A probabilistic model for the Solar System planetary distribution

Abstract

The discovery of several extrasolar systems, each one characterized by its own planetary distribution around a central star, made the scientific interest addressed to the analysis of models permitting to predict, or at least estimate, the orbital features of the extrasolar planets. The main purpose of this work is to describe a mathematical model, inspired by quantum mechanics, able to provide a probability distribution of planets placing in a star system, mainly driven by the central star mass. More in detail, for any given eigenvalue of the model discrete spectrum, a distinct probability distribution with respect to the central star distance can be built. As per the Solar System, it has been possible to prove that both inner and outer planets belongs to two different spectral sequences, each one originated by the minimum angular momentum owned by silicate/carbonate and icy planetesimals respectively. In both sequences, the peak of the probability distributions almost precisely coincided with the average planets distance from Sun; furthermore, the eigenvalue spectrum of the inner planets thickens in an accumulation point corresponding to the asteroids belt, thus showing a striking similarity to the real matter distribution in the Solar System. From this point of view, the Titius–Bode law for the Solar System planets distribution is nothing but an exponential interpolation of the eigenvalues of both inner and outer sequences.