1 Introduction

The performances of off-axis adaptive correction are of interest for astronomical observations because such an approach allows to avoid certain difficulties dealing with observations of weak stars. The approach itself assumes that some bright enough guide star (a natural or an artificial one) is chosen to perform the measurement of atmospherically distorted wavefronts, then the data obtained are applied for an adaptive correction. This method allows one to improve the images of stars located in some area around the guide star. However unlike the usual on-axis adaptive correction, the off-axis one produces some specific distortions on the corrected image.

To predict main effects arising in off-axis adaptive correction, it is sufficient to consider two stars (say, the guide and the observed one) separated by some angular distance. For this case the quality of off-axis correction is determined by the residual structure function $D_{\rm R}$ which shows the degree of correlation between the wavefronts produced by the stars at the aperture of the telescope. It has been shown (Fried 1982; Vitrichenko et al. 1984; Chassat 1989) that, even for the perfect adaptive correction, the residual structure function is an anisotropic one because the light from the stars passes through different paths in the atmosphere. As a consequence of this phenomena, the long-exposure point spread function (PSF) of the observed star will be anisotropic, i.e. the isophotes of the observed image will have a non-circular form.

The effect of interest has recently been found experimentally (Close 1998). The authors noticed that "Off-axis guide stars produce an elongation in the science object's PSF towards the direction of the guide star'' and pointed out that the anisoplanatic phenomena is responsible for this effect. In this paper we present a detailed theoretical treatment of this effect for the case of perfect off-axis adaptive correction considering how the anisotropy of corrected image depends on the observation conditions, angular separation between the stars and telescope size. The results obtained can serve for the reconstruction or calibration of wide-field images provided by systems with adaptive correction.