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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".


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