 Greetings and welcome to the Introduction to Astronomy. In this lecture we are going to talk about planetary motion and how we've been able to explain that. We looked at the early Greek explanation, but how this changed as our understanding of the universe changed during the Renaissance. And we will look at two astronomers, Tycho and Kepler, who helped to get a lot of the information needed to be able to better understand how the planets actually move. So what do we see? Well, first of all we talked about the universe being geocentric, and that was the belief of the early Greeks, because it seemed intuitive when you walk outside it does not feel like we are moving through space. And there was no measurement of parallax, which would occur if the Earth were moving. Now we talked earlier about Copernicus and his heliocentric universe, and that suggested that the Earth was a planet, however it was not obvious. There were still difficulties in explaining how the planets moved using circular orbits, and he was not able to fully eliminate epicycles because he still depended on circular orbits. So let's look a little bit about how this changed, and one of the astronomers was Tycho Brahe, and we see him at his observatory here, and he lived from 1546 to 1601. He was the last great pre-telescopic observer, so no telescope involved, as you recall. We talked about Galileo, started using the telescope around 1609 or so, and that was really the earliest advent of telescopes, so he was doing all of this by looking at the stars and using great measuring devices like the quadrant shown here to be able to measure the positions of the stars and planets. He collected decades worth of data more accurate than anyone had done before, and he found that his observations did not fit with the geocentric model. So he actually gave his own model of the universe because he also felt that his observations did not fit with the heliocentric model. So let's look briefly at what he did, and the model that he gave us was something like this. So there is the Earth at the center. So Earth was still at the center of his universe. The Sun orbited around Earth, but the planets orbited around the Sun. So what does this do? Well, it explains some of the motions of things like what Galileo would eventually find with the phases of Venus, even though that had not yet been discovered at this point. And that explained that Venus would go through a complete cycle of phases. It explained the lack of parallax because Earth was not moving. So Sun and Moon orbit Earth, the planets however orbit the Sun, and then there is the great sphere of the stars out there. So this did fit Tycho's observations and what he was able to see, and accounted for the fact that even with his extremely accurate measurements he was unable to detect parallax. Now working for Tycho was Johannes Kepler, who was the mathematician who analyzed the data. And he was able to explain planetary motion in a much simpler manner. So he was able to explain planetary motion by getting rid of these two things, getting rid of circular orbits and uniform motion, which had been the belief since the time of Aristotle. So he was getting rid of things that had been believed for well over a millennium pushing up to almost two millennia that we had believed that everything in the sky moved this way. And he gave us three laws of planetary motion that we will look at over the coming slides. So first Kepler's first law of planetary motion states that the orbits of the planets are not circles but are ellipses, with the sun at one focus. So what is an ellipse? Well we show an ellipse here. How can you draw an ellipse by putting two thumbtacks in a piece of cardboard and using a circular piece of string, so a string tied with a knot, and then putting a pencil there. So all the points on the ellipse are equally distant from these two points. So if you add this length and this length, no matter where you put that point on the circle on the ellipse, you will get those two. Now it leaves us with a number of different parts of the ellipse, the center right here at the central portion, as you might expect. There are two foci labeled F1 and F2 in the image here. And then you have two axes, whereas a circle has a diameter, an ellipse has a major axis, which is the larger region, larger axis going across, is the furthest through the center, and a minor axis, which is the smaller, shorter region that goes through the center. So when you go the shortest possible and the longest possible, you get the major and minor axes. We also have, ellipses will also have an eccentricity, depending on how squashed they are. So what do we see? Well, let's look at some ellipses here. And we have an example of some ellipses here. Let's clear that. And we have an ellipse of an eccentricity of zero is a circle. So this is what we had thought for a long time, that all of the planets orbited in circles. As the eccentricity gets larger and larger, the circle gets more and more squashed. Note how you don't really see much of a difference for a while. It's hard to tell the difference between even E equals 0.2 or 0.3. Just looking at that, it still looks like a circle. When you get up to things like 0.95, you do see these. Now most of the planets fit in this range, this first row. So in fact, all of the planets will fit in that range. Mercury being at about 0.21 is just a little bit more elliptical than this. However, we could look at orbits of things like comets, which do have much larger eccentricities. So there are things that orbit like this, but all of the planets having quite circular orbits kind of explains why we did not pick up on the fact that they were different very quickly. Now so that's his first law, said that not circles, but things orbit in ellipses. The second law stated that a line joining the planet and sun sweeps out equal areas in equal intervals of time. What does this mean? Well it means that the planet is changing its speed. So in going from 1 to 2 here takes a certain amount of time, and going from 3 to 4 takes exactly the same amount of time. The area of A here and B here are exactly the same. So we have a long, skinny sector here, a shorter, fatter sector here, and those are exactly the same amount of area. What it's really saying is that the planet is changing speed, moving faster when it is close to the sun, we call that perihelion over here, and moving slower when it is further from the sun at apheleon. So the planet is changing speed. Let's take a look at that. That works. And here we see a planet orbiting. And watch as it comes closer. You see it speed up, and then it slows down. And what Kepler found is that if you connect the two with the line, that would sweep out an equal area. So every day's worth of orbit or every month's worth of orbit would be exactly the same amount of area between the planet and the star. Now, Kepler also gave us a third law of motion, a little bit later than the first two. His third law of motion said, and we'll say it two ways here, but he said that the square of the orbital period is directly proportional to the cube of the semi-major axis. Now, I didn't tell you what a semi-major axis is. If you remember, we had something like an ellipse here. The major axis was the line going straight through. The semi-major axis is half of this. So the semi-major axis was the distance from the center to one edge of the ellipse along the major axis. That's actually the average distance between the planet and the star. Now, we can write this in equation form as p squared equals a cubed. And if we plot those out, we can see that if we look at the periods versus semi-major axes here on this logarithmic plot, that they form a straight line, that there is a pattern or a relationship between the orbital period and the semi-major axis. And we will come back to this again later when we talk about Sir Isaac Newton and how he adjusted some of Kepler's findings. So what do these mean? Well, Kepler's laws are very important because they overturned more than a millennium of focus on two things, circular orbits and uniform motion. It took a long time. Remember, Copernicus did not get rid of these. However, we have no physical basis. Why are they occurring? They are empirical laws. He found them by looking at Tycho's observations. Yet means it's coming. We will find a physical basis for these with Sir Isaac Newton, who we will discuss shortly. And he was able to derive Kepler's laws from his laws of gravitation, making them more general, as right now they apply only to objects orbiting our sun. However, Newton will apply them to everything in the universe. So let's go ahead and finish up with our summary. And what we looked at this time, we looked at Tycho Brahe and his decades of careful observations, which allowed Kepler to come up with three laws of planetary motion to explain how planets orbit in the solar system. Note that they were empirical based on observation. The physical understanding of why they were the case would have to wait for Sir Isaac Newton, who we will discuss in a coming lecture. So that concludes this lecture on the explanation of planetary motion. We'll be back again next time for another topic in astronomy. So until then, have a great day, everyone, and I will see you in class.