## Milestones in the History of Celestial Mechanics of the Planetary System

We will consider two aspects in this overview: those related to eminent scientists, summarized in Table 2.1, and those related to important discoveries in the planetary system, summarized in Table 2.3.

The history of Celestial Mechanics should start with the first attempts to observe and predict the apparent motions of the Moon, the Sun, and the planets w.r.t. the celestial sphere of fixed stars. Our story, however, begins in the 16th century with the epoch of Tycho Brahe (1546-1601) and Johannes Kepler (1571-1630). Towards the end of the 16th century Brahe had set new standards in astronomical observation techniques. First in Denmark (1576-1597), then in Prague (1599-1601), he and his collaborators observed the positions of the planets and the Sun w.r.t. the zodiacal stars with an accuracy of about 1-2 arcminutes ('). Loosely speaking, the position of a celestial body is the direction from the observer to the observed object at a certain observation epoch. The astronomical position is characterized by a unit vector (which in turn may be specified by two angles) and the corresponding observation time. The accuracy achieved by Brahe was one of the best in the pre-telescope era, close to the best that could be obtained with only mechanical observation techniques.

In view of the accuracy of the position measurements, the observation epoch had to be accurate to a few seconds of time. The best time scale then available was given by the Earth's rotation (sidereal or solar time), defined in turn by observations of the stars and the sun. This is why Brahe also needed to observe stars in his comprehensive observation program. For interpolation of time, mechanical clocks with an accuracy of few seconds over, let us say, one day already were available. These few remarks show that Brahe did not only "observe a few planets", but that he and his team accomplished a rather comprehensive observational survey in astronomy towards the end of the 16th century, lasting for a time period of about a quarter of a century. Brahe's program should be compared with current international observation programs in astronomy, organized, e.g., by the IERS (International Earth Rotation and Reference Systems Service). Brahe's main result was a very consistent, long, and complete set of positions for the sun and the planets.

Around 1600 Kepler was Landschaftsmathematiker (state surveyor) in Graz. His inclination towards astronomy was documented by his work Mysterium cosmographicum, where he tried to relate the radii of the classical planets to the five regular polyhedra. Although Brahe was not so impressed by Kepler's work he invited Kepler as a co-worker to Prague. Tycho certainly hoped that Kepler would help him to further develop his own model of the planetary system, a mixture of the systems due to Nicholas Copernicus (1473-1543) and Claudius Ptolemaeus (ca. 100-170). Kepler had different ideas, however. We know from history that Kepler tried to find a physical law governing the planetary motions. He used the law of areas and introduced ellipse-shaped orbits to reduce the calculations when processing Tycho's time series of the positions of the sun and of the planet Mars. The steps eventually leading to the so-called Kepler's first two laws are documented in the Astronomia nova (which appeared in 1609) and in the correspondence between Kepler and his teacher Michael Mastlin (1550-1631) as well as Kepler's "rival" Lon-gomontanus (1562-1647). However, the first two "laws" may not be found in the Astronomia nova because Kepler failed to confirm them by theory and observation. The third law was published only in the Harmonice mundi of 1619.

Let us include Kepler's laws in modern language:

1. The orbit of each planet around the sun is an ellipse with the Sun at one of its foci.

2. Each planet revolves so that the line joining it to the Sun sweeps out equal areas in equal (intervals of) time. (Law of areas).

3. The periods of any two planets are proportional to the 3/2 powers of their mean distances from the sun.

The first law implies that planetary motion is taking place in orbital planes, characterized, e.g., by two angles (in the planetary system, e.g., Ü the eclip-

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