The Shaw Prize in Astronomy for 2007 will be awarded to Professor Peter
Goldreich1 in recognition of his achievements in theoretical astrophysics and planetary sciences. His most important contributions include the studies on the effects of orbital resonances in the Solar System and planetary
rings.2 Resonance in the Solar System tends to have a close connection with chaos. Perhaps it is the phenomenon of chaos which dominates the formation of planets in the Solar System, causes the ¡§Mass Extinction¡¨ of creatures on Earth in the past; shapes the Kirkwood Gaps and Cassini division to what they are like today; and may even catapult the present clockwork planetary orbits round the Sun to ultimately chaotic ones!
In 1889, the French mathematician J. Henri Poincarˆm (1854 - 1912) won an international essay competition on mathematics which was held to celebrate the birthday of Oscar II, the King of Sweden. Poincarˆm¡¦s work began the study on chaotic phenomena, which challenged some of the firmly held beliefs since the advent of Newtonian
mechanics.3 Such beliefs included, for example, the notion of deterministic cause and effect and the absolute predictability of science. Poincarˆm came up with the idea that minute variations in the initial condition of a system might produce huge variations in the long run. This characteristic of chaos is often illustrated by the ¡§Butterfly effect¡¨ - the flapping wings of a foraging butterfly in South America may eventually cause a raging storm to appear in North America in a few days.
The asteroid belt lies between the orbits of Mars and Jupiter. As early as 1857, Daniel Kirkwood (1814 - 1895) noted that the seemingly empty gaps within the asteroid belt are caused by resonance between the minor planets and Jupiter. In other words, there is a simple integral ratio (e.g. 1:3) between the orbital periods of an individual asteroid and that of the giant planet. (Diagram 1) A numerical model put forward in the late 1970s by Jack Wisdom, a student of Peter Goldreich, largely explained the formation of Kirkwood¡¦s Gaps owing to the 1:3
resonance.4 Within the regions influenced by the resonance with Jupiter, the orbit of an asteroid can remain nearly circular for a million years, but then undergo drastic changes within a relatively short period of time. The orbit can become so elliptical that the asteroid may cross the orbits of Mars and Earth, and may even crash into other planets. Empty reaches in the asteroid belt appear in this way. Other cases of resonance in the Solar System can be exemplified by the Cassini division of Saturn, which is of 1:2 and 1:4 resonance with its satellites Mimas and Tethys respectively. Nevertheless, while resonance in the Solar System is apparently a complicated and volatile phenomenon, some asteroids in these regions can ¡§remain stable¡¨ all
along.5 Besides being caused by different orbital periods of two celestial bodies, resonance may also appear between the periods of rotation and revolution of a planet. For example, the ratio of the rotation period to the revolution period of Mercury happens to be 2:3.
Diagram 1: Distribution of minor planets in the asteroid belt. Kirkwood Gaps are obviously located at regions of 3:1, 5:2, 7:3 and 2:1 resonance.
A collision between the Earth and a straying asteroid far-flung from its resonance orbit will spell true calamity to all creatures on Earth! About 65 million years ago, an asteroid estimated to be around 10 km in diameter struck the Earth, resulting in an explosion equivalent to 100,000 gigatons of TNT. The planet was engulfed in a raging and sweeping conflagration, scorching many large animals to death. Most of those left over were brought down by starvation. Blanketed by thick smoke, the Earth experienced frigidity for the next few years, followed by torridity lasting several centuries. Such conditions precipitated the demise of dinosaurs, and 50 - 80% of the living organisms on Earth became extinct forever. Nevertheless, this cataclysmic impact paved the way for the subsequent rise of mammals.
