Behaviour of a weakly perturbed two-planetary system on a cosmogonic time-scale

Orbital evolution of planetary systems similar to our Solar one represents one of the most important problems of Celestial Mechanics. In the present work we use Jacobian coordinates, introduce two systems of osculating elements, construct the Hamiltonian expansions in the Poisson series in all elements for the planetary three-body problem (including the problem of Sun-Jupiter-Saturn). Further we construct the averaged Hamiltonian by the Hori-Deprit method with accuracy upto second order with respect to the small parameter, the generating function, change of variables formulae, and right-hand sides of averaged equations. The averaged equations for the Sun-Jupiter-Saturn system are integrated numerically at the time-scale of 10 Gyr. The motion turns out to be almost periodical. The low and upper limits for eccentricities are 0.016, 0.051 (Jupiter), 0.020, 0.079 (Saturn), and for inclinations to the ecliptic plane (degrees) are 1.3, 2.0 (Jupiter), 0.73, 2.5 (Saturn). It is remarkable that the evolution of the ascending node longitudes may be secular (Laplace's plane) as well as librational one (ecliptic plane). Estimates of the Liapunovian Exponents for the Sun-Jupiter-Saturn system are obtained. The corresponding Liapunovian Time turns out to be 14 Myr (Jupiter) and 10 Myr (Saturn).