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.
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 emitted by the stellar surface from the point in the direction at the wavelength is defined by the local temperature , which is assumed to be the effective temperature of the model atmosphere used for the calculation of the local line profile at the point . Integration over the stellar disk, given rotational phase and set of wavelengths, results in the residual flux which contains information on the temperature distribution on the stellar surface and, therefore, is to be compared with the observed residual flux . Thus, the integration determines the direct transformation for a subsequent inversion. A comparison of the residual fluxes determines the discrepancy function . By minimising , 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 , which usually determine special properties of the solution. Therefore, the original ill-posed minimisation problem is replaced by another, which has a unique solution:
Methods developed with such assumptions differ by the definition of . 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 . The Maximum Entropy method (MEM) first applied by Vogt et al. (1987) searches for the solution with the largest entropy, and . Generally, the two methods assume that the observed phenomena possess properties , 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 with eigenvectors one defines a new reference frame with coordinates , which are linear combinations of the unknown parameters :
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 90° (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.
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