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From: Bill (no email)
Date: Thu Oct 21 2004 - 18:22:09 EDT
Like a political candidate, my position keeps changing ;-)
My questions were, how much error from averaging is acceptable, and is there
a rule of thumb for estimating how much error there will be by looking at
declination, latitude, and altitude? I think I am coming close to a crude
working model (below).
Living and sailing in the mid latitudes I think of LAN as the point the sun
is south and at its highest, and exhibits no observable change in altitude
for a brief period of time. For me the altitude seldom exceeds 74d. Note
its azimuth is quickly changing however. Systematic error near LAN is small
and getting smaller as the date moves away from the summer solstice and the
maximum altitude falls. So this is a great time to average (for a full
reduction) to determine latitude.
As an example of the extreme where the declination and latitude are
identical (imagine yourself on the Earth's equator at equinox and ignore
change in declination over the day) the sun would rise directly in the east.
Its azimuth would remain at 90d until LAN when it passed directly overhead,
the become 270 at that moment. Trucking along approx. 1d in every 4 minutes
all day, so a relatively large error from averaging.
On the flip side of the coin, we have sunrise and sunset. Here the change
in altitude can approach 1d per 4 minutes, but the change in azimuth can be
very small, so a good opportunity for determining longitude. The problem
being the relatively large (up to .8') shift for averaging 5 sights over 4
minutes because of its rate of change and non-linearity.
Between sunrise, LAN, and sunset the "slope" or ratio between horizontal
movement and vertical changes from vertical to horizontal to vertical. So
error falls somewhere between. The above Herbert has addressed more
mathematically than I can.
To answer my own question, I came across a formula in Dutton's (article
3004) for rate of change.
Delta H per minute = 15 x cosine Lat x sine Z
Where Z is the azimuth angle of the body or its supplement.
They also have a nomogram for graphic calculation.
So it seems with a compass bearing (corrected to true) and a hand calculator
or the nomogram one could determine rate of change for 4 minutes by
multiplying the 1-minute figure by 4.
If we assume 1d (60') change over 4 minutes produces .8' averaging error, we
can mentally interpolate to estimate averaging error for the current
situation. I am guessing I am guilty of again trying to treat a non-linear
function as linear, and that is relationship will be explored as time
permits.
Then the navigator can determine whether that error is acceptable for
his/her conditions.
Bill
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