Are Planetary Systems Filled to Capacity?

Steven Soter. American Scientist. Volume 95, Issue 5. Sep/Oct 2007.

In 1605, Johannes Kepler discovered that the orbits of the planets are ellipses rather than combinations of circles, as astronomers had assumed since antiquity. Isaac Newton was then able to prove that the same force of gravity that pulls apples to the ground also keeps planets in their elliptical orbits around the Sun. But Newton was worried that the accumulated effects of the weak gravitational tugs between neighboring planets would increase their orbital eccentricities (their deviations from circularity) until their paths eventually crossed, leading to collisions and, ultimately, to the destruction of the solar system. He believed that God must intervene, making planetary course corrections from time to time so as to keep the heavens running smoothly.

By 1800, the mathematician Pierren Simon Laplace had concluded that the solar system requires no such guiding hand but is, in fact, naturally selfcorrecting and stable. He calculated that the gravitational interactions between the planets would forever produce only small oscillations of their orbital eccentricities around their mean values. When asked by his friend Napoleon why he did not mention God in his major work on celestial mechanics, Laplace is said to have replied, “Sir, I had no need for that hypothesis.” Laplace also thought that, given the exact position and momentum of every object in the solar system at any one time, it would be possible to calculate from the laws of motion precisely where everything would be at any future instant, no matter how remote.

Laplace was correct to reject the need for divine intervention to preserve the solar system, but not for the reasons he thought. His calculations of stability were in fact incorrect. In the late 19th century, Henri Poincare showed that Laplace had simplified some of his equations by removing terms he wrongly assumed to be superfluous, leading him to overlook the possibility of chaos in the solar system. Calculations with modern high-speed computers have finally provided evidence that the solar system is only marginally stable and that its detailed behavior is fundamentally unpredictable over long time periods.

Here I will outline some of the discoveries that led to current ideas about instability in the evolution of the solar system. Now is an especially promising time to consider the subject. Theorists are using powerful computer simulations to explore the formation of planetary systems under a wide range of starting conditions, while observers are rapidly discovering planetary systems around many other stars. The evidence suggests that such systems may be filled nearly to capacity. The abundance of observational data from the newly found planetary systems will stimulate and test our ideas about the delicate balance between order and chaos among the worlds.

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 shortperiod 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 coorbital 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.

Cleaning Up the Solar System

Jacques Laskar, of the Bureau des longitudes in Paris, has carried out the most extensive calculations to investigate the long-term stability of the solar system. He simulated the gravitational interactions between all eight planets over a period of 25 billion years (five times the age of the solar system). Laskar found that the eccentricities and other elements of the orbits undergo chaotic excursions, which make it impossible to predict the locations of the planets after a hundred million years. Does Laskar’s result mean that the Earth might eventually find itself in a highly elliptical orbit, taking it much closer to and farther from the Sun, or that the solar system could lose a planet?

No. Even chaos has to operate within physical limits. For example, although meteorologists cannot predict the weather (another chaotic system) as far as a month in advance, they can be quite confident that conditions will fall within a certain range, because external constraints (such as the Sun’s brightness and the length of the day) set limits on the overall system.

Laskar found that, despite the influence of chaos on the exact locations of the planets, their orbits remain relatively stable for billions of years. That is, whereas the long-term configuration is absolutely unpredictable in detail, the orbits remain sufficiently well behaved to prevent collisions between neighboring planets. An external constraint in this case is imposed by the conservation of angular momentum in the system, which limits the excursions of orbital eccentricity for bodies of planetary mass.

The orbits of the giant outer planets are the most stable. The smaller terrestrial planets, particularly Mars and Mercury, are more vigorously tossed about. The simulations show that over millions of years the terrestrial planets undergo substantial excursions in their eccentricities-large enough for those planets to clear out any debris from the intervening orbital space, but not large enough to allow collisions between them. However, Laskar found one possible exception: Mercury, the lightest planet, has a small but finite chance of colliding with Venus on a timescale of billions of years. He concluded that the solar system is “marginally stable.”

Such results suggested to Laskar that the solar system is dynamically “full” or very nearly so. That is, if you tried to squeeze another planet in between the existing ones, the resulting gravitational disturbances would dynamically excite the system, leading to a collision or ejection before the system could settle down again.

Laskar surmised that the solar system, at each stage of its evolution, was always near the edge of instability, as it appears to be today. To maintain its marginal stability, the solar system has been eliminating objects on a timescale comparable with its age at every epoch. It follows that the solar system billions of years ago may have contained more planets that it does now.

According to this view, as the solar system matured, it managed to remain stable against the breakout of largescale chaos by reducing the number of planets and increasing the spacing between them. The present number must be about as large (and their spacing about as small) as allowed by the system’s long-term stability. The solar system has increased its internal order by exporting disorder-entropy-to the rest of the Galaxy, which receives the chaotically ejected objects.

