Kepler's laws

Introduction

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Schematic representation of Kepler's second law: The areas ABF and A′B′F′ are equal and are swept out in equal intervals of time by a planet orbiting around the sun (at F).

Kepler's laws, three mathematical statements formulated by the German astronomer Johannes Kepler that accurately describe the revolutions of the planets around the sun. Kepler's laws opened the way for the development of celestial mechanics, i.e., the application of the laws of physics to the motions of heavenly bodies. His work shows the hallmarks of great scientific theories: simplicity and universality.

Sections in this article:

Introduction
Kepler's Foretelling of the Law of Gravity
Development of Kepler's Laws
Summary of Kepler's Laws

Kepler's Foretelling of the Law of Gravity

Kepler believed that the sun did not sit passively at the center of the solar system but that through some mysterious power or “virtue” actually compelled the planets to hold to their orbits. Because the planets moved slower when they were farther from the sun, this power must diminish with increasing distance. The idea that the planets were controlled by the sun was developed by Isaac Newton in his laws of motion and law of gravitation. Newton assumed that the sun continuously exerts a force on each planet that pulls the planet toward the sun. He calculated that elliptical orbits would result if the force varied inversely as the square of the distance from the sun (i.e., when the distance doubles, the force becomes four times weaker). His law of universal gravitation predicts that the planets exert small forces on each other although subject to the dominant force of the sun. These small additional forces explain most of the small departures from Kepler's laws revealed by later, more accurate observations.

Development of Kepler's Laws

Earlier theories of planetary motion, such as the geocentric Ptolemaic system and the heliocentric Copernican system, had allowed only perfect circles as orbits and were therefore compelled to combine many circular motions to reproduce the variations in the planets' motions. Kepler eliminated the epicycles and deferents that had made each planet a special case. His laws apply generally to all orbiting bodies.

Kepler's first and second laws were published in 1609 in Commentaries on the Motions of Mars. Because Mars was the planet whose motions were in greatest disagreement with existing theories, its orbit provided the critical test for his hypotheses. To do this Kepler was able to rely on the astronomical observations of his mentor, Tycho Brahe, which were much more accurate than any earlier work. The third law appeared in 1619 in Harmony of the Worlds.

Summary of Kepler's Laws

The first law states that the shape of each planet's orbit is an ellipse with the sun at one focus. The sun is thus off-center in the ellipse and the planet's distance from the sun varies as the planet moves through one orbit. The second law specifies quantitatively how the speed of a planet increases as its distance from the sun decreases. If an imaginary line is drawn from the sun to the planet, the line will sweep out areas in space that are shaped like pie slices. The second law states that the area swept out in equal periods of time is the same at all points in the orbit. When the planet is far from the sun and moving slowly, the pie slice will be long and narrow; when the planet is near the sun and moving fast, the pie slice will be short and fat. The third law establishes a relation between the average distance of the planet from the sun (the semimajor axis of the ellipse) and the time to complete one revolution around the sun (the period): the ratio of the cube of the semimajor axis to the square of the period is the same for all the planets including the earth.

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