Short times characterisations of stochasticity in nonintegrable galactic potentials

Abstract

This paper proposes an alternative characterisation of the degree of stochasticity exhibited by orbits in a fixed galactic potential. This differs from earlier work involving Liapounov exponents by focusing on the statistical properties of ensembles of trajectories, rather than individual orbits, and by restricting attention to the properties of these ensembles over time scales shorter than the age of the Universe, t_H. For many potentials, generic ensembles of initial conditions corresponding to stochastic orbits exhibit a rapid evolution towards an invariant measure $Γ$, the natural unit to consider if one is interested in self-consistent equilibria. The basic idea is to compute short time Liapounov exponents $χ(Dt)$ over time intervals $Dt$ for orbits in an ensemble that samples $Γ$, and to analyse the overall distribution of these $χ$&#39;s. One finds that time averages and ensemble averages coincide, so that the form of the distribution of short time $χ(Dt)$&#39;s for such an ensemble is actually encoded in the calculation of $χ(t)$ for a single orbit over long times $t>>t_{H}$. The distribution of short time $χ$&#39;s is analysed as a function of the energy E of orbits in the ensemble and the length of the short time sampling interval ${\D}t$. For relatively high energies, the distribution is essentially Gaussian, the dispersion decreasing with time as $t^{-p}$, with an exponent 0<p<1/2 that depends on the energy E. (to appear in Astronomy and Astrophysics)