The orbital dynamics of Oterma, a comet that is locked in a long, complex dance with the giant planet Jupiter.

In frame (a) we see a portion of Oterma’s orbit from the perspective of the solar system as a whole, using a coordinate system that has the Sun at rest in the center and Jupiter moving in a near-perfect circle once every 12 years. Oterma’s intricate path takes it both inside and outside Jupiter’s orbit. While it is on the inside it generally stays near a “3:2 resonance,” in which Oterma goes three times around the Sun for every two revolutions made by Jupiter. When it is on the outside, likewise, Oterma stays near a near a 2:3 resonance.

In frame (b) we see this same segment of Oterma’s orbit from Jupiter’s perspective, using a rotating coordinate system that keeps the giant planet stationary. This view makes it clear that Oterma’s transition between outside and inside can occur only through a narrow bottleneck around Jupiter. Frame (c) shows a closeup of the bottleneck.

This view also highlights the parallel with atomic physics: an atom moving from one molecule to another often goes through a similar transition.

Credit: Shane D. Ross, University of Southern California

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In recent years, mathematicians have discovered an almost perfect parallel between the motion of spacecraft through the solar system and the motion of atoms in a chemical reaction–a hidden unity that has led to innovative new ways to design space missions.

The celestial half of this unity arises from the theory of “dynamical systems,” which describes how a group of celestial bodies such as the Sun, the Earth and a spacecraft will move under the influence of their mutual gravity. It turns out that the tangle of gravitational forces creates tubular “highways” in the space between the bodies; if the spacecraft enters one of the highways, it will be whisked along without the need to use much propellant of its own.

The atomic half, meanwhile, arises from the theory of “transition states,” which describes how atoms are transferred from one molecule to another during the course of a chemical reaction.

The unity exists because the same mathematical equations apply in both cases–which means that insights gained from analyzing one class of problems can help analyze the other. Indeed, when NASA was preparing for its recent Genesis mission to sample the solar wind, the agency planned the spacecraft’s complex orbit with the help of specialists in atomic physics, such as mathematician Jerrold Marsden of the California Institute of Technology, engineer Shane Ross of the University of Southern California, and physicist Turgay Uzer of the Georgia Institute of Technology.

The work of all three scientists, who have received funding to study the field from the National Science Foundation (NSF), is featured in an article that appears in the Oct. 2005 issue of the Notices of the American Mathematical Society.

[Originally posted on The Love Blog]