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Originally Posted by Iain Lambert
Did it, found them to be ellipses not circles. Not terribly ellipsoidal, to be fair, but then anything big with a large eccentricity brings itself close enough to other bodies after a while that things go decidedly three-body-problem, three-body like Shoemaker-Levy 9 if they are particularly unlucky.
Apart from anything else, David, are you suggesting that there are different physical laws for the planets than for comets? Because they are rather blatently in ellipsoidal orbits.
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Thanks Iain. For good measure one could add asteroids. Those outside the belt (such as Toutatis, the official asteroid of the BABB) move in very elliptical orbits.
On a terminology side, there are two meanings to "eccectric" that are being used in this thread. One is a measure of how far apart the foci of an ellipse are. The lower the eccentricity, the closer it is to circular. The other is an eccentric circular orbit. Here the eccentricity is how far from a point the center of a circular orbit is. In addition to epicycles and other tricks, Ptolomy, Tycho, Copernicus, and every other astronomer up to Kepler used eccentric circles to describe planetary orbits. As most have noticed, actual planetary orbits are ellipses with very small eccentricities and, as such, close to circluar. What David fails to realize is that close only counts in horseshoes (and hand grenades
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As to your mathematical question, the orbital velocity varies with the inverse of the square root of the radius. The actual formula is
V = sqrt(G*Ms/R)
Where G is the gravitational constant and Ms is the mass of the sun. You can derive this from Newton's gravitational law and some elementary classical physics, so I'll leave it as a problem for the student as my profs used to say. Notice that it's independent of the mass of the orbiting object.
Now in an acutal orbit, that radius is constantly changing, so the velocity changes around the orbit. The planet speeds up and slows down as it goes around. That's what Kepler's second law points out. What is constant is the period of the orbit. Kepler's third law shows that the square of the period is proportional to the cube of the orbit's semi-major axis.