The exponential growth of infinitesimal perturbations in the long-term evolution of simulated galaxies

Abstract

Self-gravitating systems of $N$ particles are chaotic. We study how chaotic the Galaxy is, and what the consequences are. We therefore simulated the dynamical evolution of a galaxy-scale distribution of point masses in order to measure the degree of chaos in such a system. These calculations were performed using the softened gravitational $N$-body tree-code Bonsai, with up to 40 million equal-mass particles. Smaller simulations were performed to establish the scaling of the Lyapunov time $t_L$ with $N$. We established the relations between the degree of chaos, the number of particles, and the softening length in the gravitational force calculation of large-scale $N$-body simulations. The moment in which the bar forms appears to be insensitive to infinitesimal perturbations to the initial realisation. In contrast, the bar strength and its further evolution sensitively depend on these perturbations. Interestingly enough, the maximum in the run-to-run variation in the bar strength is at about the maximum bar strength, and it drops until the bar buckles. The galaxies we simulated are highly chaotic, but the softening in the simulations suppresses chaos. Still, our models show considerable variations in the macroscopic behaviour caused by infinitesimal perturbations to the initial conditions. Real galaxies, however, are expected to be orders of magnitude more chaotic than our simulations, and we are unable to quantify the consequences of this. Smooth galactic potentials for studying individual stellar orbits should be handled with caution on timescales longer than the Lyapunov time. When we extrapolate our results to the number of stars in the Galaxy without planets and other minor bodies, we conclude that Milky Way-size galaxies are chaotic on a timescale of less than a million years.