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Problem of the Week 1250

The Loopy Lunatic

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It comes as a surprise to many that the orbit of the moon around the sun has no loops in it. Indeed, it is a convex curve not very different from the orbit of Earth around the sun.

Imagine a moon orbiting a planet orbiting a star where:

  • distance planet to star = 93,000,000 miles
  • planet's orbital period = 365 days
  • moon's orbital period = 27 days
  • moon's distance to planet = L miles
  • the moon's orbit is "prograde," i.e., in the same direction as
    the planet moves around the star

What is the smallest value of L so that the orbit of the moon around the start has a loop? Is nonconvex?

Assume all orbits are circular and all lie in the same plane (so that "loop" and "convex" have clear planar meanings).


Helmer Aslaksen's The Orbit of the Moon around the Sun is Convex!

Noah Samuel Brannen, "The sun, the moon, and convexity," College Mathematics Journal 32 (2001) 268-272.

Laurent Hodges, "Why the moon's orbit is convex," College Mathematics Journal 33 (2002) 169-170.

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13 October 2017