From: peter@gunther.tcs.com (Peter Winship)
Subject: Re: Celestial Navigation Question
Organization: Teknekron Communications Inc.
Date: Thu, 31 Dec 1992 00:48:25 GMT

In article <2B42109F.1700@ics.uci.edu> omalley@kleber.ics.uci.edu (Owen O'Malley) writes:

 [stuff deleted]

>... why they [celestial navigators] don't take the bearing to the celestial
>object [azimuth] to determine their location on the circle of position.
>Surely,
>adding a handheld compass to a sextant wouldn't be difficult and it
>would allow a fix based on a single celestial object. Is this ever
>done? Is there a good reason why not or is it just tradition?

The answer might go like this:  There are three reasons why azimuth to
a star should be calculated rather than measured by the celestial
navigator.  One is due to the geometry of the situation, which causes
errors to be large and non-constant depending on observer's position,
the second to changes in magnetic direction over the surface of the
Earth, i.e., magnetic variation, and the third relates to problems in
compass measurement accuracy, i.e., magnetic deviation and personal
measurement error.

The possible radii of the circles of position obtained by celestial
altitude measurement range from 0 for a star overhead (zenith) to one
Earth radius (roughly 3434 nautical miles) for a star on the horizon.
Therefore the typical circle of position is quite big.  Because of this
large size, the Earth's magnetic field would undergo significant
amounts of variation in direction from true north over the arc of such
a circle.

Let's assume zero compass error.  In order to allow navigators to fix
their position on the circle, geodesists would have to construct a
complex table relating all possible small circles with magnetic
variation.  Speculating, I'd guess that this table would come in a
30-volume set weighing about 50kg.

For the moment, let's forget the magnetic variation and also the
increased error in compass measurements of azimuth when the star is
closer to the zenith.  Assume we have a circle of position of radius r
in nautical miles.  (0 <= r <= ~3434 nm)  The circumference of this
circle is c = 2*pi*r.  (Note that r = 3434*cos(altitude)) Now, if we
mentally go once around the circle of position, the range of azimuths
is 0-360 degrees.  Thus one degree on the compass spans c/360
nm along the circle of position.  For an altitude of 60deg., c would be
2*pi*3434*cos(60 deg.) = 10788 nm.  One degree on the compass gives,
for this circle, a span of 10788/360 ~= 30 nm.

Thus, the navigator must have an accurate determination of magnetic
variation and deviation to better than 1/30 of a degree and be very
unusually steady-handed while making measurements in order to get a fix
along this circle accurate within a mile.  60 degrees is a fairly high
altitude.  For lower altitudes, the problem gets worse.

Guessing, again, I would doubt that a water-bourne navigatior would be
able to get better than +/-3 deg. accuracy on his circle of position by
compass measurements, which, for an altitude of, say, 45 deg.,
would give a range of +/-127 nm along the circle of position.  This is
not navigational quality.

But hey, if we had a way to measure azimuth directly, you're right--
it would be easier to get a fix.

-peter

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