Analysis of the Stability of a Planetary System on Cosmogonic Time Scales

We consider the dynamical evolution of planetary systems whose structure is nearly circular and coplanar. The analysis is performed by the Hori-Deprit averaging method within the theory of the first order in planetary masses. A convenient set of canonical elements and a rarely employed variety of astrocentric coordinates are used to derive the equations of motion. Owing to the use of the chosen system of canonical elements, the expansions of the right-hand sides of the averaged equations contain a relatively small number of terms. Compared to other widespread coordinate systems, the astrocentric coordinates used by us allow a more convenient representation of the disturbing function to be obtained and do not require its expansion into a series in powers of a small parameter. On time scales similar to 10(5)-10(7) years we have studied the long-term evolution of the planetary systems HD 12661, upsilon Andromedae, and some model systems by numerical integration of the averaged equations. Possible secular resonances have been revealed in the systems considered.