Evolution of some quasicircular orbits in the planetary problem

We derive in the plane problem a new closed solution of the Lagrangian equations for resonant motion, concomitantly including zeroth order, and approximate first order and secular perturbations. A major aim is the determination of simple lower limits for the maximum eccentricities and variations of semimajor axes (intrinsic values). Applications of the general solution are made for each perturbation separately. (i) Zeroth and first order perturbations: a new closed solution for the principal zeroth order variation of semimajor axes is obtained. The maximum eccentricity and relative change of semimajor axis of any lunar orbit cannot be lower than 0.019 and 0.018, respectively. (ii) Secular perturbations: with the angular momentum integral our secular perturbations can be easily extended to the spatial problem. The planetary Lidov–Kozai problem is extended to retrograde orbits, showing that large variations of eccentricity and inclination occur for initially circular orbits, if initial mutual inclinations are between about 40° and 150°. (iii) Resonant perturbations: for first and second order resonances, and initially circular orbits our formulas generally approximate just the calculated orbital elements during the whole motion. As a new unexpected result, the numerical exploration of the asteroid belt reproduces most of its overall characteristics up to third order resonances within the restricted three-body problem and modest initial eccentricities ≤0.05≤0.05.

► I provide a new solution of the Lagrangian equations of motion.
► A new solution for zero order semimajor axis variations.
► New secular equations for retrograde orbits.
► New resonance equation.