Abstract
In this paper we propose a notion of stability, which we call varepsilon -N-stability, for systems of particles interacting via Newton’s gravitational potential, and orbiting a much bigger object. For these systems the usual thermodynamical stability condition, ensuring the possibility to perform the thermodynamical limit, fails, but one can use as relevant parameter the maximum number of particles N that guarantees the varepsilon -N-stability. With some judicious but not particularly optimized estimates, borrowed from the classical theory of equilibrium statistical mechanics, we show that our model has a good fit with the data observed in the Solar System, and it gives a reasonable interpretation of some of its global properties.
Highlights
Maybe the secret of the huge success of Kolmogorov–Arnold–Moser theory relies in the spectacular application, found out by Vladimir Igorevich Arnold, to the planetary problem
One decade after Kolmogorov’s announcement, at the 1954’s International Congress of Mathematician, of the “theorem of the conservation of the invariant torus” [14], the brilliant student of Kolmogorov—aged 27—formulated a version of Kolmogorov’s theorem suited to the planetary problem [1]. He used it to prove the “metric stability” of the simplest, albeit non-trivial, planetary system: two planets and a sun constrained on a plane
In the last part of this paper we try to apply the same techniques developed for asteroids to a system of planets, i.e., of object small with respect to the star but larger than asteroids, having orbits with very different average radii: we show that assuming for the average radius of the orbit of the i-th planet the following Titius–Bode law: Ri = b + cai with Ri the orbit’s average radius of the i-th planet, b and c fixed lengths (0.4 and 0.3 U.A., respectively, for the Solar System) and a > 1 a fixed number (a = 2 for the Solar System), and assuming N small enough, the system is ε − N stable
Summary
Maybe the secret of the huge success of Kolmogorov–Arnold–Moser (kam) theory relies in the spectacular application, found out by Vladimir Igorevich Arnold, to the planetary problem. If the total mass of asteroids depends on their number N , and it goes suitably to zero when N increases, it is possible to prove the ε − N -stability of the system In this context it is easier to outline the basic problem that one has to face: we want to fix the parameter of the system in such a way that the contributions of the collisions, in which the Hamiltonian is negative and has a large absolute value, are not too relevant for the canonical probability distribution. In this setup the asteroids have always a similar average distance from the star, but they have a well-defined distribution of the masses.
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