Application of the Undefined Frequency Method for the Analysis of a Two Planet Problemстатья
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Аннотация:The problem of motion of two bodies (planets) with comparable masses around a massive сentral body (star) is considered in proposition that planet masses are significantly less than the mass of the сentral body. It is assumed that the movement of the planets occurs in the same plane and orbits are close to circular. The motion of the planets is constructed directly in polar coordinates using the method of undefined frequencies proposed by the author, which is a modification of the method of successive approximations. The first approximation is obtained, which is the sum of uniform circular motion and small quasi-periodic terms for each planet. The latter are the sum of periodic components whose periods are equal to the periods of circular motions of both bodies, as well as their sums and differences. Herewith the periods of circular motions change in comparison with what Kepler’s third law gives: the period of the inner planet increases, and the outer one decreases. The second feature of the constructed approximate solution is the absence of secular perturbations.