Celestial Atomic Physics
. . it may happen that small differences in the initial conditions
produce very great ones in the final phenomena. A small error in the
former will produce an enormous error in the latter. Prediction becomes
impossible, and we have the fortuitous phenomenon."
French mathematician Jules Henri Poincaré (1854–1912) made this
observation in his 1908 essay "Science and Method." This remarkable
insight aptly encapsulates a key feature of nonlinear dynamics (or what
many people call, more loosely, "chaos" theory).
Sensitive dependence on initial conditions is a feature of the
motion of three or more bodies that interact gravitationally. In the
solar system, objects such as asteroids and comets, can follow chaotic
trajectories, abruptly and unpredictably shifting their orbits from
time to time. Numerical computations can pin down their paths only up
to some finite time. The distant future remains clouded.
These chaotic trajectories are particularly apparent in a
three-body system in which the mass of one body is so small that it
doesn't influence the motion of the other two bodies. In such a
"restricted" three-body system, the motion of the nearly massless
object is typically erratic. An asteroid tugged by Jupiter and the sun
falls into this category, as does a spacecraft under the influence of
Earth and sun.
Intriguingly, the same sort of uncertainty arises in atomic and
molecular systems, particularly for the motion of electrons that have
been excited to high energies. In effect, these so-called Rydberg
electrons "orbit" at large distances from their parent atoms.
"We now know that chaotic trajectories identical to those that
govern the motion of comets, asteroids, and spacecraft are traversed on
the atomic scale by highly excited Rydberg electrons," Mason Porter and
Predrag Cvitanović remark in the October Notices of the American Mathematical Society.
Indeed, the mathematics describing the motion of gravitationally
interacting bodies in space closely parallels the mathematics
describing the motion of the smallest particles of atomic and molecular
physics. As it happens, the special case of the celestial restricted
three-body problem is mathematically analogous to the situation when a
hydrogen atom loses its electron (via ionization) in crossed electric
and magnetic fields.
In celestial mechanics, understanding the nature of chaotic
trajectories is important for predicting what could happen to
populations of small celestial bodies, such as near-Earth asteroids
that could threaten the planet, and for designing gravitationally
assisted transport of spacecraft. In the case of spacecraft, the tangle
of gravitational forces creates tubular "highways" in space along which
these vehicles can proceed with little expenditure of energy (see
"Navigating Celestial Currents" at http://www.sciencenews.org/articles/20050416/bob9.asp).
Atomic physicists and chemists face a similar challenge in
modeling chemical reactions and computing ionization rates of atoms and
molecules. Among the tools that they have developed to do such
calculations is "transition state theory."
Roughly speaking, transition states are barriers the must be
crossed, from starting materials to products, for chemical reactions to
occur. Understanding the geometry of these multidimensional barriers
provides insights into how chemical reactions occur. So, using the
paraphernalia of the theory, researchers can describe how a set of
"reactants" evolves into a set of "products."
The same sort of transition occurs in the celestial realm. For
example, the comet Oterma switches, every once in a while, from a
complex trajectory outside the orbit of Jupiter to one lying within
Jupiter's orbit. To make this transition, the comet must pass through a
bottleneck near two of Jupiter's libration points—points in space where
objects maintain a fixed distance relative to the planet and the sun.
As shown by Caltech mathematician Jerrold Marsden and his coworkers,
the transition between the orbits is mathematically analogous to a
boundary between initial and final states.
"Such unanticipated connections between microscopic and
celestial phenomena are not only intellectually gratifying but also
have practical engineering applications in the aerospace and chemical
industries," Porter and Cvitanović write.
So, theory from atomic and molecular physics can be used on a
much grander, celestial scale, and celestial mechanics can provide
insights into atomic and molecular interactions. "The orbits used to
design space missions thus also determine the ionization rates of atoms
and chemical-reaction rates of molecules!" Porter and Cvitanović
One example of the value of this connection has been NASA's
Genesis mission, which was designed to collect samples of the solar
wind. In an interdisciplinary effort, Caltech's Martin W. Lo worked
with mathematician Jerrold Marsden, Georgia Tech physicist Turgay Uzer,
West Virginia University chemist Charles Jaffé, and others to design
the complex, unstable trajectory that the Genesis spacecraft required
to accomplish its mission.
It's a marriage made in the heavens.
Jaffé, C., S.D. Ross, M.W. Lo, J. Marsden, D. Farrelly, and T. Uzer. 2002. Statistical theory of asteroid escape rates. Physical Review Letters 89(July 1):011101. Abstract available at http://link.aps.org/abstract/PRL/v89/e011101.
Klarreich, E. 2005. Navigating celestial currents. Science News 167(April 16):250-251. Available at http://www.sciencenews.org/articles/20050416/bob9.asp.
Minkel, J.R. 2002. Asteroids lost in space. Physical Review Focus (June 14). Available at http://focus.aps.org/story/v9/st31.
Pegg Jr., E. 2004. Manifolds in the Genesis mission. MAA Online (Sept. 7). Available at http://www.maa.org/editorial/mathgames/mathgames_09_07_04.html.
Peterson, I. 2005. Strange orbits. Science News Online (Aug. 13). Available at http://www.sciencenews.org/articles/20050813/mathtrek.asp.
______. 1999. Prophet of chaos. Science News Online (Nov. 13). Available at http://www.sciencenews.org/pages/sn_arc99/11_13_99/mathland.htm.
Porter, M.A., and P. Cvitanović. 2005. Ground control to Niels Bohr: Exploring outer space with atomic physics. Notices of the American Mathematical Society 52(October):1020-1025.
You can learn more about the Genesis mission and spacecraft at http://genesismission.jpl.nasa.gov/.
Information about the restricted three-body problem and Lagrange points can be found at http://scienceworld.wolfram.com/physics/RestrictedThree-BodyProblem.html and http://scienceworld.wolfram.com/physics/LagrangePoints.html.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.
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to Ivars Peterson at email@example.com.
Ivars Peterson is the mathematics/computer writer and online editor
at Science News and
Science News for Kids.
He is the author of The Mathematical Tourist, Islands of Truth,
Newton's Clock, Fatal Defect, The Jungles of Randomness,
and Fragments of Infinity. He also writes for the children's
He is coauthor of the children's books Math Trek: Adventures in
the MathZone and Math Trek 2: A Mathematical Space Odyssey.