# NavList:

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

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Re: Exercise #14 Multi-Moon LOP's
From: Frank Reed
Date: 2009 May 2, 23:14 -0700

```George H wrote previously:
"Now, I don't know how that program operates, internally, and I doubt whether
Jeremy or Frank know either. It seems worthwhile, then, to ask some probing
questions about the value of such an exercise, before building on to such an
insecure foundation, as Frank has done."

Insecure foundation? You're way off, George.

As for the supposed "black box", most of these software packages do exactly
the same thing: they use the least squares solution for multiple sights
provided in many sources, including every copy of the Nautical Almanac for
some twenty-odd years.

The calculational details are not the point of all this. The POINT is that you
can get a good fix with a series of sights over a fairly brief period of
time. This is not widely known among celestial navigation
enthusiasts/practitioners.

In a (typical) rapid-fire fix, the error ellipse will be aligned with its long
axis perpendicular to the mean azimuth of the sights (quite comparable to
lat/lon at noon, by the way). The error in this direction is given
approximately by (3/2)*S/(sqrt(N)*A) where S is the standard deviation of the
observations themselves, N is the number of sights, and A is the range of
azimuth in the sights (azimuth expressed as a pure number, "in radians"). So
take Jeremy's moon sights in his "Exercise #14". First, we need the standard
deviation of observations. Jeremy has posted quite a few cases, and he gets
pretty good sights, with s.d. around 0.5 minutes of arc. In this particular
set, N is 11, A is 0.1 (5.5 degrees is a tenth of a radian). So the standard
deviation "cross-range" would be 2.3 nautical miles. Pretty good.

In the direction towards the object, the error ellipse "down-range" has a
width of about S/sqrt(N). We're not generally worried about the error in this
direction, but it's worth mentioning that you would bump up againt systematic
error fairly quickly. So if the equation says your error is 0.05 nautical
miles in that direction, don't take it seriously.

-FER
PS: The factor (3/2) above may be wrong. I got that twice, but a third time I
found (4/5). Whatever that constant is, it is "of order unity"

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