A HIGH-ORDER ANALYTICAL MODEL FOR THE SECULAR DYNAMICS OF IRREGULAR SATELLITES

We develop an analytical model for the long-term (secular) dynamics of irregular satellites of the giant planets. The disturbing potential in this model is represented by a high-order (in semimajor axis, eccentricity, and inclination) Legendre expansion. We use a third-order Horiʹs averaging method to eliminate terms in the original equations that are irrelevant for the long-term dynamics and to construct new second- and third-order secular terms. The resulting secular equations are valid for both direct and retrograde orbits (of any inclinations) and for eccentricities up to (almost equal)0.7. In the present paper we describe the mathematical background of our method and test it in several applications. The method uses a Hamiltonian formulation of dynamics. The original Hamiltonian and its high-order secular forms are represented by series that have self-similar functional forms. The coefficients of these series are calculated by using an algebraic manipulator. This approach allows us to iterate Horiʹs perturbation method to high orders. To test our method, we (1) calculate the precession frequencies of the orbits of the irregular satellites at Jupiter and (2) determine the dynamical structure of the Kozai resonance. We show that this resonance occurs at progressively larger (proper) inclinations with increasing separation of the satellite from the parent planet. These results are compared to those obtained by numerically integrating the exact equations of motion. Our theory will be particularly useful for determining the locations and strengths of secular resonances in the space occupied by distant satellite orbits. Several irregular satellites have been trapped in secular resonances by some, likely primordial, mechanism.