Treebeard's Stumper Answer

Full Moon Birthday
My birthday is this weekend, close enough to the full Harvest Moon on October 5 (at about 1 p.m.) that I'll pretend that's the day. My question is: How many years will I have to wait for my next full moon birthday? I imagine two wheels starting together, but spinning away at their own rates. I think the important facts are that successive full moons are 29.5306 days apart, and a year is 365.2422 days long. Or is it much more complicated? This is an easy question to ask, but I surely don't know the answer to this one!
After the full Harvest Moon on my birthday last Monday, I'll have to wait 19 years for my next full moon birthday, on October 5, 2017. In 19 years, there are 19 x 365.24 = 6,939.56 days. In 235 full moons, there are 235 x 29.53 = 6,939.55 days, which is almost the same. After 19 years and 235 full moons, the whole cycle of full moon dates will very nearly repeat. This 19 year Metonic Cycle has been known since Babylonian times as the common ground between lunar and solar calendars. But how did they figure it out originally?Note: I got the answer of 19 years by brute force. Pick a year, multiply by 365.24, then divide by 29.53 and check how close the result is to a whole number. 19 years is the first interval with a small enough difference that the year and closest full moon fall on the same day. Then I verified the result with an almanac. My calculations are available in a separate data table.
I'll have other near full moon birthdays. In just 3 years in 2001, the full moon will be on October 2, 3 days away. In 2006 and 2009, the full moon will be just a day or so away. After 19 years, the full moon will be just 1 1/2 hour away. To get closer, I have to look much in the future.
The cycle of full moon dates very nearly repeats after 19 years, but not quite. The full moon this month was at exactly 1:12 p.m. The full moon in 2017 will be at about 11:42 a.m., about an hour and a half earlier. That doesn't effect the date, but similar changes could change the date if the full moon was close to midnight. My real birthday is on October 4, but there is no October 4 full moon until 2074 after four 19 year cycles. If it wasn't for this regression, I would never have a full moon birthday. I assume that in the long run, all days have equal numbers of full moons. I see calendar days as a digital thing, it's the 4th or the 5th. But lunations are purely analog, so there's some wobble.
There is a math puzzle here. In the above calculation, it turns out that the full moon comes a bit before the year. But 19 x 365.2422 = 6,939.6018 days and 235 x 29.5306 = 6939.6910 days, a bit after the year. It makes a difference whether I figure the month as 29.53 or 29.5306 days. Graybear notes:
The duration of one lunation is not exactly 29.5306 days  no matter how many decimal places you use, it's still an approximation, but we will assume for the purposes of this solution that it is exact. Likewise, the length of the year is not exactly 365.2422 days.At least this shows how small differences add up. Graybear points out that leap years make the problem even worse.I suspected 19 years was the answer, since I've come across it before while reading about archeoastronomy. There are 19 stones in one of the rings at Stonehenge. There are 19 turns in the large spiral at the Anasazi Sun Dagger site at Fajada Butte in Chaco Canyon. The number 19 keeps coming up, so I tried it with my calculator right off, and it worked.
The Metonic Cycle is named for the astronomer Metros, circa 440 BC, who developed the Greek calendar that existed before the Roman Julian calendar. This was a 19 year calendar that was both lunar and solar, with leap days and leap months to keep everything working.
It's much easier to notice a new or full moon than the winter or summer solstice. But a lunar calendar quickly gets out of sync with the seasons since 12 lunar months is only 12 x 29.53 = 354 days, 11 days short of a year. Agricultural societies need a solar calendar to reckon the seasons for planting. But nomadic shepherd societies can use a lunar calendar for easy reckoning. The Hebrew and Islamic calendars are still lunar. The 19 year Metonic Cycle is the common ground.
I'm curious about the Saros Cycle of repeating eclipses. This cycle repeats in a few days over 18 years. It's also expressed as 19 orbits of the nodal points of the moon's orbit, the points of the orbit that are on the ecliptic so an eclipse is possible, about 346.6 days. 19 x 346.6 = 6585 days = 223 x 29.53, so 19 eclipse years is 223 lunar months to within a few hours, and that's the Saros cycle. My head is spinning. What is it about small prime numbers? Maybe 19 is the real answer to life, the universe, and everything!
I managed to challenge Graybear with this stumper! He also figured 19 years, but had trouble with the details:
I am now perplexed! I have, however, thought of some related questions:Thanks Graybear. I'll save the first 2 questions until January and February (and March!) next year.
What month sometimes won't have a full moon?
How often do 'blue moons' occur?
Does the moon's orbit change in relation to Earth's distance from the sun?
I got in over my head with this stumper. But it's interesting stuff, and it was cool to rediscover the Metonic Cycle with my calculator and some almanac data. I wish I had more time to explore these dusty trails through our history! Here are a few starting points on the Web:
 Fourmilab has a useful Web site with a Javascript calculator that figures dates for new and full moons as well as luner perigee and apogee for any year. You can save the page and use it offline (and study the code).
 Lunar Outreach Services show the current phase of the moon updated every minute, and has phase data for every year from 1 to 2246.
 Try a Web search on "Metonic Cycle", and you will open the door to some fascinating astronomical and ecclesiatical history. This 19 year lunar/solar cycle is still the basis for the Golden Number and the Epact that appear in Catholic Easter calculations. It's an esoteric part of our culture.
 The Catholic Encyclopedia entry for "Epact" has a fascinating but difficult discussion of how lunar cycles are used to calculate the date of Easter.
 William H. Calvin's How the Shaman Stole the Moon (Bantam Books, 1991) discusses eclipse prediction and Native American archeological sites. John Blackwell's paper on Ancient Astronomers of North America also discusses the astronomical significance of several Anasazi sites.
 Will Linden's Today Date and Time Page has an abundance of obscure data for any date.
 My MSDOS TBC program can generate lots of useful ephemeris data and maps, and can literally play the music of the spheres in MIDI! TBC is available for download (with BASIC source code) at Treebeard's Basic Vault.
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