FEATURE ARTICLE
Are Planetary Systems Filled to Capacity?
Computer simulations suggest that the answer may be yes. But observations of extrasolar systems will provide the ultimate test
Steven Soter
Gaps in Understanding
In 1866, the American astronomer Daniel Kirkwood produced the first real evidence for instability in the solar system in his studies of the asteroid belt, which lies between the orbits of Mars and Jupiter. At the time, only about 90 asteroids were known (the orbits of more than 150,000 have since been charted), but that meager population was sufficient for Kirkwood to notice several "gaps" in the distribution of their orbital periods or, equivalently, in their orbital sizes. (The orbital periods of planets, asteroids and comets increase with orbital size in a well-defined way.) Kirkwood found that no asteroid had a period near 3.9 years, which, he noted, is one-third that of Jupiter.
An asteroid that orbits the Sun exactly three times while Jupiter goes around just once would make its closest approaches to the giant planet at the same point in its own orbit and get a similar gravitational kick from its massive celestial neighbor each time. The repeated tugs Jupiter exerted would tend to add up, or resonate, from one passage to the next. Hence astronomers refer to such an asteroid as being in a 3:1 mean-motion resonance. Other gaps in the asteroid belt correspond, for example, to places where the orbital period of Jupiter would have a ratio of 5:2 or 7:3 to that of an asteroid.
A simple way to understand resonance is to push someone on a swing. If you do so at random moments, not much happens. But if you shove each time the swing returns to you, it will go higher and higher. You could also push at the same point on the arc but less frequently, say only once every two or three cycles. The swing would then take longer to reach a given height, the resonance being weaker.
An asteroid in such a resonant orbit can have its eccentricity increased until the body either collides with the Sun or a planet, or encounters a planet closely enough to be tossed into another part of the solar system. Asteroids that had been orbiting stably in the main belt are sometimes nudged into one of the resonant Kirkwood gaps, from which Jupiter eventually ejects them. These gaps are like holes through which the asteroid population is slowly draining away. Many of the meteorites that strike Earth are fragments that were ejected from the asteroid belt after straying into one of the resonant gaps.
Something similar takes place in the outer solar system. Gravitational tugs from the giant planets gradually remove icy worlds from the Kuiper belt, which lies beyond the orbit of Neptune. This process supplies the short-period comets, which enter the inner solar system for a brief time and return to it at regular intervals. In the early solar system, close encounters of small icy bodies with the growing giant planets populated the distant Oort cloud with hundreds of billions of cometary nuclei.
Such interactions also caused the orbits of the major planets to migrate. Because the growing planets Saturn, Uranus and Neptune tossed more small bodies inward toward the orbit of Jupiter than out of the solar system, those planets migrated outward, to conserve the total angular momentum. But the much more massive planet Jupiter ejected most of the small bodies it encountered into the outer solar system and beyond, and it consequently migrated inward. When the solar system was forming, the Kuiper belt contained hundreds of times more mass than it does today. The objects now in the belt represent only the small fraction that managed to survive. The same is true of the asteroid belt. Gravitational sculpting by the planets has severely depleted both populations, leaving the Kuiper and asteroid belts as remnants of the primordial planetesimal disk.
Whereas some mean-motion resonant orbits in the solar system are highly unstable, others are quite resistant to disruption. (The difference depends on subtle details of the configuration of the interacting bodies.) Many of the objects in the Kuiper belt have their orbits locked in a stable 2:3 mean-motion resonance with Neptune. They orbit the Sun twice for every three orbits of this planet. Such objects are called plutinos, after Pluto, the first one discovered. Some of them, including Pluto, cross inside the orbit of Neptune, but the geometry of their resonant orbits keeps them from making close approaches to the planet and accounts for their survival.
Thousands of small worlds called Trojan asteroids share Jupiter's orbit around the Sun, leading or following the planet by about 60 degrees. These bodies are trapped in a so-called 1:1 mean-motion resonance, the planet and asteroid having the same orbital period. This configuration inhibits close approaches to Jupiter and is relatively stable. Similar families of co-orbital asteroids accompany both Neptune and Mars around the Sun.
Gravitational tugs of the planets on one another produce cyclical motions in the spatial orientation of their orbits, causing another kind of resonance. The rotation of the orientation of an elliptical orbit takes many times longer than the orbital period of the planet itself. These slow gyrations of an entire orbit produce so-called secular resonances, which can strongly distort the orbits of smaller bodies—and not just those in the asteroid belt. The solar system is crowded with potential orbits on which objects would be subjected to secular or mean-motion resonances. Many resonant orbits overlap, and wherever that happens, small orbiting bodies are especially prone to disturbance.
Despite its orderly appearance, the solar system actually includes many elements of what mathematicians call chaos. A defining feature of chaos is the extreme sensitivity of a system to its initial conditions. The most trivial disturbance in such a system can profoundly change its large-scale configuration at a later time. A pool table provides a familiar example: Microscopic variations in the trajectory of a billiard ball, especially one involved in multiple collisions, can completely alter the outcome of the game. Chaotic systems are deterministic, in that they follow precisely the laws of classical physics, but they are fundamentally unpredictable.
The nature of chaos was not well understood until recently, when increasing computer power allowed mathematicians to explore it in sufficient detail. No one in Laplace's day imagined that the solar system, then taken as the paradigm of clockwork stability, is actually vulnerable to chaos.
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