 Greetings and welcome to the Introduction to Astronomy. In this video, we are going to discuss the explanation and the understanding of planetary motion. So this was done by two astronomers that we're going to look at in this section, and those were Tycho and Kepler. And this will really be a revolution in our understanding of how the planets moved. We looked at how Copernicus had given us the idea of perhaps a heliocentric universe, and Galileo had given us some evidence for that. But now we're actually going to look at some of the explanations and the understanding of how the planets really moved, different from what the Greeks had given us with their epicycles and the geocentric universe. In this case, we're going to look at how we can explain how things work in a heliocentric universe. So let's start off here going back a little bit just as a review. Early ideas, you know, what were they? Well, we did think that is the universe geocentric, is the Earth at the center. It seemed natural to assume that because the Earth did not appear to move, and one of the important things is that there was no measurement of parallax. Parallax is the apparent shifting of a nearby star relative to more distant stars. It would occur if the Earth were moving and observing the stars from different vantage points. However, since it was not observed, it was considered, you know, evidence for the geocentric theory. Now we've looked at Copernicus and he gave us the idea of a heliocentric universe and really suggested that the Earth was a planet just like the other planets that were known. On the other hand, this was not obvious. So it was not very obvious that this was the case. And there were still difficulties because we used circular orbits. And that is one of the keys here is that we were stuck on circular orbits. As long as you did that and you use these circular orbits, you could not easily explain the motions of the planets. And in fact, there were still difficulties in explaining the orbits. And Copernicus's followers were not able to eliminate epicycles because they needed them not to explain retrograde motion when the planets appeared to go backwards, but to explain the variations that occurred because the orbits were not truly circular. So let's look at what happened after Copernicus now. And what we find is that one of the next astronomers we want to look at is Tycho Brahe. Tycho was the last great observational astronomer in the pre-telescopic era. So he did observations and made observations of the star without the telescope. So this is a pre-telescopic observer. In fact, if you note his date of death here is 1601. That was nearly a decade, not quite, but nearly a decade before Galileo used his telescope. So he died years before the telescope was actually developed. He made some of the most accurate observations to date, maybe to the accuracy of a few arc minutes. So he was able to observe, if you remember an arc minute, 30 arc minutes would be the diameter of the full moon. So he was able to observe planetary and stellar positions to the accuracy of maybe a 15th to a 30th of the full moon, which is really good without a telescope. He also did this not for a short time, but for decades and gave us decades worth of data, observations of the planets and their positions. We found that his model did not really fit. His observations did not fit with the geocentric model. So he had some issues with the geocentric model in some cases because he observed things like comets and a supernova, a new star that occurred. Now earlier, under the Greek model, the heavens were unchanging. So things like comets and supernovae had to occur within the Earth's atmosphere. They had to be an atmospheric effect and that was because the heavens could not change, but the Earth could. So one of his problems was when he measured these, he could not measure their parallax, which he should be able to do if they were in the Earth's atmosphere. So that said that they were actually on the sphere of the stars and that the stars in the stellar sphere could change. So what he did was he actually proposed his own model of the universe that was different from both the geocentric and the heliocentric theory. He did not necessarily care for the heliocentric theory either because of parallax. He was not able to measure parallax and because he was a very good observer and able to observe positions more accurately than anyone ever had before, that if parallax could be measured, he would have been able to do it. And since he did not measure parallax, he convinced him that the Earth was not moving. So let's look briefly at his model here and what Tycho gave us was a little bit different model of the universe than the two that we've looked at before than either the geocentric or the heliocentric. In this case, the Earth is still at the center, so there's the Earth here at the center. The Earth is unmoving and that would explain the lack of parallax. However, the Sun would then orbit around the Earth and the Moon, so the Moon here and the Sun would orbit around the Earth, but all of the other planets, instead of orbiting directly around the Earth, orbit it around the Sun, so Mercury and Venus here orbiting around the Sun, Mars, Jupiter and Saturn orbiting around the Sun as well. So the Sun and the Moon are what orbit the Earth. The planets actually orbit the Sun. So this is what we have for Tycho's model of the universe, and it was very important as it was kind of a bridge between some of the changes. And it was one that actually stood up to some of Galileo's observations. Galileo, if you recall, observed the phases of Venus. Well, because Venus orbits the Sun in this model, this would predict the phases of Venus. So when Galileo made the observations, he could rule out a model like Ptolemy's that said the Earth was at the center and everything orbited around the Earth, because we would see a different set of phases for Venus. He could not rule out something like Tycho's model, even though it still had the Earth still, because the planets were going around the Sun, and we would still get those same phases. So what else came out of Tycho's observations? Let's look a little bit here at... So let's look at what Kepler did. Kepler was a mathematician, and he used Tycho's data and analyzed it. So he was the one who actually went and analyzed Tycho's data. And that was very important. Tycho had gathered all the data, but we had to see what was going on with it. And he made two great leaps that went even beyond what Copernicus had given us. If you recall, Copernicus had told us that the Sun was the center of the solar system, but he still used circular orbits and uniform speeds. Kepler was able to explain the planetary motion by getting rid of these two, and these two were the basis of Greek astronomy, that everything in the heavens moved in circles, and that everything moved at a uniform speed. So this really threw out the last remnants of Ptolemy. Copernicus had started it by putting the Sun at the center of the solar system. But Kepler kind of