Chaos might be critical to the formation of planets in the Solar System, which was originated from a nebula a few light-years across. Around
4.6 billion years ago, the Solar System was formed from the gravitational collapse of the nebula which eventually became disc-like in appearance. The tiny materials within the disc gradually coalesced to form different planets like the Earth, Jupiter and other members of the Solar System. As revealed by numerical simulation, the orbits of some tiny materials were altered by chaos during the early stages of planet formation. Such orbital deviation increased their chance of mutual collision to form planets.6 There are also other studies that employ the theory of chaos and numerical models to simulate the number and distribution of planets in the Solar
Will planetary orbits become chaotic ultimately? The French astronomer Jacques Laskar published in 1989 his result of a numerical computation on the evolution of planetary orbits over the next 200 million
years.8 Though his model was incomplete, it did involve mathematical expression containing about 150,000 algebraic terms. He found that orbits of inner planets (including the Earth) would eventually become chaotic, but this did not imply a catastrophic collision between planets such as Venus and Earth. Other studies also indicate that the Earth will continue to have a near circular orbit with low obliquity for the coming few hundred millions of
years.9 However, to confirm these findings, we have to wait for the next generation of supercomputers to carry out a more complete simulation of our Solar System.
More than 2,300 years ago, Aristotle hypothesized that celestial bodies moved in orbits. About three centuries ago, Newton published his monumental work Philosophiae Naturalis Principia Mathematica. Today, scientists have yet to fully grasp the intricate effects of Newton¡¦s gravitation among planets, let alone the evolution of planetary orbits in the Solar System. The theory of chaos has enabled us to appreciate the elegance and complexity of Mother Nature, yet also cast a shadow of doubt on our once ingrained faith in the deterministic nature of the world and the absolute predictability of science. We humans, while manifesting our ability to comprehend the Universe, would also at the same time ponder the limitation of science.
1 Peter Goldreich (1939 - ) is currently a Professor of the School of Natural
Sciences at the Institute for Advanced Study, Princeton and the Lee A. DuBridge Professor of Astrophysics and Planetary Physics at the California
Institute of Technology. He became Professor Emeritus of the Institute in 2003.
2 The relevant press release is available at http://www.shawprize.org/b5/laureates/2007/astronomy/Goldreich/release.ht
3 Newton¡¦s classical mechanics gave a solution to the ¡§two-body problem¡¨. Besides, Joseph-Louis, Comte de
Lagrange (1736 ¡V 1813) and Pierre-Simon, Marquis de Laplace (1749 ¡V 1827) had remarkable achievements in the area of the ¡§n-body problem¡¨ in astronomy by using of asymptotic methods such as perturbation in calculation.
4 Ivars Peterson (µÛ)¡A¶À±Ò©ú.¶À»ÊÃò (Ä¶)¡G¡m¤û¹y®ÉÄÁ¡G´ý¨P¤Ó¶§¨t¡n(»O¥_¡G¤û¹y¥Xª©ªÑ¥÷¦³¤½¥q¡A¶212¦Ü213¡C
5 Examples are Hildas of 3:2 resonance and Trojan asteroids of 1:1 resonance.
6 Jack J. Lissauer, Chaotic motion in the Solar System, Reviews of Modern Physics, Vol.71, No.3, 1999, pp.835-845.
7 J. Laskar, On the Spacing of Planetary Systems, Physical Review Letters, Vol.84, pp.3240-3243 (2000).
8 Ivars Peterson (µÛ)¡A¶À±Ò©ú.¶À»ÊÃò (Ä¶)¡G¡m¤û¹y®ÉÄÁ¡G´ý¨P¤Ó¶§¨t¡n(»O¥_¡G¤û¹y¥Xª©ªÑ¥÷¦³¤½¥q¡A¶212¦Ü213¡FJ. Laskar, A numerical experiment on the chaotic behaviour of the Solar System, Nature, Vol.338, pp.237-238 (1989).
9 M J Duncan, and T Quinn, The Long-Term Dynamical Evolution of the Solar System, Annual Review of Astronomy and Astrophysics, Vol. 31:265-293 (1993); J. Laskar, Large scale chaos and the spacing of the inner planets, Astronomy and Astrophysics, 317, P L75-L78 (1997).