Johannes Kepler (1571-1630) had studied astronomy long before he met Tycho: he favored the Copernican world-view and corresponded with Galileo.

Johannes Kepler

Tycho's observations included some very accurate measurements of the position of the planet Mars, which did not agree with either Ptolemy or Copernicus. When Tycho died, Kepler got hold of those observations and tried to puzzle them out. In 1609, the same magic year when Galileo first turned his telescope towards the heavens, Kepler caught a glimpse of what he thought might be the answer. That was when he published his first two laws of planetary motion:

1. Planets move along ellipses, with the Sun at one focus.
2. The line from the Sun to the planet
   covers equal areas in equal times.

Each of these statement requires some explanation.

Ellipses

The ellipse, the shape of a flattened circle, was well known to the ancient Greeks. It belonged to the family of "conic sections," of curves produced by the intersections of a plane and a cone.

The curves generated as "conic sections" when flat planes are cut across a cone.

As the drawing above on the left shows, when that plane is...

--perpendicular to the axis of the cone, the result is a circle.

--moderately inclined, an ellipse.

--inclined so much that it is parallel to one side of the cone, a parabola.

--inclined even more, a hyperbola.

All these intersections are easily produced by a flashlight in a moderately dark room (drawing below). The flashlight creates a cone of light and when that cone hits a wall, the shape produced is a conic section--the intersection of the cone of light with the flat wall.

The axis of the flashlight is also the axis of the cone of light. Aim the beam perpendicular to the wall to get a circle of light. Slant the beam: an ellipse. Slant further, to where the closing point of the ellipse is very, very far: a parabola. Slant even more, to where the two edges of the patch of light not only fail to meet again, but seem to head in completely different directions: a hyperbola.

The Third Law

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). By Kepler's formula

T = SQRT (k' a³)

where SQRT stands for "square root of" (the world-wide web does not offer more specific symbols). If T is measured in seconds and a in Earth radii (1 R_E = 6371 km = 3960 miles)

T = 5063 SQRT (a³)

Kepler's later years were not too happy. His patron, Emperor Rudolf, died in 1612, and although Kepler retained his post as court mathematician and continued to produce important work, his life was increasingly disrupted by war. That was the 30 years' war, a bitter religious battle which pitted Protestants against Catholics; it began in Prague in 1618 and engulfed all of Kepler's part of Europe.