MARGINALIA

# The American Kepler

More than a century after his death, the name of Daniel Kirkwood continues to be familiar to almost any student who takes an elementary astronomy course. There the student learns that asteroids are mainly located in a belt between the orbits of Mars and Jupiter, and that Kirkwood discovered there is a succession of gaps in that belt where no asteroids are found. This discovery made him justly famous, and it is still a staple of elementary texts and research papers alike. But ask any student—or instructor—what else Kirkwood was famous for in his own time, and you will likely be met with a puzzled frown and shake of the head. Tell them that long before the "gap" discovery, he made another that brought him worldwide attention and had prominent astronomers and the popular press hailing him as the equal of Johannes Kepler, and you will likely receive a skeptical smile. But it was so.

Adding to the extraordinariness of this event was Daniel Kirkwood's background. He was born in 1814 on a Maryland farm, and his only education as a child came from the local country school. At the age of 19, having no taste for farming, he became a schoolteacher in another such country school. One of his students wished to study algebra, and since Kirkwood knew none, he and the student sat down and worked through an elementary textbook on the subject together—a rather unlikely beginning for one who later held the chair of mathematics at a major university. From this Kirkwood realized that he had both a flair and a taste for mathematics and so went back to school himself for several years to study it. By 1849 he had worked his way up from being a mathematics instructor to become principal of an institution called the Pottsville Academy of Pennsylvania. It was about this time that he made his discovery.

During these years of development and reading, Kirkwood had slowly become intrigued by the fact that, although there is a law governing the revolution of planets (Kepler's third law), no law had as yet been discovered governing their rotations. (Astronomy makes a distinction between the terms *rotate* and *revolve*; a body rotates about an axis within itself but revolves about an axis that is exterior. Thus the earth rotates once a day but revolves about the sun once a year.) Kepler's third law of planetary motion says that if *P* is a planet's period of revolution about the sun, and *d* is its distance from the sun, then *P*^{2} is proportional to *d*^{3}. Kirkwood spent some 10 years off and on mulling over what might be a corresponding law governing planetary rotations, but with no success.

Eventually, in August of 1846, a study of Laplace's theory for the origin of the solar system led him by a process that is not at all clear to the following proposition. Consider three consecutive planets lined up in a row. There will be a point between the middle and outer planet at which a particle will experience equal gravitational force from the two planets, and another such point between the middle and inner planet. Calculate the distance, *D*, between these two points, the diameter of the middle planet's "sphere of attraction." Next, from the known periods of rotation and revolution calculate the number of rotations, *n*, that a planet makes in the course of one revolution. Kirkwood's calculations suggested to him that *n*^{2} was proportional to *D*^{3} as one went from planet to planet. He referred to his result as the analogue of Kepler's third law.

Unlike some who think they have discovered an important scientific law (Kepler himself, for instance, prancing around in paeans of ecstasy as to how God had waited 6,000 years for someone to discover what Kepler thought to be celestial harmonies among the planets), Kirkwood's behavior was exemplary. He wrote in very modest fashion to Edward Herrick at Yale, describing his discovery but noting that "perhaps it may be regarded by those better qualified to judge than myself, as a vagary not worthy of consideration." Herrick suggested that he send his letter to an astronomer at the U.S. Coast Survey, Sears Walker, then well known for his work on Neptune's orbit. Walker discussed the finding with other members of the American Philosophical Society and soon became an enthusiastic advocate, announcing that it "deserves to rank at least with Kepler's harmonies." In August 1849 Walker presented Kirkwood's letters to a meeting of the American Association for the Advancement of Science (AAAS), again concluding the result to be "the most important harmony in the solar system discovered since the time of Kepler, which, in after times, may place their names, side by side, in honorable association."

The AAAS members were impressed. Benjamin Peirce, *doyen* of astronomy at Harvard, declared it to be "the only discovery of the kind since Kepler's time, that approached near to the character of his three physical laws." Benjamin Gould, founder of what became one of the world's most important astronomy journals, said "I do not wish to express myself strongly . . . [but] nor can we consider it as very derogatory to the former to speak hereafter of Kepler and Kirkwood together as the discoverers of great planetary harmonies." Newspapers and journals soon brought the public to know of this wonderful and amazing discovery.

Walker, it would seem, was carried away by his own enthusiasm. He submitted a letter to the editor of the prestigious German journal *Astronomische Nachrichten* outlining "the discovery [as] thus announced by Mr. Kirkwood." It is only one-and-a-half pages long and contains only one table. The final column in this is labeled "Kirkwood's diameter of the Sphere of attraction, *D*," derived, one would assume, from the observational data listed in previous columns of each planet's distance from the sun, mass and rotational period. But if one uses these data to compute the period of revolution by Kepler's third law and thus *n* for each planet, and plots that against *D*, one obtains the graph shown in Figure 1. The straight line has the equation *n* = 1000**D*^{1.5} and is an amazing fit of the line to the points! Especially when, as we now know, some of the input data were wildly wrong. The mass of Mercury, for instance, was wrong by more than a factor of 2, Venus's rotation period was in error by more than a factor of 200, and most of the other data were somewhat in error. What was going on?

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