(10) Kepler and His Laws

Tycho Brahe (1546-1601)

    Tycho was a Danish nobleman interested in astronomy. In 1572 a "new star" (in today's language, a nova) appeared in the sky, not far from Polaris, outshining all others. Tycho carefully measured its position, then measured it again 12 hours later, when the the rotation of the Earth had moved the observing point to the other side of the Earth. Such a move was already known to shift the position of the Moon in the sky, helping astronomers estimate its distance. The position of the "new star" did not change, suggesting it was much more distant than the Moon (an image of its remnant is at the bottom).

Drawing of Tycho Brahe.

    This event so impressed young Tycho that he resolved to devote himself to astronomy. The king of Denmark supported him and gave him the island of Hven to build an observatory, with the taxes of the island providing him with the funding. The telescope had not yet been invented, and all measurements were done by eye, aided by open sights (like those on guns) which could be slid around circles, marked in degrees. Tycho extended such methods to their ultimate limit, the resolution of the human eye, and his star charts were far more accurate than any earlier ones. He even measured and took into account the very slight shift of star positions near the horizon, due to the bending of light in the Earth's atmosphere, similar to light bent in glass or water. And his observations of the planets became the most stringent test of the theories of Copernicus and Ptolemy.

    Concerning those theories, Tycho believed that all planets revolved around the Sun, but the Sun circled Earth. That view might have suited Denmark's Protestant church, for Martin Luther, founder of the Protestant doctrine, had rejected the views of Copernicus (who lived at the same time). Tycho's manners, however, were arrogant, and the residents of Hven complained about him, so that after the death of the king who was Tycho's patron, Tycho was forced to leave Denmark.

Johannes  Kepler

    He settled in 1599 in Prague--now the Czech capital, then the site of the court of the German emperor Rudolf--and there he became court astronomer. It was in Prague, too, where a German astronomer named Johannes Kepler was hired by Tycho to carry out his calculations. When in 1601 Tycho suddenly died, it was Kepler who continued his work.

    After Tycho's death, Kepler became the court astronomer, although the superstitious emperor was more interested in astrology than in the structure of the solar system. In 1619 Kepler published his third law: the square of the orbital period T is proportional to the cube of the mean distance a from the Sun (half the sum of greatest and smallest distances). In formula form

T²= k a³

with k some constant number, the same for all planets. Suppose we measure orbital periods in years and all distances in "astronomical units" or AUs, with 1 AU the mean distance between the Earth and the Sun. Then if a = 1 AU, T is one year, and k with these units just equals 1, i.e. T²= a³. Applying now the formula to any other planet, if T is known from the observations of many years, the planet's a, its mean distance from the Sun, is readily derived.

    Finding the value of 1 AU in miles or kilometers, that is, finding the actual scale of the solar system, is not easy. This subject is discussed in the next section. Our best values nowadays are the ones provided by space-age tools, by radar-ranging of Venus and by planetary space probes; to a good approximation, 1 AU = 150 000 000 km.

    Not only were Kepler's laws confirmed and explained by later scientists, but they apply to any orbital system of two bodies--even artificial satellites in orbit around the Earth. The constant k' for artificial satellites differs from k obtained for planets (but is the same for any satellite of Earth). By Kepler's formula

T = √ (k'a³)

where √ stands for "square root of". If T is measured in seconds and a in Earth radii (1 R_E = 6371 km = 3960 miles)

T = 5063 √ (a³)

More will be said about Kepler's first two laws in the next two sections.

Postscript: The remnant of Tycho's supernova is still visible. Above is its image made in 2005 by NASA's orbiting x-ray telescope "Chandra," (for a larger image, 296 Kb, click here; thanks to Randall Smith for pointing out that a different supernova in Cassiopeia was originally pictured on this site). The colors encode different ranges of X-ray wavelength; for more details, see here. You might compare this to an earlier picture of this supernova remnant. For more about the "Chandra," observatory, see its web site.

Exploring Further

A site about Tycho Brahe, illustrated, here

Detailed site about Kepler

Picture and links concerning Kepler

An excellent (but somewhat long) book about Kepler's life and his turbulent times is "Kepler's Witch" by James A Connor.

Willman-Bell, publishers of astronomy books (http://www.willbell.com) is offering "The Lord of Uraniborg" by Thoren, a biography of Brahe, and "Kepler" by Casper. See also the end of the preceding section, concerning "The Sleepwalkers" by Koestler.

A very readable recent book, with many details of Tycho's instruments and observations: "Tycho and Kepler" by Kitty Ferguson (Walker & Co, 2002, xiv+402 pp). My review of it here

"Kepler and the Universe" by David K.Love, Prometheus Books, 2015. A biography reviewed by Owen Gingrich in Physics Today, October 2016, p. 55.

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