1990 ), claims the solution to be with the least gradient of the parameters, and $R(T ) = grad T$ . The Maximum Entropy method (MEM) first applied by Vogt et al. (1987 ) searches for the solution with the largest entropy, and $R(T ) = T log T$ . Generally, the two methods assume that the observed phenomena possess properties $R(T )$ , which cannot be known a priori. For instance, in case of surface imaging both above assumptions cannot be proved by the observations. In fact, for choosing a unique solution, one indeed needs some additional information, and, if it is not available, it is substituted by some plausible assumptions, which lead to an apparently acceptable solution but with an unknown bias. One should note, however, that in case of data of high quality and quantity the role of the regularisation is reduced, and the solution is approaching to the maximum likelihood solution, which in this case could be also considered as an acceptable result.

An alternative Occamian approach (OA) to inverse problems was developed by Terebizh (1995 ) and applied to the Doppler imaging problem by Berdyugina (1998 ). In the Occamian approach the choice of a solution is based on the analysis of information contained in observations and the transformation model. Building the Fisher information matrix $F$ with eigenvectors $V$ one defines a new reference frame with coordinates $Y$ , which are linear combinations of the unknown parameters $T$ :

The new coordinates $Y$ comprise principal components of the solution. Small eigenvalues of $F$ indicate principal components with relatively large errors of the solution, so that the error ellipsoid is extremely elongated in these directions. Moreover, in case of a lack of data some of the eigenvalues become zero, and the corresponding parameters are linearly dependent. Therefore, only a part of the principal components $Y (p)$ completely exhausts the available information on $T$ . In such a case, the $Y (p)$ are estimated instead of $T$ , while other principal components with not enough information are assumed to be zero. Then, the transformation

leads to the desired unique and stable solution $T~$ . Thus, the solution in the Occamian approach is the one which statistically satisfactorily fits the observed data with a minimum set of $~Y(p)$ . Such a solution is unique because of the choice of $p$ and stable because of removing those principal components which contain no significant information but noise. It is not constrained with any artificial assumptions. Another advantage of the Occamian approach is that the inverse Fisher information matrix gives estimates of the variances of the solution.

A number of numerical codes for Doppler imaging of cool stars based on the Maximum Entropy method have been developed by Vogt et al. (1987 ), Rice et al. (1989 ), Brown et al. (1991 ), Collier Cameron (1992 , 1995 ), Jankov and Foing (1992 ), and Rice and Strassmeier (2000 ). Piskunov et al. (1990 ) used the Tikhonov regularisation method and Berdyugina (1998 ) the Occamian approach. A technique based on the CLEAN-like approach was suggested by Kürster (1993 ) and one based on the interferometry by Jankov et al. (2001 ). Effects of different numerical methods was investigated by Piskunov et al. (1990 ) (MEM and TR), Strassmeier et al. (1991 ) (MEM, TR, trial-and-error), Kürster (1993 ) (CLEAN and MEM), and Korhonen et al. (1999 ) (TR and OA). As mentioned above, the difference is diminished when the data used are of high quality.

Important inputs for Doppler imaging which affect significantly the result are stellar atmosphere models, atomic and molecular line lists and stellar parameters. Errors in calculations of local line profiles have a strong effect on the inversion for moderate rotators. They easily cause artificial features in maps, like polar caps and belts of cool and hot spots (Unruh and Collier Cameron, 1995 ). The same features are obtained in case of a wrong value of the stellar rotational velocity and a wrong estimate of the effective temperature of the star (Berdyugina, 1998 ). Spot latitudes strongly depend on the inclination angle of the rotational axis to the line of sight. Various tests showed that capabilities of the technique are limited in the equatorial region where spots are recovered with reduced area and contrast. Sub-equatorial spots cannot be restored, especially at lower inclinations. A recent overview of the strengths and weaknesses of the Doppler imaging technique was given by Rice (2002 ).

Doppler imaging of a single star with the inclination angle of the rotational axis to the line of sight close to $90o$ (equator-on) experiences difficulties in recovering spots on either side of the stellar equator because of the symmetry of Doppler shifts in line profiles with respect to the equator. In a binary system such an inclination angle of the binary orbit would lead to eclipses of the components. For an eclipsing binary, the equatorial symmetry problem is reduced, and the quality of the mapping can be improved (Vincent et al., 1993 ).

For the last two decades more than 60 cool stars have been studied with the surface imaging technique (see, for an overview, Strassmeier, 2002 ). Out of them 29 are single stars and 36 are components in close binaries. A dozen of stars have been monitored for several years. The total number of Doppler images is 245 as of June 2002 (http://www.aip.de/groups/activity/DI/summary/). The results obtained with the Doppler imaging techniques are discussed in Section 5. Examples of stellar Doppler images are shown in Figures 6 and 15 .