Understanding Kepler’s Second Law

Planetary Motion, Changing Speeds, Equal Areas, and the Cosmos

Kepler’s second law of planetary motion, expressed in terms of equal areas and equal times, means that the planets change speeds as they orbit the Sun.

Johannes Kepler went to work for Tycho Brahe near the end of Tycho’s life. When Tycho died, Kepler used Tycho’s data to deduce three laws of planetary motion. Because Tycho’s data were far more accurate than any previously collected data on planetary positions neither Ptolemy’s nor Copernicus’s models of the cosmos worked.

Kepler deduced three laws of planetary motion that did agree with Tycho’s data but diverged from traditional views of the cosmos.

Statement of Kepler’s Second Law

Kepler’s second law states that as a planet orbits the Sun, it sweeps out equal areas in equal times. This statement is fairly easy to memorize for an exam, but it is stated in such a way that understanding the meaning of Kepler’s second law is particularly difficult.

Sweeping out Equal Areas in Equal Times

Most people find it difficult to understand what Kepler meant by a planet sweeping out equal areas in equal times. An imaginary line between the planet and the Sun will sweep out an area in the plane of the planet’s orbit, as the planet orbits the Sun. When the planet is closer to the Sun, it sweeps out a short wide area. When it is further from the Sun it sweeps out a long narrow area. To sweep out these equal areas in equal times, the planet must move faster when it is closer to the Sun and more slowly when it is farther away. The equal times can be any amount of time less than the planet’s orbital period: a day, week, month, or whatever.

Understanding Kepler’s Second Law

Kepler’s first law states that planets move in elliptical orbit around the Sun. Hence as a planet orbits the Sun, its distance from the Sun changes.

Throw a ball into the air and watch its speed. As it moves upward away from Earth, it slows down. As it falls towards Earth, it speeds up. A planet orbiting the Sun is influenced by the Sun’s gravity, just as the ball is influenced by Earth’s gravity. So as a planet moves closer to the Sun in its orbit, it speeds up just like the falling ball. As the planet moves away from the Sun, it slows down.

As the planet orbits the Sun, it changes its speed in such a way that it sweeps out equal areas in equal times. The orbiting planet also speeds up and slows down in a way that both energy and angular momentum are conserved.

Prior to Kepler’s time, the standard scientific dogma required that planets move in circular motion at a constant speed. Kepler’s first law got rid of the circular motion requirement by allowing elliptical orbits. Kepler’s second law got rid of the uniform speed requirement, by allowing the planet to speed up or slow down as its distance from the Sun changes. Hence taken together Kepler’s first and second laws completely demolished the traditional idea that planets must follow uniform circular motion.

Kepler then developed his third law, which is a mathematical relationship between the planets’ orbital periods and distances from the Sun.

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