The role of stochastic motion in a central field with a bar-like perturbation

Abstract

The motion of a particle in a central field with a rotating barlike perturbation has been investigated by many authors, but, only a few papers deal with fixed bar perturbations which arise, for example, as a result of the radial orbit instability (e.g. D. Merritt & L.A. Aguilar 1985, MNRAS 217, 787). Our main interest is to investigate the relevance of stochastic orbits in such kind of fields. Thus, using information theory, we develop a simple method to evaluate the global degree of stochasticity of a set of orbits in a given Hamiltonian. Briefly, we obtain the Poincaré Surface of Section (PSS) for each orbit, then we choose a partition of it and, finally, we compute the entropy of the PSS defined as S = − Σ ^N_i=1 P _i In P _i, where Pi is the probability of finding an intersection of the orbit with the PSS within the i ^th partition. We expect S to have two regimes: a low-value one for regular orbits and a high-value one for stochastic orbits. To test our method, we used the well known Hénon-Heiles potential. We obtained the PSS with 2500 points for 70 orbits. Our results showed the power of the method to describe the features of the potential, the lower values of S corresponding to stability islands and the higher ones to clearly stochastic regions.

We then applies this algorithm to the isochrone potential, ϕ _I (r), plus a bar-like perturbation of the form ϕ ₁(r,g4) = −αψ(r) sin 2φ, where α is the perturbation parameter and α is a constant. We consider 90 orbits, all of them with low angular momenta. Our results show that about 50% of the orbits are stochastic, although the effective perturbation ϕ ₁/ϕ ₁ is not larger than 4%. This result is very interesting since, as the probability of finding a system with perfect spherical (or axial) symmetry is very low, there is a good chance of finding stochastic orbits for the low-angular-momentum stars that move within such a field.