### Morsel: Blue Moons

by Steve Holmes

Everyone will have heard of the phrase "once in a Blue Moon", meaning very rarely, but how many people know what it actually means? It is of course possible for the Moon to look blue in a literal sense, if its light has passed through a smoke haze, for example. However, the usual meaning is the one taken from folk-lore where a Blue Moon is an "extra" Full Moon in some sense. One early usage was to refer to the thirteenth Full Moon in a year - about one year in three has such an "extra". Years were divided into Quarters (and still are for some religious and legal purposes!) with the Quarters starting on each equinox and solstice. One Quarter would usually have three Full Moons: the first one after the start of the Quarter was called the "early" Moon, the next after that the "mid" Moon and the one just before the end of the Quarter the "late" Moon. If a fourth Full Moon appeared it was deemed a Blue Moon: by the above definitions it would be the third one in the sequence. Now, if twelve months contain thirteen Full Moons at least one of them must have two Full Moons. Due to a misinterpretation of the original meaning, Blue Moon came to refer to this second Full Moon in a (calendar) month - in the Quarter-year sense a Blue Moon could in fact have been the only one in that month.

Given the association with rarity I was intrigued to know how often a Blue Moon (in the modern sense) would actually happen, so I started by doing some rough calculations to find out.

#### How frequent are Blue Moons?

The average time interval between two Full Moons, called a lunation, is 29.53 days. Purely statistically, there is thus a (31 - 29.53) / 31 probability of there being a second Full Moon in a 31-day month, or 4.74%. In a 30-day month the probability is (30 - 29.53) / 30 = 1.57%, and of course there can be no second Full Moon in 29 or 28-day months. Over a full year, therefore, the combined probability is (7 x 4.74 + 4 x 1.57 + 1 x 0) / 12 = 3.29%. This was a sufficiently small number to justify the "happens rarely" description so I decided to improve upon the rough estimate, mainly because Full Moons do not of course happen at random, as would be presumed by a statistical approach, but at totally correlated intervals: about every 29.53 days in fact! I thus wrote a computer program which simulated a long series of Full Moons, noting whether they fell in the same month. When writing the program I made allowance for the fact that the duration of lunations varies quite a lot - from 29.27 to 29.83 days in fact, in a cyclic fashion. This variability does affect the overall numbers quite significantly.

From the data generated by the program I derived figures for all sorts of related statistics, which are summarised below.

- The overall chance of a given Full Moon being "blue" is 1 in 30.5 (3.28%).
- A year will contain at least one Blue Moon month 36.3% of the time.
- Successive Blue Moons can be as close together as 1 month but will never be more than 2 years 11 months apart.
- The occurrence of Blue Moons is highly irregular: they are either bunched (separated by 1 or 2 months) or well spaced (separated by about 2 years 8 months).

In more detail, the chance of a given Full Moon being blue and also falling in a given month varies from about 1 in 250 for months with 31 days to about 1 in 835 for those with 30 days. The chance is precisely zero for February, as the lunar cycle is longer than even a leap year February.

The question as to when these statistics will next translate themselves into reality has a perhaps unexpected twist - it all depends on where you are! A Moon that is apparently Full can be seen at some time during the relevant night from an entire hemisphere of the earth, but the exact instant of "astronomical Full Moon" (by which Blue Moons must be defined) could be before or after midnight local time for a given observer. The day in which the Full Moon falls thus depends on your time zone. For example, a Full Moon at 10pm GMT on 31st January will actually fall on 1st February for an observer whose local time is more than 2 hours ahead of GMT, and thus possibly prevent a Blue Moon happening. The dates quoted in this article are thus only guaranteed for the GMT time zone.

The next times a month will have a Blue Moon are August 2012 and July 2015: on both occasions the Full Moons fall on the 2nd & 31st of the month. Interestingly, the second Full Moon in December 2009 was also a (very small!) lunar eclipse, so for a brief moment the Moon was both red and blue simultaneously!

#### Sequences of Blue Moons

There are two ways in which Blue Moons can be separated by one lunation: if Full Moons fall on 1st & 30th January then 1st & 30th March, or 2nd & 31st January then 2nd & 31st March (the former only in a non-leap year, of course). Due to the effect of leap years and the exact time of the Full Moon, the overall likelihood for this is about 1 in 330 lunations: the last time it happened was in 1999 and the next times will be 2018 and 2037 (starting on 2nd January on each occasion). A gap of 2 lunations is also possible (1st or 2nd & 31st January then 1st & 30th April, or 2nd & 31st December then 1st & 31st March) but with much lower probability (1 in 1530) as the timing is critical: it did happen in Dec 1933 / Mar 1934 and Jan/April 1961 though. It is __just__ possible for three long lunations in a row to give a sequence of Full Moons falling on 2nd & 31st Jan, 2nd March, 1st April, 1st & 30th May, which has a gap of 3 lunations between the Blue Moons in January and May. We are now not just talking rare but very rare (1 in 9400 lunations, or 760yrs, on average) but in fact there will be an event of this type in 2113.

Note, by the way, that the interval between all the above Jan/Mar events is a multiple of 19 years. This is the Metonic cycle, after which the Moon's phases repeat on the same calendar dates and thus make Blue Moons more likely. There is not a double Blue Moon in every cycle though (1980, for example) mainly due to the effect of leap years, which disturb everything by a critical day.

