Kepler's Laws of Planetary Motion

Way back in the ancient Greek and Roman times, people believed that the Earth was at the center of the universe. Everything else revolved around it, the other planets, the moon, the sun and the stars. They were thought to be fixed to celestial spheres that were perfect and never changed. Over time that view was challenged and disproved. Copernicus was one of the first people to suggest that Earth was not at the center of everything. He said that we instead revolved around the sun, in a circular path called an orbit. Copernicus sparked a major revolution in the history of science, because his theory challenged the most fundamental ideas about how the universe worked. However, he failed to explain why the planets move in orbits. We now know that it's because of gravitational forces, as Newton explained. But the subject of this video is neither Copernicus nor Newton. Today we will look at the work of a man who saw the beauty of the solar system and found three laws to describe how it worked. That man was Johannes Kepler, and this is an introduction to Kepler's laws of planetary motion. Kepler was a mathematician by profession. In the early 1600s, he worked as an assistant and then successor to Tycho Brahe. Here is a statue of the two of them in Prague, capital of the Czech Republic. What Brahe did was collect lots of accurate data on the motion of stars and planets across the sky. He used instruments like the astrolabe to do this. The telescope had not yet been invented. After his death, Kepler inherited all that data and spent years analyzing it. He noticed some interesting things about how the planets moved. People at that time assumed that planets moved in spherical orbits, but Kepler found that this wasn't true. He came to the conclusion that planets moved in ellipses around the sun. An ellipse fitted the data, but a circle did not. An ellipse is like a circle that has been squashed in one direction. It has two focal points, or foci. According to Kepler's analysis, every planet has an orbit like this. The sun is not in the center of the orbit, it's located at one of the two foci. This is Kepler's first law of planetary motion. All planets move in elliptical orbits with the sun at one focus. Here is another diagram of an ellipse. I've drawn on two straight lines which cross at its geometric center. The longer line is called the major axis of the ellipse, which is the longest line you can draw between any two points on the outline that goes through the center. The shorter line is the minor axis, the shortest line you can draw between two points on the ellipse that goes through the center. From this point of view, a circle is really a special type of ellipse. In a circle, the two foci are in the same position, and the major and minor axis have the same length. Orbits of the planets are quite close to circular, but they are true ellipses with two foci. The sun always sits at one of these focal points. Kepler's second law takes some more explanation. It says a line that connects a planet to the sun sweeps out equal areas in equal intervals of time. What does that mean? It means that the speed of each planet changes depending on where it is in its orbit. Consider the orbit of the earth. It takes one year for earth to complete a revolution around the sun. Before Kepler came along, people thought that earth moved through space at a constant speed. It turns out that the speed of any planet changes during the year. When earth is furthest from the sun, it moves at its slowest speed. When it's closest to the sun, it achieves its highest orbital speed. Consider a line drawn between the center of the earth and the center of the sun. As earth moves in its orbit, the line moves too. We say it sweeps out an area of space. Suppose we follow that line over the course of a month. One month is 1 12th of the time it takes earth to complete an orbit. As you can see, it moves some distance around the sun, and the line between them sweeps out an area A. Then we skip ahead to a different month in a different season. The speed of the earth is slower, and it is further away from the sun. We find that the line between them sweeps out this area, B. What Kepler said is that areas A and B are exactly equal. They are the same area. This happens not just for the earth, but for all the other planets as well. A line that connects a planet to the sun sweeps out equal areas in equal intervals of time. The speeds of the planets are not constant, but this area is. Finally, we have Kepler's third law. This one is a heavily mathematical statement. The square of the period of a planet's orbit is proportional to the cube of the mean radius of its orbit. You can express the law as an equation like this, but what in the Well, the square of the period is T squared. The period is how long it takes for a planet to complete one orbit around the sun. In the case of the earth, that's 365 days. Mercury, the planet closest to the sun, has a period of 88 days. T squared means we square that period, or multiply it by itself. 3 times 3 is 9, for example, so we say that 9 is equal to 3 squared. The cube of the mean radius is R cubed. Mean radius is the average distance between a planet and the sun. For the earth, that is 150 million kilometers. For mercury, it is a mere 58 million kilometers. R cubed means that we take that number and cube it, or multiply it by itself, twice. Kepler's third law says that the two quantities are proportional. In other words, if you take T squared and divide it by R cubed, you get a number that does not change. It is a constant in the orbit of any planet, and I mean any planet. This applies to mercury just as well as it applies to earth, saturn, venus, or any other orbiting body in our solar system. What's more, the constant has the same value for all planets. Earth doesn't give you a different number from mercury, it gives you the same number. If you take T squared divided by R cubed for one planet, and take T squared over R cubed for a different planet, you get the same result. That was a fantastic revelation. This discovery may seem trivial today in a society that has seen robots explore Mars and land on comets, but it was the pinnacle of Kepler's work, the cherry on the cake that was baked from Tycho Brahe's observations. The only thing Kepler failed to explain was why the planets move in elliptical orbits, and why these mathematical constants arise. That mystery led to Newton's theory of gravitation, another breakthrough that I will save for another video. For now, have a go at the following exercises to get some further insight into Kepler's laws, and how to apply them. You are ready to tackle physics at an astronomical scale. Good luck, and thank you for watching.