1/18/2006

MathTrek - Gauss's Orbits

The first sighting of an asteroid occurred on 1801. It's observer notified his colleagues but political turmoil delayed the mail. As a result, no one else had a chance to observe it. With little else to occupy himself at that time Gauss, a 24-year-old mathematician brought his genius to bear on the problem and using only three observations of its place in the sky calculated what the asteroid's position would be when it got visible again.. On Dec. 7, 1801 the object was found only a short distance away from where Gauss had predicted it would lie. Gauss became a celebrity.

Ivars Peterson's MathTrek

Gauss's Orbits
Two centuries ago, no asteroids had yet made their appearance in astronomical catalogs of the solar system, which then included just the sun, seven planets, and the mysteriously evanescent comets.

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!"



Ivars Peterson's MathTrek - Gauss's Orbits

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