A new method of solving the equations of celestial mechanics
- Paper ID
IAF-67-07
- author
- company
Institute of Physics, Bulgarian Academy of Sciences
- country
Bulgaria
- year
1967
- abstract
The practical solution of the «-body problem in celestial mechanics represents-* the basis for investigation the motion of space vehicles. The existing methods for practical solution of this problem do not always lead sufficiently rapidly to the desired result even if one uses computers. Here we propose a new method of solving this problem which has the advantage over all known methods in this that the approximate solution of the differential equations of celestial mechanics is reduced to quadratures which can easily be calculated. The new method consists in the following. The variables of Newton’s celestial mechanics are developed in power series with respect to the constant of the gravitational interaction / (Newton’s gravitational constant). In the case of the n-body problem comparatively simple differential equations are obtained which can be easily integrated. The author1,2 has obtained the explicit solution of the n-body problem with accuracy up to terms in the developments containing the factor f1 and G. Gemidjiev3—up to terms containing the factor /3. The convergence of the new method is verified by the problem of two bodies. The calculations show that in this case even the first terms of the power series given by us converge rather quickly to Kepler’s ellipses, parabolas and hyperbolas. The remarkable thing about our approximate solution of the «-body problem is, that it gives in terms of elementary time functions and the initial conditions of movements the explicit movement of the «-bodies. Furthermore no suppositions are made respectively the masses of the bodies. As is well known almost all approximate solutions of the problem of «-bodies make use of the assumption that in the system there exists a central body with a mass considerably exceeding the masses of the.rest of the bodies. We do not use this hypothesis, and on that account our method can also be applied to such problems as the movement of a body in the gravitational field of a double, triple or quadruple star. Such problems are difficult to treat by means of classical celestial mechanics.