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# 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).

Sources:

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.