The solar system shows a multitude of periodic phenomena: from the daily rotations of the Earth, to the monthly phases of the Moon, the seasonal changes during the course of a year, the phases of Venus and other planets, the regular recurrences of comets such as Halley’s (every 76 years), the 25,700 year precession of the Earth’s rotation axis, all the way up to changes in the parameters of the Earth’s orbit, with periods of 40,000 years and more. Closer inspection reveals small modulations on top of these periods, but it is tempting to describe these by additional periods, as in the Ptolemaic theory of cycles, epicycles, and equants. However, the dynamics of the planets is governed by gravitational forces that vary as the inverse square of the distance and, hence, are strongly nonlinear. The question arises whether this nonlinearity results in some chaos.
Johannes Kepler concluded in the early 17th century that an isolated planet would move along a Kepler ellipse (such that a line joining the planet to the Sun sweeps out equal areas in equal times). For several planets their mutual perturbations leads to perihelion precessions and other variations of orbital parameters. Taking such perturbations into account was a major issue in solar mechanics in the following centuries and led to remarkable achievements. For instance, observations of the orbit of Uranus (discovered in 1781 by William Herschel) revealed significant differences to the orbit calculated in the presence of the known planets. When interpreted as due to the influence of another planet, the position of the missing planet could be predicted, and soon thereafter the efforts of Urbain V.LeVerrier, James C.Adams, and Johannes G.Galle were rewarded with the discovery of Neptune in 1846.
These results were obtained using perturbation theory. But does it actually converge? Advances in analytical mechanics in the 19th century suggested that an answer could be found and Gösta Mittag-Leffler included the issue of the stability of the solar system among the list of problems for a prize to be awarded in 1889 by King Oskar II of Sweden and Norway. Henri Poincaré was awarded the prize for his announcement that he could prove convergence, but in the course of revising his paper he noticed that there was a gap in the proof which he could not patch up. Instead, he discovered the intricate motions near a weakly perturbed hyperbolic fixed point, the so-called hyperbolic tangle, which effectively prevents quantitative continuation of trajectories that pass near a hyperbolic fixed point. Barrow-Green (1996) and Diacu & Holmes (1996) give vivid accounts of the events surrounding the prize.
A practical answer to the question of the stability of the solar system emerged in the late 20th century with the advent of powerful computers that allow integration of the equations of motions for several million years. It was then discovered that the inner planets are most susceptible to chaos, and that the Lyapunov time (inverse of the Lyapunov exponent) is about 5 million years. To illustrate the consequences of that, we quote from Laskar(1995):
A 15m uncertainty in the position of the Earth will grow to about 150m in a time of 10 million years. But it will increase to 150 million km or the mean distance of Earth from the Sun within 100 million years.
As a consequence, we have difficulty predicting the Earth’s orbit for times much larger than a few tens of millions of years. The consequences such an uncertainty can have are illustrated by numerical simulations for Mercury: by suitably selecting continuations, it is argued in Laskar (1995) that over a time of about 109 years the orbit of Mercury could change so that the planet collides with Venus and/or escapes from the solar system. It should be noted that this trajectory was especially tailored in order to show that escape is possible in principle; it does not give a clue as to how likely such an event may be.
The issue of the orbital parameters of the Earth is interesting because variations in the major rotation axis and the distance to the Sun influence our climate. In 1920, Milutin Milankovich calculated insolation data (incident solar radiation) for the Earth for long-term variations in the Earth’s orbit and suggested some relationship to the appearance of past ice ages. The uncertainty in orbital parameters over periods of more than 40 million years thus indicate that reconstructing paleoclimates will be problematic (Laskar, 1999).
Besides planets there are many other objects in the solar system. Direct evidence for a chaotic trajectory has been found, for example, for the tumbling motion of Phobos and Deimos (moons of Mars) and for Saturn’s moon Hyperion (Wisdom, 1987). Similarly, efforts to retrace the trajectory of Halley’s comet back to its earliest recorded sighting in 163 BC gave wildly diverging results. As explained by Chirikov and Vecheslavov (1989), the orbit of Halley is chaotic with an inverse Lyapunov time of about 29 returns, so that the earliest observation is just beyond predictability. The distribution of asteroids between Mars and Jupiter shows conspicuous gaps (named after the astronomer Daniel Kirkwood) near orbits with rotation periods rationally related to the 11.9 year period of Jupiter. Such resonant interactions have strong effects on the orbits and can easily lead to collisions and escape from the resonance (Laskar, 1995; Wisdom, 1987).
The relation between the Moon, the Earth, and the Sun also holds surprises. The history of the problem, and the contributions of Babylonian, Greek, and modern astronomers to the observations and mathematical tools, is reviewed in Gutzwiller (1998). Gutzwiller also gives a quantitative example for the significance of small denominators: in order to calculate the distance between the Earth and Moon with the accuracy of 10−10 achievable within the Lunar Laser Ranging project, amplitudes as small as 10−17 have to be kept because of a significant resonance. While this does not result in positive Lyapunov exponents, it is a precursor to it, a mild form of chaos, as Gutzwiller calls it.
More surprisingly, the presence of the Moon is very important for the stability of the rotation axis of Earth: with the Moon the obliquity stays within about ±1.3° of 23.3 °. Without it, the obliquity ends up in a resonance and can become as large as 60° to 90°, with catastrophic consequences for our climate and the evolution of life (Laskar et al., 1993).
Finally, it is worthwhile to point out that the presence of hyperbolic orbits in the solar system was used in connection with the satellite GENESIS (launched in August 2001) to bring it along a stable orbit close to a Lagrange point, where it will remain for a few years to collect particles in the solar wind, before being brought back to Earth along an unstable manifold (Koon et al., 2000). Exploiting trajectories that exist in the dynamical system allows the mission to be completed with minimal requirements on fuel. This mission may thus also be considered an example of chaos control, where similar ideas are being investigated.
Barrow-Green, J. 1996. Poincaré and the Three-body Problem, Providence, RI: American Mathematical Society
Chirikov, B.V. & Vecheslavov, V.V. 1989. Chaotic dynamics of comet Halley. Astronomy and Astrophysics, 221:146–154
Diacu, F. & Holmes, P. 1996. Celestial Encounters, Princeton, NJ: Princeton University Press
Gutzwiller, M.C. 1998. Moon-Earth-Sun: the oldest three-body problem. Reviews of Modern Physics, 70:589–639
Koon, W.S., Lo, M.W., Marsden, J.E. & Ross, S.D. 2000. Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos, 10: 427–469
Laskar, J. 1995. Large scale chaos and marginal stability in the solar system. In Constructive Methods and Results, Proceedings of the XIth ICMP Conference, edited by D.Iagolnitzer, Boston: International Press, pp. 75–120
Laskar, J. 1999. The limits of Earth orbital calculations for geolocical time-scale use. Philosophical Transactions Royal Society of London, Series A, 357:1735–1759
Laskar, J., Joutel, F. & Robutel, P. 1993. Stabilization of the Earth’s obliquity by the moon. Nature, 361:615–617
Wisdom, J. 1987. Chaotic motion in the solar system. Icarus, 72: 241–275
This is the complete article, containing 1,344 words
(approx. 4 pages at 300 words per page).
