On the Numerical Computability of Asteroidal Lyapunov Times

TL;DR

This paper investigates how computer hardware and integration methods affect the calculation of Lyapunov times for asteroids, introducing a new measure called the computability index to assess result stability and repeatability.

Contribution

It analyzes the impact of computational choices on chaos indicators in celestial mechanics and proposes the computability index as a novel metric for result robustness.

Findings

Differences in hardware and methods cause discrepancies in Lyapunov time calculations.

The computability index kappa quantifies the stability of computed Lyapunov times.

Exploration of phase space explains variations in published asteroid stability results.

Abstract

Chaos indicators, like the Lyapunov exponent lambda, are widely used in celestial mechanics to characterize the dynamical behavior of bodies. The stability of their orbit can be determined by the calculation of the local exponential divergence of arbitrarily close initial conditions in phase space. As the equations to calculate lambda are given, a straight prediction of the orbital stability should be possible. However, one finds in the literature a lot of discrepancies between different studies dedicated to the same object. As a possible explanation for this we investigated in the presented work the effects of the used computer hardware and integration methods on the outcome of such stability computations. Therefore we calculated the Lyapunov time of different asteroids as a measure of chaoticity. Exploring the very fine structure of the nearby phase space of the initial conditions, we are able to explain the reason of the differences in the published Lyapunov times for some objects. Applying methods of robust statistics we introduce the computability index kappa as a measure of repeatability of the results. This kappa gives an estimate, how much the obtained Lyapunov time will change, e.g. when repeating the same calculations with a different integration method.