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

TL;DR

This study uses large-scale N-body simulations to analyze the chaos in galaxy evolution, revealing that galaxies are highly chaotic with chaos timescales much shorter than previously thought.

Contribution

It establishes the relationship between chaos, particle number, and force softening in galaxy simulations, highlighting the extreme chaos in real galaxies.

Findings

Galaxies are highly chaotic with chaos timescales less than 0.1 Myr.

Softening reduces chaos in simulations but does not eliminate it.

Bar formation is insensitive to infinitesimal initial perturbations.

Abstract

Self-gravitating systems of $N$ particles are chaotic. We wonder how chaotic the Galaxy is, and what the consequences are. We therefore simulate 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 establish 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 the bar forms appears insensitive to infinitesimal perturbations to the initial realisation. In contrast, the bar strength and its further evolution sensitively depend on such perturbations. Interestingly enough, the run-to-run variation in the bar strength has its maximum around the maximum bar strength, and drops to the moment 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 due to infinitesimal perturbations to the initial conditions. Real galaxies, however, should be orders of magnitude more chaotic than our simulations, and we are unable to quantify their consequences. Smooth galactic potentials to study individual stellar orbits should be handled with caution on timescales longer than the Lyapunov time. Extrapolating to the number of stars in the Galaxy, ignoring planets and other minor bodies, we conclude that the Milky Way-size galaxies are chaotic on a timescale $\lesssim 0.1$ Myr.