Review of the works on the orbital evolution of solar system major planets

Abstract

The cognition history of the basic laws of motion of Solar system major planets is presented. Before Newton, the description of motion was purely kinematic, without relying on physics in view of its underdevelopment. From the standpoint of the modern mathematical theory of approximation, all of the models from Ptolemy’s predecessors to Kepler inclusive differ only in details. The mathematical theory worked on an infinite time scale; the motion was represented by P. Bohl’s quasi-periodic functions (a special case of H. Bohr’s quasi-periodic functions). After Newton, the mathematical description of motion came to be based on physical principles and took the form of ordinary differential equations. The advent of General Relativity (GR) and other relativistic theories of gravitation in the 20th century changed little the mathematical situation in the field under consideration. Indeed, the GR effects in the Solar system are so small that the post-post-Newtonian approximation is sufficient. Therefore, the mathematical description using ordinary differential equations is retained. Moreover, the Lagrangian and Hamiltonian forms of the equations are retained. From the 18th century until the mid-20th century, all the theories of planetary motion needed for practice were constructed analytically by the small parameter method. In the early 20th century, Lyapunov and Poincare established the convergence of the corresponding series for a sufficiently small time interval. Subsequently, K. Kholshevnikov estimated this interval to be on the order of several tens of thousands of years, which is in agreement with numerical experiments. The first works describing analytically (in the first approximation) the evolution on cosmogonic time scales appeared in the first half of the 19th century (Laplace, Lagrange, Gauss, Poisson). The averaging method was developed in the early 20th century based on these works. Powerful analytical and numerical methods that have allowed us to make significant progress in describing the orbital evolution of Solar system major planets appeared in the second half of the 20th century. This paper is devoted to their description.