Astronomy, mathematical problems of
Mathematical problems in the study of celestial objects. Special methods have been developed for the solution of a number of such problems. These methods also found use in other branches of science. On the other hand, mathematical techniques specially intended for Earth-bound problems, which may be modified as required, are extensively used in astronomy.
Astronomy is a complex science which deals with celestial objects and their systems from various aspects, which may be very remote from one another. This is why the number of mathematical problems in astronomy is very large.
A very important chapter of astronomy is astrometry, one of the main problems in which is to determine an inertial reference coordinate system in space. The coordinate systems which are traditionally employed in astronomy, geodesy and other fields of science, based on the planarity of the Earth's equator and the vernal equinox (i.e. the intersection line of the Earth's equator with the plane of the ecliptic), are not inertial and cannot be firmly fixed in space, since both these planes continuously execute a complex motion (as a result of precession, nutation and migration of the Earth's poles). For the sake of comparability, the coordinates of the stars are usually referred to by the position of the plane of the equator and of the point of vernal equinox at some fixed date (epoch), while the coordinate system itself, which has been fixed in this way, is based on very careful determinations of the coordinates of a number of stars, recorded in special catalogues (fundamental star catalogues). However, a substantial difficulty remains: In order to reproduce such a coordinate system at a moment of time other than the epoch of the catalogue, one must know the changes in the locations of the fundamental stars relative to the coordinate system, due to their proper motion. In order to overcome this difficulty, it has been attempted since the middle of the 20th century to determine the inertial coordinate system with respect to remote galaxies, the proper motion of which is negligible. In this context special importance is given in astrometry to problems involved in the calculation of the most probable values of the parameters which determine the directions of the stars on the strength of repeated observations, and to estimating the probability characteristics of these values. The solution of this problem is also important in most other branches of astronomy, since the latter is to a large extent an observational science.
Various mathematical problems must be solved in theoretical astrophysics, which on the basis of observations of celestial objects studies the structure of these objects, the physical processes taking place in them and their evolution. One of principal problems of astrophysics is the structure and evolution of stars. The theory of the internal structure of stars leads to differential equations which describe the conditions of mechanical and energetic equilibrium of the stars. The solutions of these equations can sometimes be expressed in terms of elementary functions, but most often they are so complicated that they must be solved by numerical methods.
The study of atmospheres of stars and of processes taking place in the nebulae and in the interstellar space is based on the mathematical theory of radiative transfer, which was intensively developed in astrophysics. In certain cases, e.g. in the study of the passage of radiation through a plane material layer, the transfer equation reduces to integral equations, the solution of which permits one to determine the characteristics of the radiation field inside the star, and also of the radiation emitted by the (interstellar) medium and accessible to measurements.
In studying the motion of gas masses in stars and nebulae, in studying the processes involved in the extension of gas clouds, their collisions with each other and with the interstellar space, extensive use is made of the mathematical apparatus of gas dynamics and electrodynamics.
Stellar astronomy, which is concerned with the laws governing the structure, dynamics and evolution of stellar systems, makes use of mathematical relationships between the distribution of some given characteristics of the stellar system (so-called distribution functions) and the distribution of the observed characteristics. For instance, the study of the connections (under certain supplementary assumptions) between the distribution functions of the distances of stars in a given solid angle and their absolute and apparent (observational) values yields an integral equation, the solution of which offers an explanation of the law governing the distribution of the stellar density in this solid angle. Similar equations result on comparing the distribution functions of the sought-for spatial velocities of the stars and the observed radial velocities.
In stellar kinematics the problem of determining the components of the Sun's velocity and the rotational characteristics of the Galaxy from statistical studies on the coordinates, the proper motion and the radial velocity, leads to an overdetermined system of constrained equations compiled for individual stars (or individual areas in the sky). The mathematical techniques in mechanics are used in solving problems of stellar dynamics connected with the study of star clusters, galaxies and clusters of galaxies. The individual objects which constitute the system are regarded as material points which interact in accordance with the law of gravitational attraction, with allowance for the specific features of celestial bodies. A solution of the problems of celestial mechanics, which is concerned with the motion of celestial bodies in a gravitation field, leads to systems of differential equations of motion. The solution of the most general problem of the motion of $ n $ bodies which attract each other according to the law of gravity under arbitrary initial conditions is solved by numerical integration. However, the solutions given by this method are only satisfactory during a limited period of time, and no conclusions as to the evolution of the system of objects are possible. The partial problem of three bodies was more satisfactorily solved using power series in the powers of time; however, these series converge very slowly and are therefore unsuitable for application to the observations on the motion of actual objects. Particular problems — the $ 3 $- body problem, the $ 10 $- body problem (the Sun and the 9 major planets), etc. — have also been studied.
Problems involving the motion of specific celestial bodies are solved by series expansions with respect to powers of a small parameter and subject to various assumptions that simplify the calculation.
Astrodynamics, which deals with the motion of artificial celestial objects, makes use of specific differential equations of motion. Solutions of problems involved in the motion of artificial satellites must allow for the disturbing forces due to the non-spherical shape of the Earth, the resistance of the atmosphere, the solar radiation pressure (in the case of balloon satellites) and for certain other factors.
For more details see the following entries: Astrometry, mathematical problems of; Astrophysics, mathematical problems of; Stellar astronomy, mathematical problems of; Classical celestial mechanics, mathematical problems in.
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