Mechanisms of Generation of Chaos in the Solar System

Abstract

The mechanism of generation of chaos in the Solar System is studied, by making use of a mapping model. This mapping is simple and we can study the mechanism of generation of chaos without beeing lost in unnecessary details. On the other hand, it is realistic as it contains all the main features of the original problem, i.e. the topology of its phase space is the same as that of the real system. The motion of asteroids near a resonance is used as the basic example. We start with a simple model with two degrees of freedom and we study how the appearance of chaos is affected as the model becomes more and more complex (and more realistic).

We start with the simplest nontrivial model, the restricted circular planar three body problem, with the Sun and Jupiter as primaries. A two dimensional mapping model is constructed, valid globally, and we study the topology of its phase space and the regions where chaotic motion is expected to appear. We focus our attention to the 2:1 and 3:1 resonances and their basic difference with respect to their chaotic properties is shown. Next, we proceed to the elliptic restricted planar three body problem, assuming that Jupiter is moving in a fixed elliptic orbit. The mapping now is four dimensional and we express it in the resonant action-angle variables S, σ, N, v. It is explained how the coupling between the two degrees of freedom S,σ and N,v generates chaos near a resonance. Inside the resonance we may also have ordered motion, depending on the initial phase (i.e. initial values of σ and v). Finally, we include to our model the gravitational effect of Saturn on the orbit of Jupiter and show that the appearance of chaotic motion is now widespread inside the whole chaotic zone. It is also explained why the above nentiond chaotic motion appears only inside the resonance zone.

The effect of a nonrealistic model on the evolution of the system is studied. In particular, we study the consequences of not including in our model the high eccentricity resonances. As a byproduct of the present study, we show how the knowledge of the basic families of periodic orbits of the original system can be used to check the convergence of the perturbation series and the reality of a model.