finished it up by saying that the circular orbits and uniform motion were abandoned. These, and we will find a better way of explaining planetary motion. And Kepler gave us three laws based on his studies of Tycho's data. So what did he find? Let's look at his laws here. His first law of planetary motion says that the orbits of the planets are ellipses with the Sun at one focus. So this is what we see. This is his first definition that the orbits are not circular. And this was the first time this had been done that we know of, so not circular orbits, but they were elliptical. Now we can see an ellipse here. An ellipse is kind of a squashed circle, and it has a circle has just a center. An ellipse has two foci. So one focus on this side and one focus on this side. The Sun is located at one focus, the other one is empty. And that means as the planet orbits around the Sun, sometimes it's closer to the Sun, and sometimes it's further away. Now some of the different terminology that we use for this, we've mentioned focus already. There is a center to an ellipse as well. We also have two axes, whereas a circle has a diameter and a radius. The ellipse has a major axis, which is the long axis that goes through the foci, and the minor axis that is perpendicular to that. We will also sometimes use what we call the semi-major axis. And what that is, is that is just half of the major axis. So semi-major axis would go from here to here, and that would be the semi-major axis. That's the average distance between the Sun and the planet. Now there's also another term that we need, and that is the eccentricity. The eccentricity of an ellipse says how squashed it is. The bigger the number, the more squashed the ellipses. In fact, an eccentricity of zero is a circle. So a circle would be zero in eccentricity. If you make that eccentricity closer to 1.9999, you're getting a very flattened ellipse. For the most part, the planets, the orbits, are very close to being circular. We see things with much more elliptical orbits, but those tend to be the comets. So most of the planets are not that far from a circle. So what the Greeks did wasn't that far off, but did throw off, but they were a little bit off and enough now that technology was getting good enough that we were able to start seeing the differences. Kepler's second law tells us, and we write it out here in detail, it says that a line joining the planet and the Sun sweeps out equal areas in equal intervals of time. That's a big long law there. That's what Kepler found when he made these observations. He'd map out the orbit, and he'd say that if it took one month to go from here to here, that this shaded area was the same as this shaded area as long as the time was the same, that if this was one month and this was one month, that the two areas that he measured were exactly the same. But what does this really mean for us? It really means that the planet is changing speed. So not only are they not orbiting in circles, but they are orbiting at different speeds. Sometimes they're moving faster. So we move faster when we're closer to the Sun, and we move slower when we're further from the Sun. So we call this position when it moves faster, that is the perihelion, the closest approach of the planet to the Sun, and when it moves slower is apheleon, which is when that planet is furthest from the Sun. So Kepler's second law is really telling us that the planet's speed is changing. It is not always moving at the same speed. Now Kepler's third law, he didn't come up with till a little bit later, and Kepler's third law tells us that the square of the period, the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis. Again, that's a big long definition there. We can write that as p squared equals a cubed. And what that means is that there is a relationship between the period, how long it takes the planet to go around the Sun, and its semi-major axis, which is the average distance between the planet and the Sun. So there's a relationship between these two, and that is what we see in this table here. Let's check this. We can look at this for all the planets, Mercury through Neptune, and even throwing Pluto in there. If we look at the average semi-major axis, we have the values given here. If we look at the period here, if we take this column and square it, and then take this column and cube it and divide those two, we find out that they come out very close to one, meaning that they are proportional. The little deviations here would be rounding errors. Point 39 is not precise. You'd have a few more decimal points there, and if you'd use them even more accurately, it would come out even closer to one. But you can see just from this quick calculation that the values are very close to one. So there was a relationship, and Newton will give us a little more explanation and allow us to use this as a way of being able to measure the masses of objects in space. Kepler did not have that at this point. So what did these laws actually mean? Well, what we have here is that, what is the meaning to these? And we want to look at that in terms of they're important because they overturned a millennium, a thousand years of focus on circular orbits and uniform motion. This was not just these were postulates of the Greeks, but these are also things that Copernicus thought and that all astronomers thought for thousands of years that everything in the heavens moved in circular orbits and at uniform speeds. There was not yet a physical basis to Kepler's laws. Kepler's derived them empirically based on observations. They applied only to planets going around the sun. They didn't apply to any other objects. That was all that Kepler looked at. I say yet because that's coming. Sir Isaac Newton that we'll look at will be able to derive Kepler's law based on his law of gravitation. So he will actually be able to tell us why these are all the case. So this will actually make them more general, applying them universally. So they will actually apply universally once Newton gets through with them and we'll be able to use them to look at orbits of other objects, to look at moons going around planets, to look at stars orbiting each other or stars orbiting the galaxy or galaxies orbiting each other. So let's finish up here with our summary and what we find, what we've looked at in this section was Tycho, we talked about Tycho who made many years of careful observations of the planets, extremely detailed and the most accurate to that time. His observations allowed Kepler to come up with three laws of planetary motion that we discussed and these laws were empirical. They were based on observation. We did not yet have the physical understanding that would have to wait for Sir Isaac Newton and understanding of gravity to be able to explain the physical basis for the laws that Kepler found. So that concludes our lecture on understanding the orbits of the planets and the work of Tycho and Kepler. We'll be back again next time for the next lecture. So until then, have a great day, everyone and I will see you in class.