Application of vector spherical harmonics for kinematic analysis of stars from zonal catalogues

Abstract

We solve the problem on a kinematic analysis of the three-dimensional velocity field of stars from zonal catalogues, i.e., catalogues in which the stars are presented at all right ascensions in some declination zones. We have constructed a system of vector spherical harmonics with the properties of completeness and orthogonality for a chosen declination zone. We suggest a method that allows the Ogorodnikov-Milne model parameters in the Galactic coordinate system to be estimated by analyzing the proper motions and radial velocities of stars in the equatorial coordinate system. The vector spherical harmonics are shown to have the following advantages over the standard approach based on a direct leastsquares estimation of the parameters for a specific model. First, in contrast to the standard approach, the new method can reveal all systematic components of the velocity field irrespective of a particular model. Second, it allows one to get rid of the correlation between the sought-for parameters, which presents a serious problem for the conventional method in the case of zonal catalogues. Third, the method of vector spherical harmonics allows the kinematic parameters to be estimated at least by two techniques. Comparison of these two solutions makes it possible to test the standard kinematic model for compatibility with the observational data. The developed method has been tested on the basis of numerical experiments and applied for a kinematic analysis of the proper motions of Tycho-2 stars in the southern hemisphere for which the parallaxes can be estimated using data from the Tycho-2 Spectral Type Catalogue.