|Ivars Peterson's MathTrek|
August 23, 1999
Hosted by the Exploratorium in San Francisco, the eclipse party on August 11 brought hundreds of people to the science center's first-ever all-night event. Young and old, ranging from the densely tattooed and pierced to the wildly costumed and the soberly business-suited, the celebrants camped out for the night, sprawled across the floor among the Exploratorium's exhibits.
Many people laid out sleeping bags, blankets, pads, or even air mattresses, staking out prime territory near the Webcast studio or in front of TV monitors. One group unpacked a sumptuous meal of steaming lasagna and other gourmet delights, along with the requisite bottle of vintage wine.
In one area, children created giant, iridescent soap bubbles, which jiggled and glittered brightly in the spotlights as they floated upward, before finally bursting. Roving astronomers answered questions and provided eclipse-related information.
Listening to early music from England, a French accordion band, a classical trio, an exuberant Hungarian quartet, a Romanian choir, traditional Turkish and Iranian music, and Indian (very early) morning ragas, the eclipse worshippers talked, ate, napped, and played. Dazzling performances by a sword swallower, a fire eater, and a bevy of belly dancers heightened the festive mood.
The musical program tracked the path of the moon's shadow as it cut a swath from Cornwall, England, to India. An Exploratorium team in Amasya, Turkey, provided commentary and stunning images of the solar eclipse, which reached totality in that location at roughly 4:30 a.m. Exploratorium time.
Broadcast live on the Exploratorium's Web site (http://www.exploratorium.edu/eclipse), the eclipse coverage proved a triumph for the technicians and camera operators, who captured amazing images of the sun's face blotted out by the moon, leaving just a beaded ring of brilliant light against a darkened, star-pocked sky.
The mix of ancient tradition and high-tech wizardry that marked the Exploratorium's eclipse party reminded me of the many mathematicians and scholars who had worked over millennia to develop geometric and physical models of the motions in the sky to be able to predict such awe-inspiring celestial events as solar and lunar eclipses. Indeed, precisely pinpointing the timing of these occurrences has long served as a stringent test of physical theory and mathematical model.
"Although we think primarily of the planets orbiting the sun as the fundamental issue for the origin of modern science, it is really the moon that provided the principal ideas as well as the crucial tests of our understanding of the universe," Martin C. Gutzwiller of the IBM Thomas J. Watson Research Center in Yorktown Heights, N.Y., noted last year in an article in Reviews of Modern Physics. "The moon played the role of the indispensable guide without whom we might not have found our way through the maze of possibilities."
Indeed, the daily motion of the moon through the sky has a number of idiosyncrasies that a careful observer can discover without the help of instruments. More than 3,000 years ago, Mesopotamian astronomers observed and recorded the moon's position on the horizon, in effect measuring three important characteristic frequencies of the lunar orbit (see Fractions, Cycles, and Time, Oct. 13, 1997).
"Both [the moon's] varying speed and the spread of moonrises and moonsets on the horizon proceed at their own rhythm, which is most clearly displayed in the schedule of lunar and solar eclipses," Gutzwiller says.
About 1,000 years later, Greek astronomers and mathematicians provided an explanation of those numbers and cycles in terms of a geometric model involving circular movement.
This second stage "was initiated by the early Greek philosophers, who thought of the universe as a large empty space with Earth floating at its center, the sun, the moon, and the planets moving in their various orbits around the center in front of the background of the fixed stars," Gutzwiller remarks. "This grand view may have been the single most significant achievement of the human mind."
"Without the moon, visible both during the night and during the day, it is hard to imagine how the sun could have been conceived as moving through the Zodiac just like the moon and the planets," he adds.
Toward the end of the 17th century, Isaac Newton (1643-1727) provided the physical model. His majestic, immortal opus Philosophiae naturalis principia mathematica (Mathematical principles of natural philosophy) represented the first significant endeavor to explain observations both on Earth and in the heavens on the basis of a few physical laws in the form of mathematical relations.
Newton showed how the whole clockwork mechanism in the sky operated on the basis of physical law. To demonstrate the power of his methods, he sought to derive from those laws key features of the moon's motion, including the relationship between the three different periods (or frequencies) characterizing the moon's motion.
Unsatisfied with his initial attempt to solve the problem in Principia, Newton returned to the subject of the moon's motion in 1694 for a year of intense labor. He would later comment that "his head never ached but with his studies on the moon."
Newton ultimately failed to achieve his goal, and he expressed his immense frustration in the following words: "The Irregularity of the Moon's Motion hath been all along the just Complaint of Astronomers; and indeed I have always look'd upon it as a great Misfortune that a Planet so near us as the Moon,. . . should have her Orbit so unaccountably various, that it is in a manner vain to depend on any calculation. . ., though never so accurately made."
In studying the moon's motion, Newton had been forced to confront the inevitable dynamical complexities of what is now known as the three-body problem. The moon feels the gravitational pull of not only Earth but also the sun. Those solar tugs subtly distort the moon's orbit in ways that greatly complicate the moon's movement, making precise prediction difficult.
Nearly a century before Newton, Johannes Kepler (1571-1630) had suspected something similar. When asked why a spring lunar eclipse had occurred one and a half hours later than he had predicted, Kepler had replied, after some thought, that the sun apparently had a retarding influence on the moon, especially in winter, when the sun is closest to Earth.
In the 18th century, after considerable effort, mathematicians managed to calculate key aspects of the moon's motion, succeeding where Newton had failed. Pierre-Simon de Laplace (1749-1827) eventually provided a successful, though cumbersome apparatus for doing such calculations so that their precision finally brought them into good agreement with observation.
Near the end of the 19th century, George William Hill (1838-1914), an astronomer at the U.S. Naval Almanac Office, discovered a computational trick that considerably simplified the mathematical machinery of lunar theory. Whereas previous mathematicians and astronomers had begun with ellipses and then modified these simple orbits step by step to accommodate the effect of a third body, Hill started with a particular orbit defined by a special, simple solution of the three-body problem.
Hill's technique exploited the fact that although mathematicians could not come up with a general formula describing the motions of all three bodies for all time, they could come up with precise solutions for certain special cases. Hill's starting point was a particular periodic orbit that already included the sun's perturbing influence. He then mathematically superimposed additional wiggles and shifts representing the movements of the lunar perigee and nodes to bring this main, smooth loop closer to the moon's true orbit.
In modified form, Hill's method formed the basis for all subsequent lunar calculations and, in effect, helped land the first men on the moon just 30 years ago this month.
From solar eclipse to space travel, the moon's persistently puzzling peregrinations have left quite a legacy.
Copyright 1999 by Ivars Peterson
Gutzwiller, M.C. 1998. Moon-Earth-Sun: The oldest three-body problem. Reviews of Modern Physics 70(April):589.
Nabonnand, P. 1999. The Poincaré --Mittag-Leffler relationship. Mathematical Intelligencer 21(No. 2):58.
Peterson, I. 1993. Newton's Clock: Chaos in the Solar System. New York: W.H. Freeman.
The Exploratorium in San Francisco has a Web site at http://www.exploratorium.edu.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.