Kepler's Laws

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 and so the three laws that 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 and so here's our Sun planet goes around the Sun like that the Sun is at one focus of the ellipse okay and an ellipse has to Pho sides there's one there and there's one on the other side by symmetry the Sun is that one of those both sigh all right this is a big step because earlier Tycho Brahe and other people thought that they were circular orbits and 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 all right the second law that he came up with was and let's draw our planet on their heading around the second law was the following equal areas in equal times okay 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 a 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 slingshot that you guys have heard the gravitational slingshot right when an object goes past a massive object it is moving fastest when it's closest to it and then it shoots back out all right 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 semi major axis cubed well if I take my ellipse and I draw the bisector of that ellipse and this is the semi-major axis the major axis is the entire length of the ellipse the semi major is half ax so he found that the period squared was proportional to the semi-major axis cubed which was a huge step in understanding the motion of the planets and what we're gonna see is that in fact Newton's universal law of gravitation that we just talked about you can use that to derive all of these laws all of Kepler's laws are derived from Newton's universal law of gravitation this is one of the reasons that it was such a huge step by Newton was 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 electric centricity they usually they talk about an eccentricity of one 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 okay 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 that semi-major axis is big it has to have a very long period and we know what that is T for the earth is one year but T 4 Halley's Comet what's the period of Halley's Comet it is about 76 years ok so we're gonna 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 the 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 in Montana I just looked up and there it was and now it is 76 years later it's here again so kind of rare that you would get to see it twice in your lifetime but it happens ok let's see how Kepler's third law is derive a ball from what Newton said okay so Newton said the following the force of gravity the magnitude of it is G m1 m2 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 it's attractive so let's take the 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 and sappy 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 that 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 what 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 in this case is the mass of the planet all right that looks pretty good but we don't know exactly what V is right 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 to PI R and if it goes all the way around in an amount of time T then that speed is just 2 pi R over T distance over time and that thing we got a square and we still have an R in the bottom and now we can simplify this quite a bit what do we get we've got G M sub s M sub P over R squared equals let's multiply this stuff out we've got a 4pi squared R squared we have an R on the bottom and then we end up with a capital T squared in the bottom ok and now we can cross out some stuff MP drops out if I cancel one of these ours 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 four PI squared divided by G M sub s and then I got a multiply R squared up over there so I end up with our cute but here's the deal right a circular orbit means that R is in fact the semi-major axis a major axis would be the diameter the radius is the semi-major axis and so we get T squared proportional to R cubed which is exactly what Kepler said the period squared is proportional to the semi-major 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 G M sub s this is Kepler's constant we know the mass of the Sun we know big G we obviously know four 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 Mars Jupiter and so forth all of those are going to be a similar equation with a different period and a different semi-major axis

Channel: Matt Anderson

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Id: HJ1fHPzhkZM

Length: 12min 42sec (762 seconds)

Published: Tue Jun 17 2014