Scalar and Vector Spherical Functions: Solving the Problems of Astrometry and Stellar Kinematics

Details Scalar and Vector Spherical Functions: Solving the Problems of Astrometry and Stellar Kinematics Scalar and Vector Spherical Functions: Solving the Problems of Astrometry and Stellar Kinematics

Oct 2010

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2010aipc.1283...94v&link_type=abstract

MATHEMATICS AND ASTRONOMY: A JOINT LONG JOURNEY: Proceedings of the International Conference. AIP Conference Proceedings, Volum

Astronomy and Astrophysics

Astronomy

Galactic Centre, Astronomy, Least Squares Approximations, Galactic Center, Bar, Circumnuclear Matter, And Bulge, Astronomical Catalogs, Atlases, Sky Surveys, Databases, Retrieval Systems, Archives, Etc., Interpolation, Curve Fitting

Scientific paper

The paper presents a review of the methods and results obtained in astrometry and galactic astronomy with the help of spherical harmonics. As the starting point, the representation of systematic differences in positions and proper motions via orthogonal scalar functions is considered. Proposed in the second half of XX century this method became a standard tool in comparison of ground based and space astrometry catalogues. Two further developments of this approach made by the author is reported afterwards.
The first one is deriving the parameters of mutual orientation and rotation of frames. In contrast to widely used least square method the spherical function formalism provides not only the determination of orientation and spin but (a) takes into account only the harmonics which correspond to rotation, (b) tests them for pure rotation, (c) discovers the existence of quasi-rotational terms which may smear rotation. Due to these properties the spherical harmonics yield realistic results even in the case when the observational data contain not only noise but other systematic terms that have nothing to do with rotation. The application of this method to the comparison of the catalogues FK5 and HIPPARCOS is given.
The second item is the vector spherical functions analysis of stellar kinematics. This approach allows detecting all systematic components in the stellar velocity field without being attached to a specific physical model. Comparison of the theoretical decomposition coefficients of the equations for a particular kinematical model with observational data reveals systematic components that are not described by this model. The formalism of vector spherical functions is well suited for analyzing the present and future (GAIA) catalogues containing three velocity vector components: the proper motions in both coordinates and the radial velocity. Application of this method to the proper motions of HIPPARCOS stars gives evidence that the proper motions under consideration contain systematic components that can not be interpreted in terms of the standard Ogorodnikov-Milne model. The same result is also confirmed by an analysis of the radial velocities of these stars.

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