A Canonical Approach to a Second-Order Solution for the Secular Motion of the Planets

Abstract

A second-order analytical solution for the secular motion of the planets can be efficiently developed from canonical Lie transform methods which have recently appeared in the literature. The averaging procedure side-steps the classical approach of expanding the disturbing function in terms of Laplace coefficients. Additionally, it is found through a particular choice of parameters that there is no necessity of distinguishing between perturbations due to “inner” planets and those arising from “outer” planets. Truncated power series in the planetary inclinations and eccentricities are also avoided as they can be shown to be unnecessary.

The time interval of validity of the solution will be extended over that of current theories because of the full inclusion of all second-order nonlinear contributions to the averaged Hamiltonian H̄. The original Hamiltonian is expressed in “natural” nodal-polar coordinates and averaged with respect ot the planetary arguments of latitude to give H̄ = H̄₀ + H̄₁ + H̄₂. The averaged Poisson bracket representation for H̄₂ is

$${\bar H_2} = \frac{1}{2} < \left( {{H_1};{W_1}} \right) + \left( {\left( {{H_0};{W_1}} \right);{W_1}} \right) > $$

where W₁ is the generator that removes short period contributions from H₁ (thereby producing H̄₁).

The completeness of the approach will yield a more accurate picture of the long-term evolution of the planetary elements. In particular, the improved theory will aid investigations dealing with solar system stability analyses, hybrid planetary intermediaries, and paleo-climatology models.