Geodesic
On the Conditions under Which a~Satellite Orbit Intersects the Surface of a~Central Body of Finite Radius in the Restricted Double-Averaged Three-Body Problem
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 2, Tome 259 (2007), pp. 156-173.

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This paper is concerned with the applied problem of choosing long-living orbits of artificial Earth satellites whose evolution under the influence of gravitational perturbation from the Moon and the Sun may result in the collision of the satellite with the central body, as was shown by M. L. Lidov for the well-known example of “Vertical Moon.” We use solutions of the completely integrable system of evolution equations obtained by Lidov in 1961 by averaging twice the spatial circular restricted three-body problem in the Hill approximation. In order to apply the integrability of this problem in practice, we study the foliation of the manifold of levels of first integrals and the change of motion under crossing the bifurcation manifolds separating the foliated cells. As a result, we describe the manifold of initial conditions under which the orbit evolution leads to an inevitable collision of the satellite with the central body. We also find a lower bound for the practical applicability of the results, which is determined by the presence of gravitational perturbations caused by a polar flattening of the central body. 
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V. I. Prokhorenko. On the Conditions under Which a~Satellite Orbit Intersects the Surface of a~Central Body of Finite Radius in the Restricted Double-Averaged Three-Body Problem. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 2, Tome 259 (2007), pp. 156-173. http://geodesic.mathdoc.fr/item/TM_2007_259_a10/

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