### Re: When is the next time in which a full moon falls on a leap day, Feb. 29

Date: Mon Jan 15 15:49:04 2007
Posted By: Bart Broks, Grad student, Department of Applied Physics
Area of science: Astronomy
ID: 1168071056.As
Message:

Hello Chris,

Thank you for your questions. Your first question, when the full moon falls on a leap day, should be fairly easy to answer. I can't really answer your second question, about finding out the moon cycle "in great future" as it's not very well-defined what this means- a decade, a century, a millennium?

Let us first discuss what a full moon actually is. The light you see coming from the moon is the sun's light that is reflected from the moon's surface towards the earth. The moon is called full when the whole surface of the moon that is currently visible from the Earth is reflecting light back towards the earth. This is the case when there is an imaginary line going over the sun, the earth and the moon. This phenomenon is called syzygy [1]. In case you were wondering, yes, sometimes, the earth blocks the light going from the sun to the moon. This is called a lunar eclipse, and is a fairly common, yet beautiful sight.

The orbit of the moon is a very complex movement, that can be approximated by a elliptic orbit. This gives a regular occurrence of the full moon. Jean Meeus has derived a formula that approximates the times at which a full moon occurs:

D = 20.362954 + 29.5305888531 × N + 102.19 × 10-12 × N2 [2,3]

with the D the date with a starting date at 0.00 hours at January 1st, 2000 Greenwich Median Time and N an integer number that denotes the amount of full moons that have passed. Note that there is a time difference between this time and your time in Columbus, Ohio or mine in Amsterdam, The Netherlands. This means that the answer I am deriving is only valid at longitude, although you can use the same method to compute it at any other point on the Earth's surface.

Okay, now we know when the moon is full. Next step is to figure out when it's February 29th with respect to our yardstick of January 1st, 2000. The first leap day is February 29th, 2007, which starts 59 days after the start of the 2000 and ends 60 after the start of 2000. The moon is not full at this date- it is, however, 20.4, 49.9 and 69.4 days after the start of 2000, according to our formula. So far no luck.

There is a unique property of February 29th which makes your question a little easier to solve. Between each February 29, there are exactly 3 normal years and 1 leap year, which have a total of 1461 days-unless there is turn of the century in between, because these are only leap years if the year is divisible by 400. If you would have asked another date, such as May 6th, the next May 5th could have been in 365 or 366 days, making the question a lot harder. If we discard the small quadratic term in our equation, we see that there are 49 full moons in each 1461-day period, and we have 14.00 days left of the lunar cycle. At our first leap year day, we are 9.11 days into the third lunar cycle. At our second leap year day, February 19th, 2004, we are 23.11 days into our fifty-second lunar day. The next leap day, we add another 14.00 days, and are 7.58 days into the fifty-fourth cycle. Mathematically, the days left in a cycle Lare given by:

Lx+1=mod((Lx+14.00),29,53)

The new moon is on a leap day when Lx > 28.5, i.e. at 0.00 hours on February 29th, it takes less than 24 hours for the moon to become full. Using for instance a spreadsheet, you can see that this happens at February 29th, 2048. If we now also correct for the fact that there are 2921 leap days between leap days when there is a turn of the century that is not divisible by 400, we find that this also happens in 2132, 2216,and 2376

The occurrence of a full moon on a leap day is a fairly rare occurrence-the moon is full only once every 29.5 days, and leap years occur only every 4 years. Hence, one would expect a frequency of about once per century, which is consistent with our-small-dataset.

If you would like to know the date for dates that are even further into the future, I recommend you use a computer,and for instance a small script or a spreadsheet.

Regards,

Bart Broks.

1. http://en.wikipedia.org/wiki/Syzygy, taken at January 14th 2007
2. Jean Meeus (1991). “47. Phases of the Moon”, Astronomical Algorithms (1st ed.). ISBN 0-943396-35-2.
3. http://en.wikipedia.org/wiki/Full_moon, taken at January 14th 2007

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