|Ivars Peterson's MathTrek|
The first sighting of an asteroid occurred on Jan. 1, 1801, when the monk Giuseppe Piazzi (1746-1826) noticed a faint, starlike object not included in a star catalog that he was checking. Fascinated by astronomy, Piazzi had in 1790 established and equipped an observatory in Palermo on the island of Sicily. Taking advantage of a favorable climate for astronomical viewing, he launched a lengthy project dedicated to determining precisely the astronomical coordinates of several thousand stars.
Piazzi's observations of the mysterious object on successive nights revealed that it moved slowly against its starry backdrop, first drifting backward, then reversing direction and overtaking the background stars. Unsure whether the object was a comet or a planet, Piazzi watched it regularly until Feb. 11, when he fell ill. By the time he recovered a few days later, he was able to make only one more observation before the object advanced sufficiently close to the sun to disappear in its glare. Piazzi named the tiny new planet Ceres.
Piazzi had already begun notifying colleagues in other parts of Europe of his discovery, but political turmoil in Italy delayed the mail. As a result, no one else had a chance to observe the object. Only one-tenth the brightness of Uranus and on the fringe of visibility in most telescopes of the time, this faint speck had no telltale planetary disk to make it easier to locate.
To recover the object once it emerged from the sun's glare several months later, astronomers needed to know its orbit. Piazzi's observations, however, covered a period of just 41 days, during which time the object had moved through an arc of only 3 degrees across the sky. Any attempt to compute the orbit of such an inconspicuous object from this meager set of data appeared futile.
To Carl Friedrich Gauss (1777-1855), a 24-year-old mathematician who early in life had displayed a prodigious talent for mathematics and a remarkable facility for highly involved mental arithmetic, this problem presented an enticing challenge. Having completed his studies at the University of Göttingen, Gauss was living on a small allowance granted by his patron, the Duke of Brunswick.
With a major mathematical work just published and little else to occupy his time during the latter part of 1801, Gauss brought his formidable powers to bear on celestial mechanics. Like a skillful mechanic, he systematically disassembled the creaky, ponderous engine that had long been used for determining approximate orbits and rebuilt it into an efficient, streamlined machine that could function reliably given even minimal data.
Assuming that Piazzi's object circumnavigated the sun on an elliptical course and using only three observations of its place in the sky to compute its preliminary orbit, Gauss calculated what its position would be when the time came to resume observations. In December, after three months of labor, he delivered his prediction to the Hungarian astronomer Franz Xaver von Zach (1754-1832), who had organized a self-proclaimed "celestial police" to track down a "missing" planet between the orbits of Mars and Jupiter.
Any hope of locating Piazzi's celestial mote after a lapse of nearly a year rested on the reliability of Gauss's innovative methods and the accuracy of his calculations. On Dec. 7, von Zach relocated the object only a short distance away from where Gauss had predicted it would lie. Gauss became a celebrity.
Historical accounts typically omit the mathematical details of how Gauss solved the problem of determining the orbit of Ceres. In an illuminating article in the April Mathematics Magazine, Donald Teets and Karen Whitehead of the South Dakota School of Mines and Technology in Rapid City fill in that gap.
"Gauss's work offers a rare instance of solving an historically great problem in applied mathematics using only the most modest mathematical tools," Teets and Whitehead remark. "It is a complicated problem, involving over 80 variables in three different coordinate systems, yet the tools that Gauss uses are largely high school algebra and trigonometry!"
"Gauss achieves greatness in this work not through deep, abstract mathematical thinking, but rather through an incredible vision of how the various quantities in the problem are related, a vision that guides him through extraordinary computations that others would likely abandon as futile," they add. Indeed, it's often difficult to see how the various computational steps Gauss undertook might reasonably lead to the final goal.
In essence, Gauss confronted the problem of how to determine two sun-centered vectors approximating the planet's position at two different times, given three Earth-centered observations of the object's latitude and longitude. Solving that problem then allowed him to determine the planet's orbital plane and the shape and orientation of its elliptical orbit within this plane.
After Piazzi's discovery, astronomers quickly found additional minor planets. To Gauss, the discovery of one asteroid after another furnished new opportunities for testing the efficiency and generality of his methods.
Gauss's computational triumphs brought him immediate, lasting recognition as Europe's top mathematician and a comfortable position as a professor of astronomy and director of the observatory at Göttingen, where he lived modestly for the rest of his long, productive life.
Never in a rush to see his ideas in print, whether in pure mathematics, astronomy, or physics, Gauss relentlessly reworked his results again and again until they were polished to perfection. Clarifying his thoughts step by step and eliminating everything but the essential elements, he would obliterate all traces of the path he had followed to arrive at his insights. No scaffolding ever marred the elegant mathematical structures he constructed so patiently.
Gauss spent years refining his techniques for handling planetary and cometary orbits. Published in 1809 in a long paper called "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" ("Theory of the motion of the heavenly bodies moving about the sun in conic sections"), this collection of methods still plays an important role in modern astronomical computation and celestial mechanics.
Gauss himself noted with pride that "scarcely any trace of resemblance remains between the method in which the orbit of Ceres was first computed, and the form given in this work."
Copyright 1999 by Ivars Peterson
Cunningham, C.J. 1992. Giuseppe Piazzi and the "missing planet." Sky & Telescope (September):274.
Peterson, I. 1993. Newton's Clock: Chaos in the Solar System. New York: W.H. Freeman.
Stewart, I. 1977. Gauss. Scientific American (July):123.
Teets, D., and K. Whitehead. 1999. The discovery of Ceres: How Gauss became famous. Mathematics Magazine 72(April):83.
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