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An astronomer once told me that Nature really hates to solve the Three-Body Problem, as evidenced by the fact that She avoids it whenever She can.  He was referring to the large number of binary star systems in the universe (two stars in relatively close orbits), and the relative rarity of triple-star systems (three stars with relatively small mutual separations).

The Three-Body Problem to which he was referring is a mathematical conundrum that has been around since the time of Newton .  While one can easily solve for the orbits of two mutually gravitating bodies like stars, there is no known way of exactly solving the same problem when three bodies are involved.  There are good approximations when one of the bodies is much smaller than the other two, or when one is very far away from the others, but the general problem has no analytic solution.

One of the approximate solutions, the so-called “Trojan solution”, is of some interest in science fiction.  It turns out that when two of the bodies are much more massive than the third, the smaller object can be locked into an orbit such that the three objects always form an equilateral triangle.  Nature has used this solution in our own Solar System.  At the L₄ and L₅ Lagrange points 60 degrees ahead of and behind the position of Jupiter in its orbit, there are a collection of “Trojan” asteroids that lead and trail in the orbit of the giant planet.  Astronomers decided to name those asteroids ahead of Jupiter with an index number and the name of a Greek hero of the Trojan War (e.g., 588 Achilles, 659 Nestor, 911 Agamemnon, 1143 Odysseus, 1404 Ajax, 1437 Diomedes, etc.) while the asteroids trailing behind Jupiter were named for the combatants from Troy (884 Priamus, 1172 Aneas, 1173 Anchises, etc.)  However, because they were named before this convention was established, the Greek-named 617 Patroclus was put in with the Trojans and 624 Hektor was put with the Greeks.  These Trojan asteroids are “herded” around the solar system by Jupiter.  Perhaps some future planet-faring civilization may find these solutions of the three-body problem to be a useful source of raw materials or a good stable location for man-made space environments.

The algebraic difficulties of the three-body problem are not a major impediment to the study of planetary orbits.  There are good numerical methods for solving the three-body problem to good accuracy, so with modern computers we can calculate orbits of multi-body systems to whatever precision we are willing to expend the resources to obtain.  However, when we do such calculations we find that most of the orbits for close three-body systems are unstable.  After a few orbits, one of the bodies is often ejected from the system, leaving behind a simpler two-body system.  Also, such solutions are usually “chaotic”, so that minute differences in the initial conditions of the system can produce dramatically different final orbital results.

The intrinsic instability and chaos of most close three-body orbits raises the question of whether binary star systems can be expected to have planets at all, and in particular, to have Earth-like planets in stable orbits around them.  This question is of particular interest because more than half of the stars in our galactic neighborhood are binary or multiple-star systems.

One leading example is our nearest stellar neighbor, Alpha Centauri, which consists of a close binary of Sol-like stars, with a third stellar companion orbiting much further out.  The two primary stars are Alpha Centauri A, a spectral type G2 star (like our Sun) with a mass of 1.09 solar masses, and Alpha Centauri B, a smaller and dimmer type K1 star with a mass of 0.90 solar masses.  Proxima, the third star of the group, is type M5 star with a mass of about 0.1 solar masses.  Alpha Centauri A and B are in an elliptical orbit with a period of 80 years, approaching each other to as close as 11 AU and receding to as far as 35 AU as they orbit.  Here, 1 AU (astronomical unit) is defined as the distance from the Sun to the Earth, 11 AU is roughly the distance from our Sun to the orbit of Saturn, and 35 AU is the distance from our Sun to somewhere between the orbits of Neptune and Pluto.  Proxima is a light-weight and somewhat unstable “flare star”.  It orbits about 13,000 AU (about 1/5 of a light year) from A and B., a distance so large that it is uncertain whether Proxima is even gravitationally bound to its larger companions or whether it will eventually wander away.

