# Trying to Calculate Basic Lunar Probabilities

I'm trying to do what I thought was a basic problem but doesn't seem to be working out properly. I'm looking at disproving a claim that tides cause earthquakes, where the main mechanism that people claim is that it happens during a new or full moon, and especially during a perigee moon (when it's closest to Earth).

People like to give ±1 week windows which is basically half a lunar cycle, so it seems basic that the probability an earthquake would happen, given random chance, would be 50% that it would happen within a week of, e.g., a full moon. Similarly, since perigee/apogee happens roughly on the same timescale, it seems as though it would be 50% chance that an earthquake would happen within 1 week of a perigee moon. Put the two together and you have a 25% chance that an earthquake would happen within both a perigee and full moon, correct?

(Important note, updated: Actual period between new/full moons averages 29.52 days, actual time between perigee moons averages 27.56 days. However, apogee and perigee are NOT normally distributed. Apogee happens a mode of 27.78 days (mean is 27.55±0.27), while perigee is much more asymmmetric, having a mode of 28.4 days at the peak of approximately a Lorentzian. Half the max of the Lorentzian is ±0.08 days. But, the mean is 27.56 with a standard deviation of 1.12; the range is 24.6-28.6 days. I'm thinking this could throw off the modeling?)

Assuming that's correct, I seem to be running into issues when trying to figure out the probability that an earthquake would happen by chance within ±X days of BOTH a perigee and full moon. I thought the equation would be simply (2*X/(# days in lunar month, 29.5))*(2*X/(# days between perigee, 27.5)).

However, when I do a monte carlo simulation with 500,000 randomly chosen dates within the time period of 1933 to 2012 (just happens to be when I have earthquake data), the fractions do not line up. For example, the simulation shows that 14.4% should be within 5 days of both a full and perigee moon, but my above math says it's only 12.3%.

I have checked the results of my simulation against days from perigee, apogee, new moon, and full moon times. As expected from a random distribution with a large $N$, the number of times that the simulated earthquake is a given time period away from maximum perigee/apogee/new/full moon is even. Except for perigee, where I see a fall-off for $>|±12.5|$ days from when it's closest to perigee. I'm thinking this has to do with the non-Gaussian distribution of perigee times? And could that account for the 2% difference at the 5-day example?

Is the best way to approach this, because these perigee times are a bit crazy, to simply go with the Monte Carlo results?

P.S. This has been updated to better reflect a correction I made in my data. When I initially posted this, I had some incorrect full/new moon dates in my table that were throwing some results off.

• I'm confused. Why are you estimating the proportion of days that are at different phases of the moon? You have actual data - that is, you know, for each day, where the moon was in its cycle. Aug 28 '12 at 20:59
• Given what you've got, I would not simulate data at all. I would look into time series analysis. This would let you account for the fact that earthquakes in a given time period are not independent of earthquakes in the previous time period. It would also make it possible to look at moonphase (probably a variation on seasonality). Aug 28 '12 at 21:15
• When I do the calculation described in the question (by sampling daily for $80$ years) I get $12.3$% for the answer. In R: days <- 1:(80*365.25); k <- length(days[days %% 27.56 < 2*5 & days %% 29.52 < 2*5]); k/length(days). Varying the starting phases changes the result by less than $0.03$%. The $15.2$% must be a computational error.
– whuber
Aug 28 '12 at 22:52
• And yes, I know that there isn't a correlation, but I'm actually doing this for a podcast I run, "Exposing PseudoAstronomy," and this is the next episode topic. So I wanted to do the analysis myself and then compare that with what the handful of pseudoscientists claim. I find going through the exercise oneself is often enlightening, and I'll be in a better position to inform my listeners. Aug 29 '12 at 2:23
• If your code doesn't produce the same answer as whuber, then you probably have a bug in your code, and there isn't much we can do. Aug 29 '12 at 6:13

There is an oddity that $29.52/27.56$ is unusually close to $15/14.$ So, even though the proportion of times within $5$ days of both will eventually be about $\frac{2X}{29.52} \frac {2X}{27.56}$, it might take longer to converge than you expect. Instead of trying to calculate by a formula, you can simply ask a computer to count how many minutes are within $X$ of the perigee and full moon over that whole range. However, I don't think this effect is large enough to explain the difference between $15.2\%$ and $12.3\%$.
A Monte Carlo simulation normally would mean that you choose times randomly, not evenly spaced. One possibility is that you chose a gap which is too close to a simple rational times one of the periods. $79$ years/$43000$ is $1.49$ days, which might be too close to $29.52/20 = 1.48$. If you used exactly $29.52/20$ then instead of $10/29.52 = 33.88\%$ samples within $5$ days you would get $7/20 = 35\%.$ Again, this doesn't seem to be large enough to explain the discrepancy, but why add significant errors of this type?
• Thank you for the reply. I've updated my question - I had an error in my tables for new/full moon times where some were crossed, resulting in non-uniform results when I corrected the Monte Carlo to large, random $N$ times. Aug 29 '12 at 1:22