A random walk through mathematics -- mostly through the random part.
Fun question: does there exist a natural or man-made satellite whose orbit around the sun does loop back on itself?
Let's see. So the moon travels 3% of the speed of the earth. We seek an orbital body which is 30x faster.The moon takes 27 siderial days to go around the earth (about). So the answer to your question is the same as the question "is there a satellite whose orbit around the earth is less than a day"?The answer is of course yes.
The Moon does not orbit the Earth, it orbits the Sun and the Earth perturbs its path. This can be verified by working out the relative gravitational pull the Moon receives from the Earth and Sun (ie, the Sun's is twice as great as the Earth's).
CarlBrannen: your analysis is incorrect. It would be correct if these satellites that orbit around the earth in less than a day took the same orbit as the moon. They don't.The fastest satellites, those in low-Earth orbit (period ~ 90 min), achieve less than 10 times the speed of the moon. Thus they, too, do not loop back on themselves.So, there are no man-made satellites (orbiting Earth) that fit the bill. Are there any natural ones?
For the sake of anyone still reading this post or reading it in the future, the answer is that there are satellites whose orbits around the sun loop back on themselves.There are no man-made ones, but there are some natural ones. A good example is Jupiter's innermost moon Io. It orbits Jupiter at ~17km/s, while Jupiter orbits the sun at ~13km/s.I think it's not that clear a priori that any such satellites should exist. The intuition for why they do is that if you take a planet really far away from the sun, it will be orbiting rather slowly; if you take an inner moon, it should orbit somewhat quickly. For a far enough planet and close enough moon, you expect looping.
Boris,thanks for looking that up.Next question, since I'm a probabilist: choose a planetary system at random. What is the probability that it contains Io-type satellites?Unfortunately, I don't think we know what the underlying measure on the space of possible planetary systems is. The space itself would actually be reasonably easy to define; the orbit of each body is specified by finitely many numbers. The hard part is either:a) knowing enough about how planetary systems form to figure out the distribution theoretically, orb) observing enough planetary systems to figure out the distribution empirically.(I'm not sure whether (a) or (b) is harder.)
There could be made-man satellites whose orbit loops back on themselves, it just requires a very elliptical orbit. Somebody who is keener than I might like to check the eccenticities of the Molniya and Sirius satellites (the maximum speed of an elliptical orbit is proportional to sqrt((1+e)/(1-e)), where e is the eccentricity.
Ok, ok, so the maximum speed of a satellite is the escape velocity of the planet it orbits (which in the case of the Earth is slower than the speed the Earth orbits the Sun). So we need to wait for NASA to put a satellite into an elliptical orbit around one of the gas giants.
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