Probably the leading question concerning the Alpha Centauri system is whether either Alpha Centauri A or Alpha Centauri B (or both) could have habitable planets in orbit around them.  Up to now, conventional wisdom would have answered that question “probably not”.  The reason is that, while either major member of the Alpha Centauri system could probably have planets in stable orbits out to about 2 AU before the perturbations of the other star produced chaotic orbits, it was thought that the process of planet formation itself would be greatly impeded in a binary system.  The view was that the proto-planetary dust cloud from which planets were formed should collapse inward from distances on the order of 100 AU under the friction of collisions, and the sweeping action of the binary system members would eject material, frustrate this process, and suppress the formation of planets.  Moreover, it was expected that shock waves produced in the gas cloud around one star from the passage of the other would heat and vaporize ice crystals, dispersing the cloud, and preventing accretion.  Now, however, there are reasons to modify these views.

On the observational front, recent successes in astronomical searches for planets orbiting stars outside our solar system have found a number of examples of Jupiter-like gas giant planets orbiting in binary star systems with separation distances ranging from about 12 to 1000 AU.  On the theoretical front, Dr. Alan G. Boss of the Carnegie Institution in Washington , DC , has developed a numerical model of the proto-planetary gas cloud in a binary star system, which removes the artificial viscosity effects of previous models, but includes vertical motion in the disk and convective cooling.  His calculations indicate that planet formation may actually be enhanced in a binary star system.

Boss found that the shock wave heating I binary star systems can be rather weak, and in these cases gas-giant planets can emerge in the planet-forming disk of gas and dust in the same way they do around single stars.  Ice grains can combine through the process of core accretion and grow into solid cores of several Earth-mass sizes.

But in addition to core accretion, there is another planet forming mechanism that may be even more important in binary systems.  The disc of gas and dust orbiting a new star, if it is massive enough, is intrinsically unstable to gravitational attraction because once a region of higher-than-average density appears, it tends to grow progressively larger.  Boss has shown that in cases where a proto-planetary disc around one of the stars is just massive enough to be on the edge of such an instability, the passage of the binary companion, with a time scale of around 1000 years, can act as a trigger to precipitate planet formation.  When the binary minimum separation distance is more than 50 AU, Boss found that the companions each formed proto-planetary disks of around 20 AU which were relatively unaffected by the perturbations of the companion.  However, when the binary minimum separation distance is less than 50 AU, the proto-planetary disks of each star formed spiral arms that typically evolved into dense self-gravitating clumps, a major step to planet formation.

On the basis of Boss’ calculations, there seems to be a distinct difference between the processes of planet formation in single star systems and binary systems, with the latter actually “pumped” toward planet formation earlier and perhaps with different location probabilities.  Thus, when improved resolution with interferometric techniques, etc. permits us to determine in detail the planetary structure of the star systems in our neighborhood, we may be in for some surprised when we look at binary systems.

The implications of this work for SF are fairly clear, and it vindicates the work of many SF authors.  Writers should have no reservations about placing Earth-like planets around binary star systems, or having scenes with two “suns” in the sky, etc.  Moreover, consider the scenario of an elliptical binary system having a long period between close passes (~ 10 AU), and suppose both stars had Earth-like planets with planet-faring civilizations.  This setup has interesting socio-political implications, with trade and contact between two planetary civilizations punctuated by the periodic close passages, etc.  It makes one feel rather sad and lonely to be isolated in a star system with only one Sun and only one Earth-like planet.

AV Columns On-line: Electronic reprints of over 120 "The Alternate View" columns by John G. Cramer, previously published in Analog , are available on-line at: http://www.npl.washington.edu/av.  Preprints referenced below can be obtained at: http://www.arxiv.org .

Reference:

Planet Formation in Binary Systems:

“Gas Giant Protoplanets Formed by Disk Instability in Binary Star Systems”, Alan G. Boss, Astrophysics Journal 641 20 (2006); ArXiv preprint astro-ph/0512477.

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