Are Planetary Systems Filled to Capacity? (part 2)
Summary (Aug 20, 2007): In this essay, Steven Soter examines computer experiments that simulate the gravitational interactions among planets over billions of years. These models suggest that the solar system is only marginally stable and is dynamically full, or nearly so. Adding another planet between the existing ones would make the system unstable, resulting in a collision or ejection of a planet.

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Steven Soter is a research associate in the Department of Astrophysics at the American Museum of Natural History in New York City, and scientist-in-residence at New York University, where he teaches on subjects ranging from life in the universe to geology and antiquity in the Mediterranean region. His research interests include planetary astronomy and geoarchaeology. He collaborated with Carl Sagan and Ann Druyan to create the acclaimed Cosmos television series in 1980.

In part one of this essay, Soter described how the traditional view of a stable clockwork solar system broke down after the discovery of chaotic behavior and instabilities in the long-term orbital motions. Here he describes computer experiments simulating the gravitational interactions among the planets over billions of years. These models suggest that the solar system is only marginally stable and is dynamically full, or nearly so. That is, if one tried to insert another planet between the existing ones, the system would become unstable, resulting in a collision or ejection of a planet.

This essay also appears in the September/October edition of American Scientist magazine.

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Are Planetary Systems Filled to Capacity?
Conjectures on Order and Chaos among the Worlds

By Steven Soter


Making Worlds Is a Messy Business

The prevailing theory of solar system formation was 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.

Numerical simulation reveals how the inner part of a planet-forming disk evolves. Initially, such a disk is composed of numerous planetesimals in near-circular (low-eccentricity) orbits (top). Within a few million years, orbital eccentricities grow to appreciable size for most of the smaller bodies, and planetary embryos form as smaller objects coalesce. As time goes on, the smaller bodies are swept up or scattered away, leaving a few planets in low-eccentricity orbits (bottom). (Adapted from Chambers 2001.) Click image to enlarge.

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, re-enter 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 the Earth.

Astronomers are now getting the chance to check whether such simulations reflect reality. For more than a decade, they 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.

Numerical simulations suggest the wide variety of outer planetary systems produced by planetesimal disks. The outer solar system is shown for comparison (a). The simulated planetary systems that resulted from the 11 experimental runs range from having just one (b) to as many as seven (f) outer planets of varying mass (indicated above each planet in Earth masses). The different outcomes depend on the initial number and distribution of planetesimals and the chaotic interactions between them. (Adapted from Levison et al. 1998.)

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.

Studies of extrasolar systems offer astronomers increasing opportunity to test their ideas about planet formation. The three innermost planets of star 55 Cancri, for example, have orbits smaller than that of Mercury. These three planets are separated from a much more massive world by a region of apparent stability, which is predicted to harbor another planet. This region encompasses the star’s habitable zone, where surface temperatures would allow a suitable planet to support liquid water. The numbers above each planet in the bottom panel show its minimum mass, expressed in Earth masses.

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.

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SELECTED BIBLIOGRAPHY

F.C. Adams & G. Laughlin. 2003. Migration and dynamical relaxation in crowded systems of giant planets. Icarus 163, 290-306.

R. Barnes & T. Quinn. 2004. The (in)stability of planetary systems. Astrophysical J. 611, 494-516.

R. Barnes & S.N. Raymond. 2004. Predicting planets in known extrasolar planetary systems. I. Test particle simulations. Astrophysical J. 617, 569-574.

J.E. Chambers. 2001. Making more terrestrial planets. Icarus 152, 205-224.

J.E. Chambers. 2004. Planetary accretion in the inner solar system. Earth & Planetary Science Letters 223, 241-252.

Juric´, M. & S. Tremaine. 2007. Dynamical origin of extrasolar planet eccentricity distribution. Submitted to Astrophysical J.

J. Laskar. 1996. Large scale chaos and marginal stability in the solar system. Celestial Mechanics & Dynamical Astronomy 64, 115-162.

M. Lecar, F.A. Franklin, M. Holman & N.W. Murray. 2001. Chaos in the solar system. Annual Review Astronomy & Astrophysics 39, 581-631.

H.F. Levison, J.J. Lissauer & M.J. Duncan. 1998. Modeling the diversity of outer planetary systems. Astronomical J. 116, 1998-2014.

J.J. Lissauer. 1999. Chaotic motion in the solar system. Reviews Modern Physics 71, 835-845.

J.J. Lissauer et al. 2001. The effect of a planet in the asteroid belt on the orbital stability of the terrestrial planets. Icarus 154, 449-458.

A. Morbidelli. 2002. Modern integrations of solar system dynamics. Annual Review Earth & Planetary Sciences 30, 89-112.

D.P. O’Brien, A. Morbidelli & H.F. Levison. 2006. Terrestrial planet formation with strong dynamical friction. Icarus 184, 39-58.

S.N. Raymond & R. Barnes. 2005. Predicting planets in known extrasolar planetary systems. II. Testing for Saturn mass planets. Astrophysical J. 619, 549-557.

S. Soter. 2006. What is a planet? Astronomical J. 132, 2513-1519.

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Read Steven Soter's other Astrobiology Magazine essay:

SETI and the Cosmic Quarantine Hypothesis

Note: Stellar Evolution

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Monday, August 20, 2007