Of moons and months and mathematical manipulation

This morning in Ethnomathematics we finally had a chance to spend some time trying to reconcile the "lunar" and "solar" calendars one might use were one stationed on Jupiter and decided to use the largest of the Galilean satellites, Ganymede, to reckon your "months."

A Ganymedean month (the time it takes to make one transit around Jupiter) is roughly 7.15 (Earth) days; meanwhile, Jupiter's tropical year (the time it takes the planet to make one transit around the sun) is 4331.57 (Earth days). Just as on Earth, where a tropical year doesn't contain an even number of lunar months, resulting in a "drift" between lunar-based and solar-based calendars, there are an uneven number of Ganymedean months in a Jovian year. What to do?

After a bit of numerical piddling and fiddling, we figured out two reasonably accurate ways of making the numbers jibe. Both rely on the fact that each Jovian year we have a "remainder" of 5.82 days that don't quite make up a full Ganymedean month. If we let 11 Jovian years pass, we've saved up 64.02 spare days...this figure is very close to 9 full Ganymedean months, which give us 64.35 days. Therefore, if we add 9 months every 11 years (which can be done in some systematic fashion, much as is done with the Jewish lunar calendar), we've add only 0.33 extra days. This overage is tantalizingly close to 1/3...so why not simply take away one day every three 11-year cycles? This day too can be chosen systematically, removed from the middle of the 17th year of the 33-year cycle it corresponds to, for instance.

Neat!

I'll leave it to my readers (I'm sure your curiosity is now piqued) to puzzle through the details of the other solution we arrived at, which had much the same flavor and involved slightly more frequent adjustments.

Fun stuff...now if we can only figure out how to make these systems fit nicely with the 365.24-day Earth tropical year, which some of our Jovian emigrés still insist on using...