Kepler's Laws of Planetary Motion

The first known proposal that the sun is the center of the solar system was in 270BC by a Greek named Aristarchus of Samos. Nearly 1600 years elapsed until Polish astronomer Nicolaus Copernicus proposed the same - allegedly by reading the ancients and not by observations of the heavens. There were heavy debates going on regarding the structure and dynamics of the solar system through the 1500s and 1600s. Johannes Kepler was a German mathematician born in 1571. He was hired by Tyco Brahe (a Danish astronomer and nobleman who was living in Prague in the current day Czech republic) to analyze some of the best astronomical data in existence at the time.  There is a detailed account of the interesting history at this Wikipedia site: Kepler History. Ultimately it took Kepler over a decade to work out the math of the heavenly bodies by hand. This labor led Kepler to propose three laws of planetary motion.  The first two were published in 1609, and the third law in 1619. The laws were: 1. Planetary orbits are elliptical with the sun residing at one focus of the ellipse. 2. The planetary orbits sweep out equal areas in equal time.  See this graphic: Graphic. We will see in a future chapter that this amounts to conservation of angular momentum for orbiting systems. 3. Planetary orbital periods are related to the semi-major axis of their orbital trajectory by , where 'T' is the orbital period and 'a' is the semi-major axis of the orbit. This means that any planet orbiting our sun should have the same ratio as any other planet. NOTE: This third law relationship is very nearly true, but not exactly. We know that gravity holds planets in their orbits. By far, the strongest source of gravity is the sun, but for outer planets, the inner planets always help the sun by pulling inward. In this sense outer planets get pulled inward by a larger mass than the inner ones. The farther out they are, the more inner planets there are to help. So in a real sense, outer planets orbit as if the sun's mass were a little a larger. Without this extra 'help', all the planets orbiting the same sun with the same mass would have the same ratio. Kepler came up with these laws a few decades before Newton. It was convincing work, and therefore Newton's burden, when he came up with a law of universal gravitation, was to prove that his law of gravitation could reproduce these laws of Kepler. To do this, Newton needed to invent the mathematics of calculus (and differential equations), which he did. It was really this work that prompted Newton to publish his treatise on his laws of motion after two decades during which he just kept them to himself! THE FIRST LAW: We will not prove Kepler's laws using calculus and differential equations, since the required math is beyond the scope of this class. Without that mathematics, we cannot prove the first law - or that planetary orbits are elliptical. Rather we will assume planetary orbits are circular, and starting there see if we can come up with the second and third laws. Since a circle is an ellipse of zero eccentricity, this is a fine thing to do. It should also be noted that the eccentricities of the planetary orbits are quite small. From mercury to Neptune they are 0.2, 0.01, 0.02, 0.1, 0.05, 0.06, 0.05, 0.01. Since a circle has an eccentricity of zero, what we can take away from this is that the planetary orbits are all very nearly circular rather than being elongated ellipses. THE SECOND LAW: Regarding the second law for circular orbits, it is trivial. In a circular orbit, as we saw in the last section, there is only a centripetal force of gravity. That means the speed of the orbiting planet remains constant while it makes a steady turn around the sun. Since the orbit is circular, the distance to the sun is constant. So area swept out can be written as the radius of the orbit times the arc length over two (like little triangles), or The rate of change of this area (or the rate at which it sweeps) if we assume radius is constant, is just Since both r and v are constants, the planet sweeps out constant area in time. In the event of non-circular orbits, what we see is that the closer the planet is to the sun (r), the faster it travels (v) and vice versa. THE THIRD LAW: You will see that our result from the moon's orbit in the last section already proves this. Our expression with the sun's central mass tugging the planets inward instead of the earth's tugging on the moon is:

We use the mass of the sun since that mass represents the central object around which the planets are orbiting. Since all the planets orbit the same central sun, the left side of the expression is the same for every planet. That means the right side must be the same as well. In other words, we may write Kepler's third law as:

In doing such calculations, it is common to use earth-based units, or ones which call the period 1 yr (earth year) and the radius 1 AU (astronomical unit), which is the distance from earth to sun, by definition. Using such units, and describing other planetary periods and radii in them, makes the calculations trivial.