2 00 6 Long - term evolution of the asteroid orbits at the 3 : 1 mean motion resonance with Jupiter ( planar problem )

The 3:1 mean-motion resonance of the planar elliptic restricted three body problem (Sun-Jupiter-asteroid) is considered. The double numeric averaging is used to obtain the evolutionary equations which describe the long-term behavior of the asteroid's argument of pericentre and eccentricity. The existence of the adiabatic chaos area in the system's phase space is shown. The 3:1 mean-motion resonance in the planar elliptic restricted three-body problem (Sun–Jupiter–asteroid) has long attracted considerable attention of specialists [1–16]. In order to find secular effects, the equations of motion can be averaged over fast variables, namely, over mean longitudes of the asteroid and Jupiter (see, for example, [3, 12]). Upon averaging, a nonintegrable system appears which describe the " fast " and " slow " components of secular evolution. The " fast " evolution consists in changing resonance phase (Delaunay variable) D = λ − 3λ ′ , where λ and λ ′ are the mean longitudes of the asteroid and Jupiter, respectively. The " slow " evolution reveals itself in a gradual change of perihelion longitudes of resonance asteroid orbits. In [4–6,8,11] a model Hamiltonian system was considered with Hamiltonian H p which was a principal part of the Hamiltonian of a planar elliptic three-body problem averaged over λ and λ ′ taking the resonance into account. Such a truncation of the Hamiltonian is justified at small orbit eccentricities of the asteroid and Jupiter. Processes described by this model system are also separated into " fast " and " slow " processes. In order to analyze different variants of the " slow " evolution, one can make yet another averaging: averaging over fast processes [6,17]. In this paper double averaging is used for studying the " slow " evolution without restrictions on the orbit eccentricity of an asteroid. The results of this study allow one to understand, under which initial conditions the system with Hamiltonian H p correctly describes secular effects at the resonance under discussion. 1 2. Averaging over mean longitudes We assume that the semimajor axis of the orbit of Jupiter can be taken as the unit length, while the sum of masses of the Sun and Jupiter is the unit mass. The unit time is chosen so that the period of revolution of Jupiter around the Sun is equal to 2π. We write the equations of motion of the asteroid in the variables where x, y, and L are the elements of …