This process, called dynamical relaxation, operates in star clusters and in entire galaxies as well as in evolving planetary systems. As such systems expel their most unstable members, the orbits of the remaining objects become more compact.

Extensive computer simulations show that the eight planets greatly disturb the motions of test particles placed on circular orbits at most locations in the solar system. Such particles are sent into close encounters with the planets, which remove them in only a few million years, a small fraction of the age of the solar system. But these simulations also identify several regions where objects can survive for far longer times. One such region is a broad zone centered about halfway between the orbits of Mars and Jupiter-the asteroid belt. Computer simulations by Jack Lissauer and colleagues at NASA Ames Research Center and at Queen’s University, Ontario, showed that if an Earth-sized planet had formed there, it could remain in a stable orbit for billions of years. This result is not too surprising, because the zone of the asteroid belt is well populated and must therefore be relatively immune to disturbance. The same study found, however, that a giant planet in the asteroid belt would soon become unstable.

The Kuiper belt is another region of stability, as there are no other planets to stir up the neighborhood beyond the orbit of Neptune. The Trojan asteroids of Mars, Jupiter and Neptune occupy other protected interplanetary niches.

Aside from such islands of stability, interplanetary space is remarkably empty. Most of the small objects orbiting between the planets (such as Earth-crossing asteroids and shortperiod comets) are transient interlopers, which recently leaked into the neighborhood from the asteroid and Kuiper belts. The planets will soon eject them or sweep them up in collisions. Indeed, a planet is now defined by the requirement that the object has cleared its orbital neighborhood of other material. Were it not for the leaky reservoirs that supply a steady trickle of debris to their vicinity, the planets would have thoroughly cleaned out most of the orbital space between them.

Making Worlds Is a Messy Business

These ideas fit naturally into the prevailing theory of solar system formation, originally proposed by the philosopher Immanuel Kant in 1755. According to his nebular accretion theory, the solar system and other planetary systems formed by the condensation and accumulation of dust and gas in flattened disks of debris orbiting around young stars. The theory has found strong support in modern observations: Astronomers today routinely detect such debris disks around newborn stars.

The dust-sized particles in such a disk first coagulate to form trillions of rocky asteroids and icy comets a few kilometers in diameter, called planetesimals. These objects in turn gently collide and grow to produce scores to hundreds of Moon- to Mars-sized bodies called planetary embryos, orbiting amid the swarm of remaining planetesimals. Some embryos in the outer parts of the disk grow large enough for their gravity to capture the abundant gas from the nebula, giving rise to giant planets.

As long as the planetesimals retain most of the mass in the disk, their gravity locally exerts a damping effect (called dynamical friction) on the motion of the larger embedded embryos, and the whole system remains dynamically well behaved. The embryos grow by capturing material from so-called feeding zones in the disk, and their orbits become rather evenly spaced. But once the embryos have swept up most of the mass from the disk, the damping effect becomes too feeble to keep the system under control. The gravitational tugs that the embryos exert on one another can then pump up their orbital eccentricities without limit. At that point, to use the vernacular, all hell breaks loose. In this final stage of planet formation, the orbits of the planetary embryos begin to intersect, and the whole system erupts into large-scale anarchy. Entire worlds collide and merge, while others are flung capriciously out into the Galaxy.

The observational evidence makes it clear that the worlds formed in the young solar system were subjected to intense bombardment, their surfaces being saturated with craters. Many of them are still covered with enormous impact scars. Some moons and asteroids look like they were entirely blown apart and reassembled from fragments. A Mars-sized planetary embryo evidently collided with and entirely melted the proto-Earth, explosively throwing off a great splash of debris, some part of which reassembled to form the Moon.

As the growing planets swallowed up planetesimals from the debris disk, they were also ejecting countless others to great distances. Many of those objects had enough energy to escape to interstellar space, where they now drift between the stars. Others, flung without quite enough velocity to escape, reached the outermost fringes of the solar system, where the gravitational influence of nearby stars and the Galaxy itself could circularize their orbits. Hundreds of billions of these icy objects now populate the distant Oort cloud, loosely bound by the Sun’s gravity. Some of them, further nudged by passing stars and galactic tides, reenter the inner solar system as spectacular long-period comets.

Theorists today use computer models to simulate the late stages of planetary formation. They can follow the dynamical evolution of such systems, using a range of starting conditions to represent different debris disks. Some of the simulations generate planets with orbits and masses that resemble those in our solar system. Others produce systems with giant planets in more eccentric orbits. In such simulations, collisions and ejections reduce the number of growing planets and increase the average spacing between them. The planets effectively compete for space, “elbowing” each other apart.

