Application of the Hori–Deprit Method to the Analysis of the Cosmogonic Evolution of Weakly Perturbed Planetary Systems

The dynamical evolution of nonresonant two-planet systems whose structure is nearly circular and coplanar is studied by the Hori–Deprit averaging method. The Poincaré astrocentric coordinates and the complex form of the second system of Poincaré canonical elements are used. The equations are averaged to the second order in planetary masses. The averaged system is integrated using one model two-planet system and the real two-planet system HD 12661 as examples. The constructed solution is compared with the solution of the averaged system of the first approximation and with the solution of the exact equations of motion in rectangular coordinates. In the case of both planetary systems, the second approximation is shown to agree better with the solution of the exact equations. According to the exact equations, the equations of the second approximation, and the equations of the first approximation, the oscillation period of the eccentricities in the HD 12661 system is 26 175, 26 309, and 26 391 years, respectively.