An example of stable chaos in the Solar System

Abstract

MANY planets have been shown to have chaotic instabilities in their orbital motions, but the long-term significance of this is not fully understood¹. The eccentricity of Mercury, for example, changes by about 25% of its value over 40 times the Lyapunov time^2,3(the e-folding time for divergence of nearby orbits), but the orbit of Pluto, in an integration lasting 50 Lyapunov times⁴, shows no significant change. Here we show that the orbit of the near-Jupiter asteroid 522 Helga is chaotic, with an unusually short Lyapunov time of 6,900 yr. We integrate its motion, including perturbations from the outer giant planets, over a period 1,000 times longer than this, and find no significant instability. Chaos in the orbit of 522 Helga is caused by a 7:12 resonance with the orbit of Jupiter, but the size of the chaotic region in phase space is small; stability is ensured because the eccentricity and precession of the orbit are such that it avoids close encounters with Jupiter. Asteroid orbits with larger proper eccentricity would, we suggest, be genuinely unstable, consistent with the sparse asteroid population near Helga. Although Helga is the first clear-cut example of a stable chaotic orbit, we argue that 'stable chaos' may be a rather common feature of Solar System dynamics.