Archimedes to Hawking: Laws of Science and the Great Minds behind Them

By Clifford A. Pickover | Go to book overview

BODE’S LAW OF PLANETARY DISTANCES

Germany, 1766. The mean distances of planets from the Sun can be predicted using a simple numerical relationship.

CROSS REFERENCE: ASTRONOMER JOHANN DANIEL TITIUS, THE TITIUS-BODE LAW, AND SWISS-GERMAN PHYSICIST JOHANN HEINRICH LAMBERT.

In 1766, two English surveyors, Charles Mason and Jeremiah
Dixon, drew the Mason-Dixon Line between Pennsylvania and
Maryland, which would later be used to mark the boundary
between free and slave regions of the United States. In this
same year, the British Parliament repealed the unpopular Stamp
Act that had taxed all documents in the American colonies by
requiring the documents to carry a tax stamp. English chemist
John Dalton (see “Dalton’s Law of Partial Pressures” in part III)
was born.

Bode’s Law, also known as the Titius-Bode Law, expresses a relationship that describes the mean distances of the planets from the Sun. Consider the simple sequence 0, 3, 6, 12, 24, …, in which each successive number is twice the previous number. Next, add 4 to each number and divide by 10 to form the sequence 0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10.0, 19.6, 38.8, 77.2, … Remarkably (and perhaps strangely!), Bode’s Law states that this sequence gives the mean distances D of the known and yet-to-be-discovered planets from the Sun, expressed in astronomical units (AU). An AU is the mean distance between Earth and the Sun, which is approximately 92,960,000 miles (149,604,970 kilometers). For example, Mercury is approximately onethird of an AU from the Sun, and Pluto is about 39 AU from the Sun. Bode’s Law can be expressed by

where N = 0, 3, 6, 12, 24, 48 …. We also sometimes see the law expressed as

where A = 0.4, B = 0.3, C = 2,and n = 0, 1, 2, 3….

This relationship was discovered in 1766 by the German astronomer Johann Daniel Titius (1729–1796) of Wittenberg and published by Bode six years later. At the time, the law gave a remarkably good estimate for the mean distances of the planets that were then known—Mercury (0.39), Venus (0.72), Earth (1.0), Mars (1.52), Jupiter (5.2), and Saturn (9.55). Uranus, discovered in 1781, has a mean orbital distance of 19.2, which also

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