The greatest astronomer of the early Renaissance was Tycho Brahe. For 30 years, from 1570 to 1600, Tycho compiled the most accurate astronomical observations the world had known. The invention of the telescope was still to come, but Tycho developed ingenious mechanical sighting devices that allowed him to determine the positions of stars and planets in the sky with unprecedented accuracy.

Tycho had a young mathematical assistant named Johannes Kepler. Kepler had become one of the first outspoken defenders of Copernicus, and his goal was to find evidence for circular planetary orbits in Tycho’s records. To appreciate the difficulty of this task, keep in mind that Kepler was working before the development of graphs or of calculus—and certainly before calculators! His mathematical tools were algebra, geometry, and trigonometry, and he was faced with thousands upon thousands of individual observations of planetary positions measured as angles above the horizon.

Many years of work led Kepler to discover that the orbits are not circles, as Copernicus claimed, but ellipses. Furthermore, the speed of a planet is not constant but varies as it moves around the ellipse.

Kepler’s laws, as we call them today, state that

1. Planets move in elliptical orbits, with the sun at one focus of the ellipse.

2. A line drawn between the sun and a planet sweeps out equal areas during equal intervals of time.

3. The square of a planet’s orbital period is proportional to the cube of the semimajor-axis length.

FIGURE 13.1a shows that an ellipse has two foci (plural of focus), and the sun occupies one of these. The long axis of the ellipse is the major axis, and half the length of this axis is called the semimajor-axis length. As the planet moves, a line drawn from the sun to the planet “sweeps out” an area. FIGURE 13.1b shows two such areas. Kepler’s discovery that the areas are equal for equal $\text{Δ}t$ implies that the planet moves faster when near the sun, slower when farther away.

FIGURE 13.1 The elliptical orbit of a planet about the sun.

Full Alternative Text

All the planets except Mercury have elliptical orbits that are only very slightly distorted circles. As FIGURE 13.2 shows, a circle is an ellipse in which the two foci move to the center, effectively making one focus, and the semimajor-axis length becomes the radius. Because the mathematics of ellipses is difficult, this chapter will focus on circular orbits.

FIGURE 13.2 A circular orbit is a special case of an elliptical orbit.

Kepler published the first two of his laws in 1609, the same year in which Galileo first turned a telescope to the heavens. Through his telescope Galileo could see moons orbiting Jupiter, just as Copernicus had suggested the planets orbit the sun. He could see that Venus has phases, like the moon, which implied its orbital motion about the sun. By the time of Galileo’s death in 1642, the Copernican revolution was complete.