Chaotic motion around prolate deformed bodies

Abstract

The motion of particles in the field of forces associated to an axially symmetric attraction center modeled by a monopolar term plus a prolate quadrupole deformation are studied using Poincaré surface of sections and Lyapunov characteristic numbers. We find chaotic motion for certain values of the parameters, and that the instability of the orbits increases when the quadrupole parameter increases. A general relativistic analogue is briefly discussed.PACS numbers: 05.45.+b, 95.10.Fh, 95.10.Ce, 04.20.Jb, 03.20.+i Attraction forces represented by a monopolar plus a prolate quadrupolar distribution of masses (charges) are a good approximation for elongated massive (charged) bodies. Examples range from astrophysics to nuclear physics. There are many observed galaxy clusters with a cigar like shape [1]. Also, the nuclear charge of light gold atoms has been reported as having a large prolate deformation [2]. Most of the Dwarf Galaxies in the Virgo Cluster may obey the "prolate hypothesis", i.e., they probably have a prolate spheroidal shape [3]. Asteroids also have a prolate shape, but usually they are not axisymmetric. Merrit [4] found, from detailed modeling of triaxial galaxies, that most of the galaxies must be nearly axisymmetric, either prolate or oblate.Classical, as well as, quantum chaos have been studied in a variety of axially symmetric fields of forces. In particular, attraction centers described by potentials that are the sum of two terms: a monopolar term and a quadrupolar deformation. Furthermore this center is "perturbed" by an external distribution of masses (charges) represented by its external multipolar moments, i.e.,Sometimes the monopolar term is changed by the potential of a spring [5]. In general, in all these cases the terms that originate the chaos are the external multipolar moments. We shall consider the simplest, albeit, important case of a particle moving in the field of a monopole plus a quadrupole deformation. This deformation is usually considered to be the major deviation from spherical symmetry. In cylindrical coordinates, (r, ϕ, z), the field takes the generic form,where α is a constant that may be associated with the central body mass (charge). It is instructive to have a special model in mind, consider two equal masses located on the z-axis symmetrically, at z = −a and z = +a. The gravitational potential of the above mass configuration up to the order a 3 is (3) with q = 2αa 2 . We shall use α = 1 without loss of generality. Note that we are not considering external multipolar moments (V P = 0), i.e., only deformed cores will be studied.We can distinguish two cases depending on the sign of q. The oblate deformation case, q < 0. This is the common case for bodies deformed by rotation and has been analyzed in astronomy for more than two hundred years. The integrability of the Newton equations for a particle moving in the gravitational field of an axially symmetric oblate body is an unsolved problem. It is known as the classical problem of the existence of the third isolating inte...

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