# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Lunars - Even Easier**

**From:**Frank Reed

**Date:**2008 Jul 01, 17:01 -0400

I've discovered another "clearing miracle"... We all know that if the two bodies in a lunar observation are perfectly lined up vertically, then clearing the lunar distance is not substantially different from correcting ordinary sextant altitudes: |z1 - z2| = LD. The difference in the zenith distances is the lunar distance, so "clearing" a lunar distance is no different from correcting the altitudes (or zenith distances) of the two bodies. We have to be more careful with small details in the altitude corrections, as is always the case with lunars, but the math is essentially the same. No trig is required to correct altitudes. Likewise, though not quite as obvious, when the two bodies are in exactly opposite azimuths, the correction is nearly as simple: z1 + z2 = LD. The sum of the zenith distances is the lunar distance. This was the case with the lunar shot by Jeremy a few weeks ago, and as I've already described, it means we can clear it without trig. But sometimes a miracle happens. Even when the objects aren't perfectly aligned, the same almost trivial math applies. The math doesn't care whether the objects are really aligned so long as there is an equivalent case where they are in fact aligned. For example, the two objects could be separated in azimuth by 170 or even 165 degrees instead of 180 degrees, and under the right conditions we can "pretend" that they are separated in azimuth by 180 degrees, and it all works out correctly. How can this be?! Some kind of crazy voodoo? No, just good old rigorous math. It works because the altitude of the Moon doesn't matter much, especially when the observed lunar distance is close to 90 degrees or the Moon's altitude is near the zenith. So we can ignore the Moon's real altitude (as observed) and substitute an altitude that turns the math into one of those special "aligned" cases. Assuming we have observed a lunar where the objects are nearly aligned, we use the mathematical condition of an aligned lunar to replace the Moon's altitude. Suppose we see the Sun and Moon roughly on opposite sides of the sky. Suppose we observe the lunar distance is 85 degrees (center-to-center) and the Sun's zenith distance is 45 degrees (after taking out dip and semi-diameter) and the Moon's observed zenith distance is 38 degrees (also after dip and SD). The sum of the zenith distances is 83 degrees which is less than the lunar distance so they're not aligned. But let's pretend they're aligned and drop the observed zenith distance of the Moon. We replace it by 40 degrees. Then we can work the clearing process by simple addition and subtraction of the altitude corrections --as if the two bodies were aligned perfectly. No trig at all! So how much of a change in the Moon's altitude can we tolerate? Well, we already know that. The allowable error in the Moon's altitude is given by dh = (6')*tan(LD)/cos(h_moon). This is the allowable error in the sense that if you make an error smaller than this in the Moon's altitude, the error in the clearing process will be smaller than 0.1 minutes of arc. If this doesn't look familiar to you, it's something I discovered a few years ago. It's not in the literature. You can prove it with a little calculus. Note that this formula applies under the assumption that refraction can be ignored compared to parallax. At very low altitudes, it doesn't work quantitatively, but the behavior is qualitatively similar (the zero error zone is skewed to larger distances). All of this implies that there are a rather large number of practical cases, at least in the tropics, where you can clear lunars without any tables at all, except the standard almanac altitude correction tables. So chew on that, kids! -FER PS: the corresponding "allowable error" for the other body's error is dh = (6')*sin(LD)/cos(h_body). That's not relevant to this story and I'm including it here only for "completeness". --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---