These numerical experiments confirm that the formation of planets is exquisitely sensitive to initial conditions. For example, the displacement of only one in a hundred starting embryos along its orbit by only one meter, keeping everything else the same in a simulation, can make the difference between ending up with three terrestrial planets or five. Such results strongly suggest that a trivial chance encounter determined the very existence of Earth.

Astronomers are now getting the chance to check whether such simulations reflect reality. For more than a decade, observers have been discovering and charting the configuration of other planetary systems, which were long assumed to exist. Planet hunters have already detected more than 240 worlds orbiting around other stars, more than 60 of them in systems having two or more known planets. So far, the observing techniques are limited to detecting giant planets, in most cases at least 10 times more massive than Earth. Smaller terrestrial planets undoubtedly exist around many of those stars, but current measurements cannot yet reveal them.

Astronomers were surprised to learn that most of the known extrasolar planets have orbits much more eccentric than those of the giant planets in our solar system. It was generally assumed that the other systems would resemble our own, with planets in nearly circular orbits. Perhaps, some argued, our solar system is exceptional and most planetary systems were formed in a different way. This now looks unlikely.

Mario Juric and Scott Tremaine at Princeton University recently ran thousands of computer simulations to follow the dynamical evolution of 10 or more giant planets in a disk undergoing collisions, mergers and ejections. For simulations that begin with planets relatively close together, the ones that survive to the end have a distribution of orbital eccentricities that beautifully matches the data for the observed extrasolar planets. For simulations that begin with the planets farther apart, leading to fewer interactions, the surviving giant planets have lower orbital eccentricities, more like our own solar system. Most of the simulations end up with two or three giant planets, after the ejection of at least half of the initial population. This result suggests that free-floating planets, unattached to any star, are very common in the Galaxy.

Other studies confirm that many of the worlds initially populating a planet-forming disk, if not most of them, end up being tossed out into interstellar space. The largest worlds left behind continue to grow by sweeping up smaller objects that remain bound to the central star. Making planets thus seems to be an extremely messy business. A growing planetary system resembles an overly energetic infant learning to eat cereal with a spoon: Some is consumed, but much of it ends up on the floor, walls and ceiling.

Most of the known extrasolar planets are more massive and have shorter periods and more eccentric orbits than the planets of our solar system. However, that does not necessarily mean that our system is anomalous. Current observational techniques strongly favor the discovery of massive planets with orbital periods of only a few years or less, and even the giant planets of our solar system, with their longer orbital periods, would be near the limits of detection if observed from the distance of a nearby star.

Worlds on the Edge

A few years ago, Rory Barnes and Thomas Quinn at the University of Washington used computer simulations to examine the stability of extrasolar systems having two or more planets. They found that almost all systems with planets that are close enough to affect one another gravitationally lie near the edge of instability. The simulations showed that small alterations in the orbits of the planets in those systems would lead to catastrophic disruptions.

This remarkable result might seem surprising. But the prevalence of such marginally stable systems makes sense, Barnes and Quinn concluded, if planets form within unstable systems that become more stable by ejecting massive bodies. The investigators remarked, “As unsettling as it may be, it seems that a large fraction of planetary systems, including our own, lie dangerously close to instability.”

Barnes, now at the University of Arizona, and Sean N. Raymond, at the University of Colorado, went on to hypothesize that all planetary systems are packed as tightly as possible, as Laskar had suggested earlier. In some of the observed extrasolar systems, Barnes and Raymond identified apparently empty regions of stability around the central star. Those regions, they predict, contain planets small enough to have evaded detection.

For example, the star 55 Cancri has four known giant planets, three of them close in with short orbital periods and a more distant planet with a period of nearly 15 years. Between the inner three and the outermost planet lies a large area in which, Barnes and Raymond predict, one or more new planets will eventually be found. This region includes the “habitable zone,” where a planet’s surface temperature would allow liquid water to exist.

What we have here is a fascinating new hypothesis, which posits that our solar system and other mature planetary systems are filled nearly to capacity. The present configurations of such systems contain about as many planets as they can hold, spaced about as closely together as stability allows. Such is the expected outcome of the chaotic process that makes planets. A family of planetary embryos grows by feeding on a vast swarm of smaller objects in a debris disk until the system loses its brakes. Global instability then erupts, and the larger worlds consume or eject the more erratic ones until the system settles down into the mature state of marginal stability. The process is one of self-organization, increasing order within the system by exporting disorder to the external environment, in this case the Galaxy.

Like any good scientific hypothesis, this one makes testable predictions. Astronomers will search for new planets in the stable regions in other systems. This process may take a long time, because the smaller planets are very difficult to detect, but as observational methods continue to improve, we will eventually find out whether the idea of “packed planetary systems” stands up to critical scrutiny.