Linearized averaged resonant equations and their solution for dust particles

Abstract

The averaged resonant equations of motion for the planar circular restricted three-body problem are solved on the linearization basis taking into account also non-gravitational effects. The averaged resonant equations are derived from Lagrange's planetary equations with additional Gauss's terms caused by the non-gravitational effects. The time depending solution has the standard form with exponential, quadratic, linear and constant terms. The existence of a rotational symmetry in the action of the non-gravitational effects around the star determines the order of a characteristic equation of the linearized system. In the symmetrical case (order 3) the considered non-gravitational effects are the stellar electromagnetic radiation and the radial stellar wind (stellar radiation). In the asymmetrical case (order 4) the stellar radiation and interstellar gas flow are considered. It is investigated how well the linearization solution describes real solution obtained from an equation of motion by a comparison of the resonant libration frequency found analytically and numerically. It is found that from initial values of the evolving orbital parameters (semimajor axis, eccentricity, longitude of pericenter, and resonant angular variable) in the averaged phase space the linearization frequency depends most sensitively on the initial value of the resonant angular variable. For small libration amplitudes of the resonant angular variable the best match of the real libration frequency and the linearization frequency is located approximately at the solution of the resonant condition ($da / dt$ $=$ 0). If the initial averaged conditions are chosen close to the solution of resonant condition, then the linearization frequency for practically all simple oscillatory evolutions matches the real libration frequency and the linearization solution very well approximates the real evolution.