Diffusion characters of the orbits in the asteroid motion (Q5947703)

The problem of the gaps in the distribution of asteroids of the main belt in relation to their mean motions is one of the famous problem of celestial mechanics in spite of its relatively young age (about 70 years). A special attention is attracted to the fact that, for some commensurabilities between the mean motions of asteroids and Jupiter of type \(4/3\), \(3/2\) there are populations of asteroids, but for others, \(2/1\) for example, there is a gap (the famous Hecuba gap). It was understood that the problem arises due to different character of stability of motion of asteroids. There are many investigations trying to explain the mechanism of formation of gaps in the asteroid distribution. In the last two decades the progress was mainly achieved by studying processes of diffusion in dynamical systems, and by investigating their regular and chaotic behavior. Here the authors apply to so-called symplectic mapping method to the restricted three-body problem in order to explain the differences in asteroid distribution for resonances \(4/3\), \(3/2\) and \(2/1\). The exponential diffusion law is established in the developed chaotic region, and an algebraic law in the mixing region. Also, the authors detect a region where the diffusion follows a logarithmic law. It is shown in the vicinity of an island that the logarithm of escape time decreases linearly as the initial position moves away from the island. But when closely approaching the island, the escape time grows fast, consistent with the superexponential stability of invariant curve. When applied to the motion of asteroids, the fixed points of this mapping and their stability give an explanation of the observed distribution of asteroids.