Welcome back. Today, we're going to be leaving the solar system, and starting to talk about some of the planets that have been discovered around nearby stars. But before we do so, we're going to need to review some basic astronomy and physics, talk about Copernicus and Kepler, and Newton's Laws. because these are the tools that astronomers use when they analyze their data from the radial velocity searches that have led to discovery of planets around nearby stars. So let's begin with Copernicus. It was Copernicus who shifted our view of our own solar system. Before Copernicus, most people believe that the Earth was the center of the universe, and that the Sun and the planets all revolved around the Earth. Not an unreasonable thing to see, to think about when you see the stars rotate in the sky around the Earth. Of course our picture today is different. And while the key people who helped change that picture was Nicolaus Copernicus. It was Copernicus who put the Sun in the center of the system. With the Earth, just one of many planets, orbiting around the Sun. For his work, Copernicus made a bank note. And here he is and here is his Copernican System. Next, we turn to Kepler. And it was Kepler who realized that the planets don't move on circles, but they move on ellipses. An ellipse has two foci centered around here. And we talk about an ellipse. We talk about the minor axis, the shortest distance between the planet and the star and the major axis. When we apply this to a planetary system, we have the point of closest approach and furthest approach as the planet moves around the star. Now, historically, there were astronomers back in Greek times, classical times, who thought about these ideas that the Sun was the center of the solar system. That perhaps even the idea that planets may have moved on elliptical orbits. I'll refer you to a fascinating film called Agora. It's a Hollywood movie that focuses on historical figure, Hypatia, really the first female mathematician of note, and an astronomer as well. And in that movie, they suggest that she developed the idea of ellipses before Kepler. She was one of the key people in developing our notions of circles, ellipses, and parabolas. And I won't ruin the story, but it's a fascinating one and a fun movie to download if you can get a copy of it. Let's go back to Kepler. One of the things that Kepler realized was that, as a planet moved along its elliptical orbit, its speed would change. And it would do so in a way in which it kept its velocity times the distance from the Sun, or its host star, constant. This means it's moving at constant angular momentum. So it was Kepler who realized that V times R is constant, so that the star would be moving fastest when it was closest to the star. R is small, V is big. Star moves, planets move fast here, slowest here. That the planet would move around the star, sweeping out equal areas in equal time. So that in order to cover this area here in the same time as this area here, it would move faster here, slower here. When we talk about ellipses, let me just remind you of these basic properties of how we relate. The Sun, the average, the semi major axis to the ellipse, to the aphelion and perihelion distances. Kepler also realized there's irregularity between the period of the planets orbit and its distance from the star. This is Kepler's Third Law. What's shown here is the data on the solar system planets. Kepler, of course, only had this data. He didn't know about Uranus, Neptune, and Pluto. And it was not until Galileo that we knew about Jupiter's satellites, but what we're showing here is all the modern data. What's plotted here is the relationship between the semi major axis, the characteristic distance the planet is from the star, and its period. And you see as the semi major axis increases, the period increases. And Kepler's Law says the periods squared scale as the distance cubed. So that we can put in say, the Earth's period and the Earth's distance. So that's 1 AU and, so the Earth spends a year going around the Sun, Jupiter. We can plug in Jupiter's distance and work out its period. And it will scale as period squared goes as distance cubed. This relationship holds not just for planets orbiting the Sun, but for moons orbiting a planet. So what you see is that Io, the closest in-planet, has the shortest period. Europa, whose period, as we talked about earlier, is double Io, its distance, we can plug that in, is going to be 2 to the 3, to the 2 3rds power times Io's distance. This 2 to the 2 3rds will give us back the right factor. Ganymede is at 4 times the distance, so Gam, 4 times the period, so its distance is 4 to the 2 3rds power. So now, let's take these laws and apply them to Eris. And that's what I'd like you to do next. And then we'll move on and talk about Newton's laws and get an understanding of where those Kepler's laws, come from, and how we can derive them from our understanding of gravity.