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Subject: It works - within limits.
From: Herbert Prinz (hprinz@XXX.XXX)
Date: Thu Apr 11 2002 - 21:18:05 EDT
To find GMT and our position simultaneously we need the observation of the
altitude of any two celestial bodies, the distance of the Moon from any suitable
body, and the time intervals between these three observations.
The simplest case from a mathematical point of view is to measure the altitudes
of the Moon and second body themselves (because they are needed anyway for
clearing the distance), and to measure all three quantities at exactly the same
moment. One can cheat a little on the latter by bracketing the distance
observation with the altitude observations and subsequent averaging.
While it's true that the required altitudes can be computed, we also know that
there is no free lunch. To use computed altitudes merely means that we have
already observed some (other) altitudes at an earlier stage (for the purpose of
finding time and latitude). The solution by this method is, therefore, a running
fix.
Consider the case where double altitudes of the Sun, or some such method is used
during the day to establish local time and latitude. If a vessel sailing from New
York towards the Azores in and out of the meanders of the Gulf Stream observes
Sun altitudes around 10:00 and 14:00 to establish local time and then takes a
lunar around 20:00, the dead reckoning of longitude made good between the former
and the later observations can easily be off by 10 nm, and hence its local time
be off by 40s. So the error in computed Moon altitude could be up to 10' of arc,
hence the error of computed parallax up to 10" of arc, which in turn could
translate to an error of as much as 20s in GMT or 5nm in longitude for the final
fix. This is not much in the scheme of things. Most navigators were and will be
happy to get GMT within a minute. I am only emphasizing that this is an
additional error that does not appear if altitudes for clearing the distance are
measured directly and that cannot possibly be detected or eliminated by any
mathematical tricks.
Bruce Stark soft-pedals the impact of DR error on the accuracy of the final fix
in his message "It works", of April, 2. The reason why it works for Bruce even
"when both GMT and longitude are wildly uncertain" is that in his example, he
does not depend on measuring local time at all; he computes it from accurate data
and only THEN shifts assumed GMT and assumed longitude in sync with each other,
so as to not upset their relation (defining local time). Naturally, after 2
iterations, one gets the correct GMT and longitude from the lunar distance. But
this is tautological. In the real world, however, local time is only as good as
your dead reckoning since the time you established it. The "wildly uncertain" DR
does, indeed, not matter up to the moment where we start with the first
observation for time. But any subsequent error in dead reckoning will have its
inevitable effect on the resulting fix for GMT from the final lunar observation.
Of course, there is nothing special about lunars here. This is a general problem
with the running fix. Take the standard timed altitude observation of two stars
as an example. When starting from a wildly wrong DR position, St. Hilaire will
get you the right fix, albeit only after 2 or 3 iterations. That's not
surprising, because direct methods will not even need a DR. The same is true for
a running fix, if and only if you are absolutely sure about your dead reckoning
between observations. But if you screw up on the DR (e.g. by getting into a
current) no method will tell you. All of them will result in the same wrong fix.
There is another, minor problem with computed altitudes. I am not sure whether
this has been mentioned already. They rely on an unverified assumption about
there being ideal atmospheric conditions in the direction of the Moon and second
object. But if there are unusual circumstances, the effect can result in up to
12s error in GMT for every 0.1' deviation from standard refraction. If one
measures the altitudes, this error is automatically eliminated, at the cost of
only the corresponding negligible positional error. Using computed altitudes is
thus inherently less safe than measuring them.
All numbers I gave are worst case scenarios that I could THINK of. I never lost
time at sea and never had to depend on a lunar distance. So, I really don't know
what I am talking about. All I have ever done were isolated experiments of
various kinds.
Herbert Prinz
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