Astron. Astrophys. 332, 541-560 (1998)

3. Spot modelling technique

The reconstruction of a surface map of an active star using photometric data alone is an ill-posed problem. The wide band flux modulation provides information only on the distribution of the projected spotted area vs. phase, i.e., stellar longitude, whereas a unique mapping requires information both on longitude and latitude. In the particular case of an eclipsing binary system, eclipses increase the information on the hemisphere of the active star being occulted, but the reconstruction problem still remains ill-posed even in the most favourable cases.

As a matter of fact, one can obtain a unique solution in most cases by selecting the surface map which minimizes the , but such an approach is highly unsatisfactory because most of the structures appearing in the map will result from the modelling of the noise present in the data and are not related to any real pattern on the star. Moreover, the solution which minimizes the is usually highly unstable, i.e., small variations in the input data induce very large changes in the solution. It is possible to reduce the influence of the noise on the map and increase that of the signal by establishing a priori a limit for the fit, greater than the minimum. The main drawback of this approach is that the solution is not unique, i.e., many (virtually infinite) maps fit the light curve within the given limit.

It is possible to select a unique and stable solution if some a priori assumption on the properties of the picture elements (pixels) of the map is introduced (e. g., Nityananda & Narayan 1982, Tikhonov & Goncharsky 1987).

There are two main a priori assumptions which have been widely used in the field of active star mapping: the maximum entropy criterion (hereinafter ME, e. g., Gull & Skilling 1984, Vogt et al. 1987) and the Tikhonov criterion (hereinafter T, e. g., Piskunov et al. 1990).

The implementation of these regularization methods has been made starting from the computational tools applied for the analysis of the light curves of RS CVn (Rodonò et al. 1995). For calculation purposes, the photosphere of each of the two stars is divided into N surface squared elements (hereinafter pixels) of side s. Therefore pixels from 1 to N map the first component whereas pixels from to map the second component. The contribution of the pixel i to the observed wide-band flux at orbital phase j can be written as: , where is an element of the response function matrix relating the observed wide-band flux to the photospheric brightness of the components (e. g., Vogt et al. 1987) and is the specific intensity of the i -th surface element at the given isophotal wavelength (Golay 1974). Therefore, the observed flux at phase j is:

The elements of the response function matrix are computed using the computer code for the synthesis of binary system light curves described by Lanza et al. (1994). It assumes that stellar photospheres are triaxial ellipsoids of semi-axes , and (), respectively, and treats ellipsoidicity, gravity darkening and the geometry of the reflection effect according to Kopal (1959). The physics of the reflection effect is treated assuming black-body re-irradiation with given bolometric albedo.

Once the matrix R has been evaluated, a suitable algorithm must be applied to solve for the brightness vector I given the ill-conditioned nature of the problem defined by Eq. (1) alone. We construct a regularized solution by finding the appropriate extreme value of the entropy or Tikhonov functionals, subject to the condition of a given limit (e. g., Press et al. 1992, Cameron 1992). As a mapping parameter, we choose the spot covering factor , that is the fraction of the i -th pixel covered by spots. It is more adequate as a mapping parameter than the temperature because it guarantees the additivity of the image entropy (Cameron 1992).

The surface intensity of the i -th pixel is a function of its spot covering factor according to:

where is the brightness of the spotted photosphere of the i -th pixel and the corresponding unspotted brightness. They are computed using the empirical photospheric flux distributions given by Poe & Eaton (1985), including also the effects of the limb and gravity darkenings.

The corresponding to a given surface distribution of the spot covering factor is defined as:

where M is the total number of observations in the light curve, the flux at the phase j -th computed by Eq. (1) adopting the surface intensity resulting from Eq. (2), and is the observed flux at the phase j -th with standard deviation .

The functional form we have assumed for the entropy of the surface map of star k () is:

where the sum is extended over the pixels belonging to the k -th star, is the relative area of the i -th pixel of the star (total area of the star ), m the default spot covering factor which determines the limiting values for : . In all our computations we have adopted: .

The Tikhonov functional T measures the smoothness of the map of a star and is defined in the case of a continuous covering factor as:

where and l are the colatitude and longitude and the integration is extended over the photosphere of the star (Piskunov et al. 1990). In our model the numerical expression of the Tikhonov functional for the k -th component is:

where the external sum is over the star's pixels, while the inner one is over the neighbour pixels of the i -th pixel. A generic pixel has up to four neighbours, two with the same latitude and two with the same longitude. The factor is given by: when the pixels i -th and n -th are at the same longitude, and when the pixels are at the same colatitude .

The regularized solution is obtained by a constrained minimization of the functionals (cf., e. g., Vincent et al. 1993):

for the ME criterion, and:

for the T criterion, where or is the Lagrange multiplier. The adopted algorithm is a version of the conjugate gradient method developed by Byrd et al. (1994) and implemented by Zhu et al. (1994).

When the solution minimizes the , but it is severely affected by noise and instability. By increasing the value of , the increases and the solution becomes unique and stable. The features of the map are determined as a combination of the information coming from the data and the a priori assumptions, while the value of quantifies the relative weights of such two sources of information (see, e. g., Titterington 1985, Narayan & Nityananda 1986). We select the most appropriate value of by a careful inspection of the fit, because the usual criterion of fixing a limiting is not completely satisfactory in the case of a wide-band light curve, whose noise level is not exactly known a priori. Specifically, we consider the distribution of the residuals between the computed and the observed fluxes, which, due to the a priori assumptions, is not that expected from the pure statistics, the difference becoming the larger, the greater the value of (Bryan & Skilling 1980). From the analysis of the residual distributions obtained for different 's, we determine the appropriate Lagrange multiplier as the greatest value giving a distribution which does not appreciably differ from the normal one (i.e., by more than ), particularly during the course of eclipses when the information content of the data is higher.

An estimate of the errors on the computed 's is made difficult by the use of a regularization scheme. In fact it is based on an a priori constraint and hence systematic errors prevail in the present approach. They can be estimated only through a comparison of solutions obtained with different regularization criteria.

For a given regularization criterion and the corresponding solution, it is only possible to estimate the variations of the 's produced by a variation of the observed flux at constant , obtaining an estimate of the stability of the given solution. In particular, we shall estimate the variations of the 's for the ME solutions, for which an analytical expression can be derived assuming that the variations of the observed fluxes are uncorrelated and that is constant (see Sect. 5.1).

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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