A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2016 Sep 11, 14:29 -0700
Another way to estimate the Moon-Sun lunar distance is from a source that gives the Moon's phase. For those of you with smartphones, you probably have a weather app that has a "Sun and Moon data" page, and often these will give the phase of the Moon as the "fraction illuminated" to the nearest percent. Supposeyour weather app tells you that the Moon's phase is 62%. That number is calculated from phase = [1-cos(elongation)]/2. It's a neat math puzzle to work out why that is so, but let's just take it as a given. Now, the "elongation" of the Moon is identical to the Sun-Moon lunar distance. And you'll note that the combination of terms here, [1-cos(x)]/2 is defined by navigators as the "haversine". So, if this mnemonic works for you, you can remember this rule as
phase = hav(LD).
If the phase is 62% or 0.62 then, expanding the haversine and solving for cos(LD), we get
cos(LD) = -0.24.
You can do that in your head easily. But to go further, at this point you may want to grab a calculator, or, maybe you remember that for small angles near 90°, the value of the cosine is nearly equal to the angular excess from 90° converted to radians. Multiply 0.24 by 57.3 and you get 13.8°. Add that onto 90 and the lunar distance is 103.8°. Given that there are only two significant digits in the phase, we should only expect two significant digits in the angle above 90, so really I should quote the result as 104°. That's close enough to preset your sextant.
Note I have myself used this trick to get a quick estimate of the Moon-Sun lunar distance. I had better distances available with some poking around on my phone, but the weather app's Moon phase was right in front of me.