MIT OpenCourseWare http://ocw.mit.edu 8.01 Physics I: Classical Mechanics, Fall 1999 Please use the following citation format: Walter Lewin, 8.01 Physics I: Classical Mechanics, Fall 1999. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsMIT OpenCourseWare http://ocw.mit.edu 8.01 Physics I: Classical Mechanics, Fall 1999 Transcript – Lecture 22 Today, I will talk to you about elliptical orbits and Kepler's famous laws. I first want to review with you briefly what we know about circular orbits, so I wrote on the blackboard everything we know about circular orbits. There's an object mass little m going in a circle around capital M. This could be the Sun; it could be the Earth. It has radius R, circular. We know there in equation one how to derive the time that it takes to go around. The way we found that was by setting the centripetal force onto little m the same as the gravitational force. Also, the velocity in orbit-- maybe I should say speed in orbit-- also follows through the same kind of reasoning. Then we have the conservation of mechanical energy-- the sum of kinetic energy and potential energy. It's a constant; it's not changing. You see there first the component of the kinetic energy, which is the one-half mv-squared, and then you see the term which is the potential energy. We have defined potential energy to be zero at infinity, and that is why all bound orbits have negative total energy. If the total energy is positive, the orbit is not bound. And when you add these two up, you have an amazing coincidence that we have discussed before. We get here a very simple answer. The escape velocity you find by setting this E total to be zero, so this part of the equation is zero. Out pops that speed with which you can escape the gravitational pull of capital M, which is the square root of two times larger than this V.And I want to remind you that for near Earth orbits, the period to go around the Earth is about 90 minutes, and the speed-- this velocity, then, that you see in equation two--is about eight kilometers per second, and the escape velocity from that orbit would be about 11.2 kilometers per second. And for the Earth going around the Sun, the period would be about 365 days, and the speed of the Earth in orbit is about 30 kilometers per second, just to refresh your memory. Now, circular orbits are special. In general, bound orbits are ellipses, even though I must add to it that most orbits of our planets in our solar system--very close to circular, but not precisely circular. But the general solutions call for a elliptical orbit. And I first want to discuss with you the three famous laws by Kepler from the early 17th century. These were brilliant statements that he made. The interesting thing is that before he made these brilliant statements, he published more nonsense than anyone else. But finally he arrived at two... three golden eggs. And the first golden egg then is that the orbits are ellipses-- he talked always about planets-- and the Sun is at one focus. That's Kepler's law number one. These are from around 1618 or so. The second... Kepler's second law is-- quite bizarre how he found that out, an amazing accomplishment. If you take an ellipse, and you put the Sun here at a focus-- this is highly exaggerated because I told you that most orbits look sort of circular-- and the planet goes from here to here in a certain amount of time, and you compare that with the planet going from here to here in a certain amount of time, then Kepler found out that if this area here is the same as that area here, that the time to go from here to here is the same as to go from there to there. An amazing accomplishment to come up with that idea. And this is called "equal areas, equal times." Somehow, it has the smell of some conservation of angular momentum.And then his third law was that if you take the orbital period of an ellipse, that is proportional to the third power of the mean distance to the Sun. And he was so pleased with that result that he wrote jubilantly about it. I will show you here the data that Kepler had available in 1618, largely from the work done by, of course, astronomers, observers like Tycho Brahe and others. You see here the six planets that were known at the time, and the mean distance to the Sun. For the Earth, it is one because we work in astronomical units. Everything is referenced to the distance of the Earth. This is 150 million kilometers. And it takes the Earth 365 days to go around the Sun; Jupiter, about 12 years; and Saturn, about 30 years. And then when he takes this number to the power of three and this number squared, and he divides the two, then he gets numbers which are amazingly constant. And that is his third law. The third law leads immediately to the inverse square dependence of gravity, which he was not aware of, but Newton later put that all together. But he very jubilantly writes: And he wrote that in 1619. So, orbits in general are ellipses. And now I want to review with you what I have there on the blackboard about ellipses. You see an ellipse there? Capital M-- could be the Earth, could be the Sun-- is at location Q. The ellipse has a semimajor axis A, so the distance P to A-- perigee to apogee-- is 2a. If M were the Earth, capital M, then we would call the point of closest approach "perigee," and the point farthest away from the Earth, we would call that "apogee." If capital M were the Sun, we would call that "aphelion" and "perihelion." So you see the little m going in orbit; you see the position vector r of q. It has a certain velocity v. And so the total mechanical energy is conserved. The sum of kinetic energy and potential energy doesn't change. The first term is the kinetic energy-- one-half mv squared, and the second term is the potential energy--no different from equation three for circular orbits, except that now capital R, which was a fixed number in a circle, is now a little r, and little r changes, of course, with time. Also that velocity, v, in that equation number five will also change with time because it's an elliptical orbit. It will not change in time in equation number two and in equation number three. Now I give you a
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