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Understanding Kepler's Third LawPlanetary Motion, Orbital Period, Distance from Sun, and Mass
Kepler's third law of planetary motion is a mathematical relation between orbital periods and distances. It allows astronomers to measure masses of celestial objects.
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 Third LawKepler's third law, which is often called the harmonic law, is a mathematical relationship between the time it takes the planet to orbit the Sun and the distance between the planet and the Sun. The time it takes for a planet to orbit the Sun is its orbital period, which is often simply called its period. For the average distance between the planet and the Sun, Kepler used what we call the semi-major axis of the ellipse. The semi-major axis is half the major axis, which is the longest distance across the ellipse. Think of it as the longest radius of the ellipse. Kepler's third law states that the square of the period, P, is proportional to the cube of the semi-major axis, a. In equation form Kepler expressed the third law as: P^2=ka^3. k is the proportionality constant. To Kepler it was just a number that he determined from the data. Kepler did not know why this law worked. He found it by playing with the numbers. Newton's Form of Kepler's Third LawUsing Newton's laws it is possible to show why Kepler's third law works. For circular orbits, the centripetal force required to keep the planet moving in a circular path equals the gravitational force between the Sun and planet. For elliptical orbits, the idea is similar but a little more complex. Because the gravitational force depends on the mass it turns out that the proportionality constant in Kepler's third law involves the mass of the Sun or other object being orbited. See the figure for the equation for Newton's form of Kepler's third law. In the case of a planet and a star, the mass of the planet is negligible and can be dropped from the equation. In the case of two stars or other orbiting objects of similar mass, both masses must be included. Significance of Kepler's Third LawKepler's third law is extremely important to astronomers. Because it involves the mass it allows astronomers to find the mass of any astronomical object with something orbiting it. Astronomers find the masses of all astronomical objects by applying Kepler's third law to orbits. They measure the mass of the Sun by studying the orbits of the planets. They measure the mass of the planets by studying the orbits of their moons. Moons have nothing orbiting them, so to find the mass of the moons astronomers need to send a probe to be affected by their gravity. Astronomers find the masses of stars by studying the orbits of stars in binary systems. They can not measure the masses of stars that are not in binary systems. In all these cases astronomers use Kepler's third law. Kepler's third law is the only way to measure the masses of astronomical objects. Further ReadingZeilik, M., Astronomy The Evolving Universe 9th ed. Cambridge, 2002. Understanding Kepler's Second Law
The copyright of the article Understanding Kepler's Third Law in Mechanical Physics is owned by Paul A. Heckert. Permission to republish Understanding Kepler's Third Law in print or online must be granted by the author in writing.
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