Is the outer Solar System chaotic?

Abstract

One-sentence summary: Current observational uncertainty in the positions of the Jovian planets precludes deciding whether or not the outer Solar System is chaotic. 100 word technical summary: The existence of chaos in the system of Jovian planets has been in question for the past 15 years. Various investigators have found Lyapunov times ranging from about 5 millions years upwards to infinity, with no clear reason for the discrepancy. In this paper, we resolve the issue. The position of the outer planets is known to only a few parts in 10 million. We show that, within that observational uncertainty, there exist Lyapunov timescales in the full range listed above. Thus, the "true" Lyapunov timescale of the outer Solar System cannot be resolved using current observations. 100 word summary for general public: The orbits of the inner planets (Mercury, Venus, Earth, and Mars) are practically stable in the sense that none of them will collide or be ejected from the Solar System for the next few billion years. However, their orbits are chaotic in the sense that we cannot predict their angular positions within those stable orbits for more than about 20 million years. The picture is less clear for the outer planets (Jupiter, Saturn, Uranus and Neptune). Again their orbits are practically stable, but it is not known for how long we can accurately predict their positions within those orbits.The existence of chaos among the Jovian planets is a contested issue. There exists apparently unassailable evidence both that the outer Solar System is chaotic 1, 2 , and that it is not 3, 4, 5 . The discrepancy is particularly disturbing given that computed chaos is sometimes due to numerical artifacts 6,7 . In this paper we discount the possibility of numerical artifacts, and demonstrate that the discrepancy seen between various investigators is real. It is caused by observational uncertainty in the orbital positions of the Jovian planets, which is currently a few parts in 10 million. Within that observational uncertainty, there exist clearly chaotic 1

Surfing on the edge: chaos versus near-integrability in the system of Jovian planets

“…The author has reported related results for integrations lasting 10 9 yr (Hayes 2007). The essential conclusion is the same, in that even after 10 9 yr, there remain some ICs (about 10 per cent) that show no evidence of chaos, although some of the ICs appearing as regular over 200 Myr develop exponential divergence later.…”

The interplay of chaos between the terrestrial and giant planets

“…The effects of the inner planets are crudely taken into account by adding their masses and momenta to those of the Sun. It has been shown, albeit with a small number of samples, that this system contains some initial conditions within observational uncertainty that show chaos on a Lyapunov time-scale of a few Myr, while other initial conditions also within observational uncertainty show no observable chaos on time-scales ranging from 200 Myr to 5 Gyr (Hayes 2007(Hayes , 2008. The goal of our first experiment is simply to explore more fully the structure of Lyapunov times that exist in this system, with initial conditions that are (mostly) within observational error of their true positions.…”

“…The chaos in the inner SS was soon verified by full numerical integration of all the planets, but here chaos was also observed in the outer planets (Sussman & Wisdom 1992). This led to uncertainty about the existence of chaos in the outer SS that was resolved by demonstrating that both chaotic and apparently regular orbits exist within observational uncertainty (Hayes 2007(Hayes , 2008.…”

“…During the past few decades there have been many studies of chaos in the SS, a survey of which is beyond the scope of this paper. The work in this paper stems from the recent observation that within the observational error of the positions of the Jovian planets, there exist some initial conditions that result in chaos in this system, and other initial conditions that lead to no observable chaos over durations ranging from 200 Myr to 5 Gyr (Hayes 2007(Hayes , 2008. We study two questions arising from this observation.…”

High-precision computation: Mathematical physics and dynamics

“…Researchers have found that double-double or quad-double arithmetic is required to avoid severe numerical inaccuracies, even if other techniques are employed to reduce numerical error [45]. We also mention the recent studies of W. Hayes [43], where some comparisons of the stability (see Figure 1) of the solar system are performed using various numerical ordinary differential equation (ODE) integrators and checked to higher precision using a Taylor series integrator, performed using 19-digit Intel Extended precision.…”