UPDATE (6 February): Some solvers have asked me if I consider
searching in Wikipedia (or elsewhere) for how to solve the inverse square
equation in preparation for solving the inverse cubic equation to be in
violation of what I consider an "original" solution. The answer is that
originality of a solution is as defined here,
a link that I point to every month. I believe it covers the current situation
quite explicitly, and the answer is: yes, it is in violation. If you looked up
how to solve for the inverse square case after reading this riddle, you are
most welcomed to enjoy the riddle, but don't ask to be listed alongside people
who have solved it entirely by themselves.
One fine evening, on January of 1684, three men came out of a meeting of The Royal Society and headed out to a nearby coffee shop. Based on the conversation that ensued and the outrageous wager the three later made over a sum of money, one can only suspect that what they drank was somewhat stronger than coffee.
The three were Edmond Halley, Christopher Wren and Robert Hooke, and the topic under discussion was the hot one of the day: finding a formulation of gravity that would explain Johannes Kepler's laws of motion (which were published earlier in the same century).
Hooke bet that he could do it. Wren put up cash money to say that he can't.
Ultimately, Hooke never was able to collect on that bet, but Halley had the good sense to turn this problem to someone else, Isaac Newton, who managed to provide the solution: if a planet orbits a stationary star, and if the force of gravity applied by the star on the planet is directed towards the star and its magnitude is inversely proportional to the square of the distance between the planet and the star, the planet will follow Kepler's laws of motion. Namely, it will orbit in an elliptical path. (In the process, Newton also invented both Calculus and science as we know it. However, history books are silent on the question of whether he made Wren cough up any cash.)
Newton's discovery is nowadays referred to as "the inverse square law of gravity".
This month's question: suppose gravity would not have worked according to an inverse square law. Suppose, instead, that gravity would have worked according to an inverse cubic law. What stable orbits would planets around stationary stars hold, in such a universe?
To be clear, "stable orbit" means here a locus in space that describes a closed path around the star (not hitting into the star) that the planet will follow. There is no requirement for this orbit to be "stable" in the sense of being a "stable equilibrium".
Solvers can get an asterisk next to their name if they can, additionally, provide substantial additional characterization of the motions of such planets.
As usual, all answers should be accompanied by proofs.
List of solvers:Thomas Mack (2 February 16:35)
Dieter Beckerle (15 February 09:55)
Kan Shen (15 February 18:42)
Zijie Zhu (17 February 23:31)
Elegant and original solutions can be submitted to the puzzlemaster at riddlesbrand.scso.com. Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.
The solution will be published at the end of the month.
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