This paper is devoted to the memory of prominent scientist-mechanician, laureate of Lenin reward, Professor M.L. Lidov. His works on the evolution of satellite orbits which were carried out fifty years ago have not lost significance up to day. They were developed after the recent discoveries of numerous distant satellites of the giant-planets. Among them few so called apsidal- librating orbits which evaluate in the conditions of known Lidov ? Kozai resonance are discovered. In 1961 the model of a double- averaged Hill problem was studied by M.L. Lidov in details for the investigation of the evolution of satellite orbits under the influence of gravitational perturbations of external bodies. Later his improvement was carried out by constructing the solutions of 3rd order in regard to small parameter ? the relation of mean motions of the planet and the satellite. One of the first integrals is received in Lidov's form and on its basis the qualitative analysis of evolutionary equations and the intersect conditions satellite orbits with the surface of spherical planet having finite radii was carried out. Then the new constructive-analytic solution of the evolutionary problem was proposed. In contrast to previous solutions it takes into account approximately the set of 4th power additive in the secular part of perturbing function. With the help of constructed solution it is possible to describe analytically (more precisely and on longer time interval) the evolution of satellite orbits which have great apocentric distance comparable with the radii of planetary Hill sphere with regard to the Sun
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