Astrometry, mathematical problems of

Problems in astronomy related to the creation of an inertial reference coordinate system in space and a unification of the totality of fundamental astronomic constants by determining the coordinates of celestial bodies and by studying the rotation of the Earth. Methods in astrometry are based on the theory and the practice of geometrical measurements on the celestial sphere and the determination of distances of stars.

An important element in astrometric studies is the most accurate possible determination of the direction of the straight line "observer–star" at the moment of the observation. Since the study of the mutual location of points is more convenient and more illustrative than the study of mutual location of directions, an auxiliary spherical surface (the so-called celestial sphere) is introduced in the treatment, and all objects are assumed to be at an equal distance from the observer and located on this sphere. Spherical trigonometry makes it possible to use various coordinate systems on the celestial sphere and to determine numerous relations between the angles and the arcs of configurations of celestial objects. These relations determine a large proportion of astrometric methods of observation and the geometrical foundation of astrometric telescopes.

The phenomena studied in astrometry are related to various irregularities in the rotation of the Earth (the irregular rotation of the Earth, migration of poles, variation of the rotation axis due to precession and nutation, etc.) on one hand, and to the actual displacements of celestial bodies on the other. These considerations determine the temporal and the qualitative structure of the series of observations which are performed and necessitate the use of specific mathematical methods for their analysis. Hence the importance and the urgency of solving and studying large systems of linear equations involving thousands of equations with hundreds of unknowns, which is complicated even further by the fact that some of them can only be insufficiently determined. A typical example of the genesis of such problems is the processing of observations made on the bodies of the solar system (mainly small planets) in order to determine the origin of the fundamental coordinate system. Another example is the derivation of the constituents of the gravitation field of the Earth from the observations of the Earth made by artificial satellites, realized by the worldwide network of observation stations.

The complexity of the rotation of the Earth, the difference between the real Earth, which is a visco-elastic body, from the model as an absolutely solid makes it necessary to search for latent relationships by methods such as correlation-spectral analysis and various smoothing methods. Such studies include the studies of the variation in latitudes and longitudes and the resulting pole migration, as well as evaluations of fine and complicated fluctuations in the rate of rotation of the Earth.

Owing to the nature of the observed phenomena it is often impossible to complete the observations in a single cycle, and it is then necessary to work with partly overlapping segments of observations and with a variable origin. Such observations are processed by methods of solving finite-difference equations. This is typical of instrumental research in astrometry and of the problems in astrometry which involve the measurement of some linear combinations of errors in the coordinates of celestial objects.

Methods of mathematical statistics (mostly linear methods) are employed to allow for errors in astrometric observations.