On chaos in the oscillations of galaxies

This paper describes a search for chaotic behaviour in a low-dimensional model of the oscillations of a galaxy. The governing equations of the model are the complete set of moment equations of the second order associated with the collisionless Boltzmann equation. The moment equations reduce to the tensor virial equations and equations governing the evolution of the kinetic energy tensor of the system. The classical integrals of motion of a gravitational many-body system supplement the governing equations. Closure of the system of moment equations requires a specification of the density distribution in the system and the velocity field describing the mean motions of the stars. In this paper, we close the system for a galaxy possessing triplanar symmetry in the case that the planes of symmetry remain fixed in an inertial frame of reference. We model the galaxy as a heterogeneous ellipsoid with an arbitrary stratification of the density distribution, and we represent the mean motions of the stars in terms of a velocity field that sustains that density distribution consistently with the equation of continuity. The evolution of the model is quasi-homologous in the sense that the semi-axes of an ellipsoid characterizing the size and shape of the galaxy are functions of the time, whereas the stratification of the density does not change. With the aid of additional, non-classical integrals of the motion, which were discovered by Som Sunder & Kochhar, we eliminate the kinetic energy tensor from the virial equations and reduce the virial equations to a closed system of equations of motion that govern the semi-axes of the characteristic ellipsoid. Those equations of motion represent the dynamics of a galaxy as the dynamics of an oscillator with just three degrees of freedom, and they admit of a Hamiltonian formulation. We investigate chaotic behaviour in the model for axisymmetric galaxies. The oscillator that represents the dynamics of an axisymmetric galaxy has only two degrees of freedom, and the constancy of the energy integral enables an investigation of the motion with the aid of Poincare return maps. Our criteria for chaotic behaviour are dissolution of invariant curves in the return maps and, more generally, sensitivity of the motion to small changes in the initial conditions. We survey axisymmetric oscillations of a spherically symmetric model and of oblate and prolate, axisymmetric models with the geometries of an elliptical galaxy of Hubble type E5. Oscillations with excitation energies up to the order of 40 per cent of the binding energy of an equilibrium configuration are found to be regular. Substantial families of chaotic oscillations appear at excitation energies of the order of 40-50 per cent of the binding energy. Lyapunov times of these chaotic oscillations are generally in the range of 50-150 dynamical units of time. For galaxies with typical sizes and masses, the Lyapunov times would be of the order of 1.5-4.5 Gyr.