History of Astronomy : Kepler's Laws and Beyond

Professor Dave here, let's talk about Kepler. The Copernican Revolution, which placed the Sun at the center of the solar system rather than the Earth, was a major event during the European Renaissance, a time when scientific thought truly began to flourish. In this climate, a figure like Danish astronomer Tycho Brahe was motivated to wield his status and wealth to build the most sophisticated instruments that had ever been built for the purpose of studying the heavens, and employ them to collect the best astronomical data yet collected at the time. Through meticulous observation of the planets and other objects, the Copernican model was corroborated much further, but the stage was set for further refinement of the model as well. Copernicus thought that the planets traveled in perfectly circular orbits around the Sun, but this changed when Brahe's young assistant, Johannes Kepler, got his hands on this new data. He used it to determine that planets do not trace circles as they go around the Sun. They trace ellipses. These shapes are discussed at great length in the algebra portion of my mathematics playlist, so if you want to know a lot about ellipses, click on this card now, but otherwise it will be enough to simply understand that unlike a circle with its one center, an ellipse has two foci, and the sum of the distances from the foci to a particular point on the ellipse is a constant. When these foci get very close together, it starts to resemble a circle, since a circle is just an ellipse, with foci that coincide. This is why the orbits of the planets seem circular, and why we operated under that assumption for so long. The brilliance of Kepler was to realize that the planets actually follow elliptical paths, with the Sun acting as one focus of the ellipse. The point in a planet's orbit that is closest to the Sun is called perihelion, and the point furthest away is called aphelion. The distances to these points are very close for most of the planets, because the ellipses have very low eccentricity, but nevertheless, they are not circles, they are ellipses. This fact is the first of three laws, known as Kepler's laws, which essentially marked the birth of celestial mechanics. The second of Kepler's laws says that the orbital speed of a planet will vary according to its distance from the Sun. It slows down when further away, and speeds up when closer in. A little more formally, it says that a planet will sweep out equal areas over equal time intervals. So if we take this area, A, of the ellipse involving a part of the orbit that is closer to the Sun, and this other equal area, A, of the ellipse involving a part of the orbit that is further away from the Sun, the distances traveled by the planet, either C to D or M to N, in order to sweep out these areas, are of different magnitudes. However, the planet moves faster when close, and slower when far away, in a manner that results in precisely equivalent time intervals. And Kepler's third law states that the amount of time a planet takes to orbit the Sun, or its period, is related to the length of the semi-major axis. That's this length here, or half the major axis, which is the longer axis of the ellipse. Specifically, P squared is proportional to A cubed, when measured in years and astronomical units, and the precise relationship will depend on the mass of the planet. These laws are derived from observation, and when we use these laws to predict the positions of the planets at any future moment, we find that they are extremely precise, which is all that any good science can hope to do. These laws were used to calculate the relative distances of all the planets from the Sun, and had other practical applications as well, but the impact of these laws was much deeper than the merely practical. These laws represented the first time that a handful of extremely simple mathematical formulas were able to describe and predict the motions of the heavens to the utmost precision. It was undeniable proof that the universe obeys mathematical principles that are decipherable by mankind, which is the basis of the revolution that produced modern scientific thought. We were no longer helpless victims of the natural world, tumbling in darkness. We ignited a spark that lit the candle, that shined a light, such that we began to understand our surroundings. Roughly simultaneous to Kepler, an Italian scientist, Galileo Galilei, was observing the same night sky, but focusing on entirely different observations. He used the best telescopes of the time to see that the Moon had distinct terrestrial features, like mountains and craters, just like the Earth. This transformed the Moon into a world unto itself, rather than just a mysterious glowing disk. He noticed the dark sunspots on the Sun, which changed position over time, thereby deducing the rotation of the Sun. He observed Jupiter's moons, and as they orbit Jupiter, this proved that not everything in the solar system orbits around the Sun or the Earth, no matter which of the two you put at the center. He saw what we would later understand to be Saturn's rings. He observed the phases of Venus, demonstrating that it must be in orbit around the Sun, just as the Moon is in orbit around the Earth, the final nail in the coffin for the geocentric model. And he observed far more stars than were previously visible to the naked eye, indicating that the universe is much bigger than was thought at the time. Although he is often better remembered for being the father of the scientific method, as well as all of his contributions to classical mechanics, Galileo did as much as anyone else ever did to rewrite our perception of the cosmos. This earned him the same contentious status with the Catholic Church as Giordano Bruno, but Galileo was a little less stubborn, recanting his work under pressure, and he was sentenced only to house arrest, living the last of his years confined to his home. The year that Galileo died was the same year that Sir Isaac Newton was born, and his contributions to science were immeasurable. Beyond his laws of motion, he also provided the first accurate description of the gravitational force, which was then used in a stroke of genius to demonstrate that objects fall to the Earth for the same reason that the Earth goes around the Sun, and he thereby demonstrated that all three of Kepler's laws were consequences of his own laws of motion and universal gravitation. All of this is described in detail in my Classical Physics series, so head there for more information if you're interested. Otherwise, let's continue our story. From Ptolemy to Newton, our picture of the solar system changed dramatically, leaving us with a perception about the universe that did not change much until the 20th century. We discovered two more planets beyond Saturn, as well as a host of other small objects, but heliocentrism and the elliptical paths of the planets around the Sun were locked into place. With some historical perspective taken care of, let's move forward and take a closer look at each major object in the solar system, so that we can become better acquainted with these beautiful worlds.