Chaotic Dynamics of Comet 1P/Halley; Lyapunov Exponent and Survival Time Expectancy

Abstract

The orbital elements of comet Halley are known to a very high precision, suggesting that the calculation of its future dynamical evolution is straightforward. In this paper we seek to characterize the chaotic nature of the present day orbit of comet Halley and to quantify the timescale over which its motion can be predicted confidently. In addition, we attempt to determine the timescale over which its present day orbit will remain stable. Numerical simulations of the dynamics of test particles in orbits similar to that of comet Halley are carried out with the Mercury 6.2 code. On the basis of these we construct survival time maps to assess the absolute stability of Halley's orbit, frequency analysis maps, to study the variability of the orbit and we calculate the Lyapunov exponent for the orbit for variations in initial conditions at the level of the present day uncertainties in our knowledge of its orbital parameters. On the basis of our calculations of the Lyapunov exponent for comet Halley, the chaotic nature of its motion is demonstrated. The e-folding timescale for the divergence of initially very similar orbits is approximately 70 years. The sensitivity of the dynamics on initial conditions is also evident in the self-similarity character of the survival time and frequency analysis maps in the vicinity of Halley's orbit, which indicates that, on average, it is unstable on a timescale of hundreds of thousands of years. The chaotic nature of Halley's present day orbit implies that a precise determination of its motion, at the level of the present day observational uncertainty, is difficult to predict on a timescale of approximately 100 years. Furthermore, we also find that the ejection of Halley from the solar system or its collision with another body could occur on a timescale as short as 10,000 years.