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If the Moon orbited Earth twice as fast, but in the same orbit, the frequency of solar eclipses would (a) double; (b) be cut in half; (c) stay the same.

The answer key states that the correct answer is (c), but I don't see how this can be correct. Setting aside the violation of Kepler's 3rd law in the statement above, the synodic month would be 14.8 days if the Moon orbited twice as fast. Under normal circumstances, at least two eclipses (solar or lunar) can occur during an eclipse season, which is a period of 31-37 days occurring every six months, where the Moon's orbit and Sun's path (differing by 5º) overlap. At least one new moon and one full moon will occur during an eclipse season, since a synodic month (29.5 days) is shorter than an eclipse season. This means that at least one solar eclipse (formed during a new moon) and one lunar eclipse (formed during a full moon) will occur during an eclipse season. Now, if the Moon orbits twice as fast, shouldn't there be twice as many eclipses? (There would be at least 2 full moons and 2 new moons occurring in 31-37 days). Am I missing something?