Fractal escapes in Newtonian and relativistic multipole gravitational fields

We study the planar motion of test particles in gravitational fields produc ed by an external material halo. Both the Newtonian and the general-relativ istic dynamics are examined, and in the relativistic case the dynamics of b oth massive and massless particles are investigated. The halo field is give n in general by a multipole expansion; we restrict ourselves to multipole f ields of pure order, whose Newtonian potentials are homogeneous polynomials in cartesian coordinates. A pure n-pole field has n different escapes, one of which is chosen by the particle according to its initial conditions. We find that the escape has a fractal dependency on the initial conditions fo r n > 2 both in the Newtonian and the relativistic cases for massive test p articles. but with important differences between them. The relativistic mot ion of massless particles, however, was found to be regular for all the fie lds we could study. The box-counting dimension was used in each case to qua ntify the sensitivity to initial conditions which arises from the fractalit y of the escape route. (C) 2000 Published by Elsevier Science B.V. All righ ts reserved.