Abstract
Analytical solutions to a variety of simplified versions of the restricted three-body problem in celestial mechanics possess long running history that encompasses several centuries. Most of the successes were limited either to the planar configuration of the three bodies, to the motion around the Lagrange points, or to the Kozai–Lidov effect. We review some analytical advances obtained by separating rapid and slow subsystems as presented in recently published papers concerning the non-planar motion of the three bodies unrelated to the Lagrange points and to the Kozai–Lidov effect. Most (but not all) of the discussed advances correspond to the bound motion in the considered celestial systems.
Highlights
Rapid and Slow Subsystems inIt is well-known that the classical three-body problem, which has the history of several centuries, does not have a general analytical solution
One of the analytical advances covered in this review is the discovery of conic-helical orbits of the light body in three body systems
Examples are presented for the motion of a planet around a binary star and for the motion of a moon in the planet-star systems
Summary
It is well-known that the classical three-body problem, which has the history of several centuries, does not have a general analytical solution. In the last several years, there have been published papers presenting analytical results, obtained by separating rapid and slow subsystems on the non-planar motion of the three bodies unrelated to the Lagrange points and to the Kozai–Lidov effect. These advances in the restricted three body problem concern both the bound motion and the unbound motion in the considered celestial systems. A, we relax the assumption of the circular orbits of the stars in the binary—the assumption used in Section 5—and present analytical results on the effect of a low-eccentricity of the orbits of the stars on the motion of the circumbinary planet
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