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Where Is There Chaos in the Solar System?

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.

Hyperion

One of the first pieces of chaos seen in the solar system was Hyperion, a moon of Saturn. When Voyager 1 passed by the moon in August 1981, scientists saw some weird stuff in the shape of it. But it was already a weird object. According to analysis by Jack Wisdom (University of California at Santa Barbara), the moon was not tidally locked with the planet, which it should be because of its size and proximity to Saturn. Gravity should have robbed enough angular momentum by this point and create a severe tidal bulge and frictional forces inside the moon should further slow it down, but no dice. What people learned from Voyager 1 was that Hyperion is an oblong object with dimensions of 240 miles by 140 miles, meaning its density can be different and not spherically distributed, so gravity pulls are not consistent. Using chaos theory, Wisdom along with Stanton Peale and Francois Midnard in 1988 were able to model the motion of the moon, which doesn’t spin on any conventional axis but instead tumbles around once every 13 days and completes an orbit every 21 days. Saturn was tugging on the moon, but as it turns out another moon was also: Titan. Hyperion and Titan are in a 4:3 resonance and so lining up for a nice severe pull can be tricky and cause the chaotic motion seen. For Hyperion to be stable, simulations and Poincare sections showed that 1:2 or 2:1 resonances would be needed (Parker 161, 181-6; Stewart 120).

Triton.

Triton.

Triton

This work from Hyperion inspired scientists to look at Triton, a moon of Neptune. Peter Goldreich (California Institute of Technology modelled Triton’s history in an attempt to find out. Triton did orbit the Sun but was captured by Neptune based off its retrograde motion. In the process of capturing the moon, chaotic perturbations were present that impacted the current moon’s orbits, causing several to move to be between Triton and Neptune. Voyager 2 data did support this, with 6 moons stuck inside that orbital range (Parker 162).

Asteroid Belt

In 1866, after plotting the orbits of the then known 87 asteroids, Daniel Kirkwood (Indiana University) found gaps in the Asteroid Belt that would have 3:1 resonances with Jupiter. The gap he spotted was not random, and he further uncovered a 2:1 and a 5:2 class as well. He also uncovered a class of meteorites that would have come from such a zone, and began to wonder if chaotic perturbations from Jupiter’s orbit would cause any asteroids on the outer regions of the resonance to be kicked out upon a close encounter with Jupiter. Poincare did an averaging method to try and find a solution but to no avail. Then in 1973 R. Griffen used a computer to look at the 2:1 resonance and did see mathematical evidence for chaos, but what was causing it? Jupiter’s movement wasn’t as directly the cause as scientists had hoped. Simulations in 1976 by C. Froescke and in 1981 by H. School into 20,000 years from now yielded no insights either. Something was missing (162, 168-172).

Jack Wisdom took a look at the 3:1 group, which was different from the 2:1 group in that perihelion and aphelion didn’t line up nice. But when you stack both groups and look at the Poincare sections together, the differential equations do show that something does happen – after a few million years. The eccentricity of the 3:1 group grows but then returns to a circular motion but not until after everything in the system has moved around and is now differentiated from where it started. When the eccentricity changes again, it pushes some of the asteroids to Mars orbit and beyond, where gravity interactions stack up and out goes the asteroids. Jupiter was not the direct cause but did play an indirect role in this strange grouping (173-6).

The early solar system.

The early solar system.

Proto-disc Formation

Scientists used to think that the solar system formed according to a model developed by Laplace, where a disc of material spun around and slowly formed rings which condensed into planets around the Sun. But upon closer examination, the math didn’t check out. James Clark Maxwell showed if that the Laplace model was used, the biggest objects possible would be an asteroid. Progress was made on this issue in the 1940s when C.F. on Weizacher added turbulence to the gas in the Laplace model, wondering if the vortices arising from chaos would help. They sure did, and further refinements by Kuiper added randomness and the accretion of matter led to better results still (163).

Solar System Stability

The planets and moons orbiting each other can make the question of long-term predictions tough, and a key piece to that kind of data is the stability of the solar system. Laplace in his Treatise on Celestial Mechanics gathered a planetary dynamics compendium, which was built off of perturbation theory. Poincare was able to take this work and make graphs of the behavior in phase space, finding that quasiperiodic and double frequency behavior was spotted. He found this led to a series solution but was unable to find the convergence or divergence of it, which would then reveal how stable this all is. Birkoff followed up by looking at the cross sections of the phase space diagrams and found evidence that the desired state of the solar system for stability involves lots of small planets. So the inner solar system should be okay, but how about the outer? Simulations of up to 100 million years of the past and the future done by Gerald Sussman (Caltech/MIT) using Digital Orrery, a supercomputer, found … nothing … sort of (Parker 201-4, Stewart 119).

Pluto, then a planet, was known for being an oddball, but the simulation showed that the 3:2 resonance with Neptune, the angle Pluto makes with the ecliptic will vary from 14.6 to 16.9 degrees over a 34-million-year period. It should be noted however that the simulation had round off stack errors and the size between each calculation was over a month each time. When a new run of the simulation was done, a 845-million-year range with step of 5 months each time still found no changes for Jupiter through Neptune but Pluto showed that accurately placing its orbit after 100 million years is impossible (Parker 205-8).

Works Cited

Parker, Barry. Chaos in the Cosmos. Plenum Press, New York. 1996. Print. 161-3, 168-176, 181-6, 201-8.

Stewart, Ian. Calculating the Cosmos. Basic Books, New York 2016. Print. 119-120.

© 2019 Leonard Kelley

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