8. Centripetal Forces & Gravitation

Overview of Kepler's Laws

# Anderson Video - Kepler's Laws

Professor Anderson

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>> Hello class. Let's talk a little bit about Kepler's laws. Remember Johannes Kepler was Tycho Brahe's assistant. And Tycho Brahe collected all this data on the universe, the stars, the planets, all these various motions. And he was trying to make sense of planetary orbits. But he was never able to fully complete the model and his assistant, Johannes Kepler, picked up where he left off and was able to come up with a model for planetary orbits. So the three laws he came up with we now know them as Kepler's Laws, are the following. The first is that orbits are elliptical. And they are elliptical with the sun at one focus. Okay, so here's our sun. Planet goes around the sun like that. The sun is at one focus of the ellipse. An ellipse has two foci. There is one there and there is one on the other side by symmetry. The sun is at one of those foci. Alright, this is a big step because earlier Tycho Brahe and other people thought that they were circular orbits. The reason they thought they were circular orbits is because when you look at the orbit of the earth around the sun it is nearly circular. Okay. But other planets have much more eccentric orbits. Alright, the second law that he came up with was -- and let's draw our planet on there heading around. The second law was the following. Equal areas in equal times. So what does that mean on our picture here? What it means is as this planet sweeps out this ellipse about the sun, if I take a stopwatch and I measure how far it moves in some amount of time and then I shade that area, I would get some number. But if I take the stopwatch and I do the same amount of time at the far end of its elliptical orbit and I map out that area, I get the exact same number. Equal areas in equal times. So if this is one month of an orbit, you get some number. One month of an orbit you get the same number. This necessarily tells you that the planet is moving slowest out here in order to get the same area as this region where it is moving fastest. And the planet kind of zips in towards the sun, goes back out, slows down a whole bunch and then comes and does it again. And this is sort of like the sling shot that you guys have heard. The gravitational sling shot, right? When can object goes past a massive object it is moving fastest when it's closest to it and then it shoots back out. Alright, the third law that Kepler came up with, which is a little bit more difficult to see is the following. The period squared is proportional to the semi-major axis cubed. In words, that's what it is. In math it looks like that. T squared is proportional to A cubed. T is the period. So for the earth the period would be one year. What is this semimajor axis cubed? Well, if I take my ellipse and I draw the bisector of that ellipse, then this is the semimajor axis. The major axis is the entire length of the ellipse. The semimajor is half of that. So he found that the period squared was proportional to the semimajor axis cubed which was a huge step in understanding the motion of the planets. And what we're going to see is that in fact Newton's Universal Law of Gravitation that we just talked about, you can use that to derive all these laws. All of Kepler's laws are derivable from Newton's Universal Law of Gravitation. This is one of the reasons that it was such a huge step by Newton, he sort of tied all the planets together. He tied our entire solar system together, with one law. Which is pretty remarkable. Okay. What we said was highly elliptical orbits are very eccentric. Circular orbits have low eccentricity. Usually they talk about an eccentricity of 1. So, if I think about orbits that are reasonably circular that would be stuff like the earth. So if this is our sun, the earth has an orbit that looks nearly circular. It's still elliptical but it's nearly circular. Whereas Halley's Comet comes in very steep and shoots way back out. This might be something like Halley's Comet. And, Halley's Comet has a long period. Right? If this semimajor axis is big it has to have a long period. And we know what that is. T for the earth is one year. But T for Halley's Comet, what's the period of Halley's Comet? It is about 76 years. So we're going to see it again in 2060. You guys will likely be around to take a look for Halley's Comet in 2060 something. And definitely try to catch that with your telescope if you're around, it's a really nice sight to see. I saw it back in 80s. And in fact my grandfather, who was alive at the time, he looked at it with me when Halley's Comet came by and he said, "Oh yeah, there it is again". And I said, "What"? He goes "Yeah, I saw it when I was 8 years old standing in the fields of Montana. I just looked up and there it was. And now 76 years later it's here again". It's kind of rare that you would get to see it twice in your lifetime but it happen. Okay. Let's see how Kepler's third law is derivable from what Newton said, okay? So, Newton said the following, "The force of gravity, the magnitude of it, is GM1M2 over R squared. Any two masses are gravitationally attracted together by that magnitude. Okay, usually we put a minus sign in front of it but this just means that it is attracted. So let's take the example of a planet orbiting the sun and let's put it in a circular orbit. So the sun has mass M sub S. The planet has mass M sub P. And now this thing's going to go about the sun in a circular orbit. And it's a distance R from the sun. What do we know? What we know is there has to be a force on this planet to keep it moving in a circle. That force is just gravity. And so it's G mass of the sun, mass of the planet, divided by R squared. But we know if this thing is moving in a circle, those forces don't add up to zero. They add up to something else. What do they add up to? Mv squared over R. And the M is this case is the mass of the planet. Alright. That looks pretty good. But we don't know exactly what V is. V is the speed of this thing. But what we do know is that if it goes all the way around, it goes a distance 2 pi R. And if it goes all the way around in amount of time T then this speed is just 2 pi R over T. Distance over time. And that thing we're going to square and we still have an R in the bottom. And now we simplify this quite a bit. What do we get? We've got GM sub S M sub P over R squared, equals. Let's multiply this stuff out. We've got a 4, pi squared, R squared, we have an R on the bottom. And then we end up with a capital T squared in the bottom. And now we can cross out some stuff. MP drops out. If I cancel one of these R's I can do that. And if I multiply across by R cubed I can write a very nice formula, which is the following. T squared, let's multiply T squared up over there. Equals 4 pi squared, divided by GM sub S. And then I've got to the multiply R squared up over there so I end up with R cubed. But here's the deal, right. A circular orbit means that R is in fact the semimajor axis. The major axis would be the diameter, the radius is the semimajor axis. And so we get T squared proportional to R cubed. Which is exactly what Kepler said. The period squared is proportional to the semimajor axis cubed. And in fact, we can write down the following. It's equal to K sub S, times A cubed, where K sub S is called the "Kepler Constant" and it has a value of 4 pi squared divided by GM sub S. This is Kepler's constant. We know the mass of the sun. We know big G. We obviously know 4 and pi. And so we can write down a specific value for Kepler's Constant and it applies to everything in our solar system. Not only the earth but Mercury and Venus and Mars and Jupiter and so forth. All of those are going to obey a similar equation with a different period and a different semimajor axis.

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