The above are the only cases when a month (February) can have no Full Moon at all, at a combined probability of 4.69% per annum i.e. 4 or 5 times a century. All other months must always have at least one Full Moon. It is not possible to have a gap of between 4 and 29 lunations, and the longest period with no Blue Moons is 36 lunations (about 2 years 11 months). The most common gap is 34 lunations: this can be understood from a theoretical standpoint as it takes an average of 33.6 lunations for a Full Moon to move through all the days in a (mean) month and get back to the beginning again, thus making a Blue Moon likely once more.

So, that's clear then - "once in a Blue Moon" means, on average, once every 2 years 8 months!

#### 'Black Suns'

There is, of course, nothing statistically significant about Full Moon - one could just as well repeat the above analysis for any other phase of the Moon. In particular, two __New__ Moons happen in a given month with the same frequency as Full Moons. One might think this was not very interesting though, as a New Moon is not actually visible, but New Moon is when solar eclipses occur! The question thus arises as to the frequency of two solar eclipses in the same calendar month. Roughly, this should be the Blue Moon probability (3.28%) multiplied by the percentage of pairs of solar eclipses one lunation apart (10.33%), giving 0.34% of all eclipses. In fact there are six occasions in the 1,645 solar eclipses from 1900 to 2599 AD, or 0.36%: pretty close to the estimate! The last time it happened was July 2000 but the next is not until December 2206. On all these occasions both eclipses are partial, however - in fact this is generally true for almost all pairs separated by one lunation. There are only 15 cases from 2000 BC to 3400 AD where one of the eclipses is not partial (next in July/August 2195) but none of these cases is within the same calendar month and the partial of each pair is of very small magnitude (no more than 0.061, where 1.000 would be just total).

I would suggest that a new saying be invented "once in a Double Black Sun", meaning about once a century, but I don't think it would catch on somehow!!

#### Eclipsed Blue Moons

We now know that a Blue Moon is quite unusual, and of course a lunar eclipse is relatively unusual, so how about the two happening together? I already knew that 31st December 2009 was one such occasion so I checked for instances after that and found that the next times a Blue Moon will be eclipsed are 31st January 2018, 31st December 2028 & 31st January 2037 - but these will all be total eclipses. There is then a very long gap, as the next eclipsed Blue Moons after that are not until 31st July 2129 (which is the next partial) & 30th August 2137 (total). A partially eclipsed Blue Moon is thus quite a rare event - maybe even more so than the "Double Black Sun" mentioned above!

There is more, however! It's a bit complicated though.

I noted above that after 31st December 2009 the next eclipsed Blue Moons will be on 31st January 2018, 31st December 2028 & 31st January 2037. Note the absolutely exact 19 year interval between the December dates and between the January dates - this is another example of the Metonic cycle noted previously. The cycle only works exactly if there are 4 leap years in the 19 year interval, however - there would have been a further eclipsed Blue Moon on 31st December 2047 but, due to there being only 3 leap years in this 19 year interval rather than 4, things get 1 day out and so the (total) eclipse is actually on 1st January 2048. Also note the intervals of 8 years 1 month (Dec to Jan) and 10 years 11 months (Jan to Dec) which add up to one Metonic cycle. These intervals happen because each of them is equal to an exact number of lunations: 8 years 1 month is 100 lunations, 10 years 11 months is 135 lunations. As with the full Metonic cycle, this means that the Moon will be at the same phase after these intervals. Not only that, but each interval is also (nearly!) equal to an exact number of "node to node" lunar cycles, plus a half: 108.52 and 146.52 respectively. The nodes are the two points at which the Moon's orbit crosses the plane of the Earth's orbit - more critically, they are the points near to which all eclipses must occur. An extra half orbit will just move the point of interest from one node to the other so it is not really important, and the value does not have to be exactly 0.5 (or zero, of course) for an eclipse to happen - a slight error is acceptable.

Taking these two facts together, after each of these intervals not only will the Moon be full on the same calendar date plus or minus one month, which will make it very likely to be a Blue Moon again, but if it was close to a node (i.e. eclipsed) the first time then it will be reasonably close to the other node the second time and thus will probably also be eclipsed again. This explains the periodic nature of eclipsed Blue Moons. The long gap is caused by the fact that because the "extra half" is actually quite a bit larger than 0.5 the Moon quickly gets too far from a node for an eclipse to happen. There is then quite a time to wait before the alignments correct themselves again. In fact, during this time there are several instances of the __first__ Full Moon in a Blue Moon month being the one that is eclipsed.

The difference between the 235 lunations and the 255 "node to node" cycles in a full Metonic period is 0.57 days. In this time the Moon will travel 7.57° away from the node which, due to the tilt of its orbit, will shift it 0.67° higher or lower than the eclipse in the previous Metonic cycle. Given that the width of the Earth's (umbral) shadow is about 1.45°, that means it only takes fractionally more than 2 shifts for the eclipse position to move right through the shadow i.e. there will be at most 3 eclipses in a Metonic-period sequence of eclipsed Blue Moons, as found above. The overall sequence can be doubled by taking account of the 8 year and 11 year cycles of course, as these are just formed from interlinked Metonic cycles, one at each node. Each set of 3 eclipses will either be small partial - total - total (as at present) or large partial - total - large partial (as on, for example, 31st March 2238, 30th March 2257 & 30th March 2276) so the "relative rarity" of partially-eclipsed Blue Moons will change as the centuries go by.