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I used Google to easily locate <>
which looks like a good start for anyone interested in this
application.  I am certain that many more interesting links may be
discovered along the way.  I personally have never looked at
Maxima but probably should do so one day.

How did you manage with the computer 'basic' matter raised
sometime in the dim past?

Frank J.
Rochester, NY

From:               "George Huxtable" 
To:                 
Subject:            [NavList 1751] Re: Latitudes by lunar distance. was: Lunars with and
    without altitudes
Date sent:          Sun, 26 Nov 2006 15:47:09 -0000
Send reply to:      NavList@fer3.com

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DW has sent an interesting posting, [NavList 1750], with
threadname
"Lunars with and without altitudes". In fact, though, it's more
relevant to the original thread, "Latitudes by lunar distance", so I
have tried reverting its threadname back again.

It may indeed be the answer to my request for a detailed and
numerical
account of how to work Frank's proposed method.

But there are some snags...

One is the unfamiliarity (to me, and no doubt, to others) of the "CA"
programming language he has used. I can see it has some virtues,
in
allowing algebraic expressions to be constructed using numerator
and
denominator.

Though no programming pundit, I have managed to deduce much of
what
that program does, as others have too, perhaps. But it would be
useful
to have available some definition of the syntax of that language. Can
DW point us to where that can be found? It would also be helpful to
see some comments added to the prgram, to flesh out the
description he
has given. A particular difficulty, for me, was in expression (%i10),
which I didn't manage to translate into its equivalent in normal
algebra, without knowing the detailed rules for the notation..

DW's final results are in the form of two position-lines, based on his
assumed position of N38d, W70d. One, using star1, Pollux, gives a
position line at right angles to a displacement through 0.61094 d in
the direction 63.83353 degrees. The other, using star2, Aldebaran,
gives a position line at right angles to a displacement through
0.69511 degrees in the direction 5.789598 degrees. By roughly
plotting
I get their intersection to be at about N38 39', W69 33', not that far
from Frank's own deduction of N38 46', W69 25'. But it's somethig
of a
surprise that there should be any difference at all, based as they are
on the same observations.

One likely reason for that difference is the use of Astronomical
coordinates taken from the Nautical Almanac, given there to the
nearest 0.1 arc-minutes. Where Frank's data comes from, he hasn't
told
us yet, but my guess is that he is presuming that more precise
coordinates have been somehow made available to an observer.

But there are other details, that need to be thought about, when
such
great precision is being sought.

1 Moon semidiameter. DW has appled a correction for this, outside
what
his program does, before he has entered the details. If the HP is as
stated, 58.4', then the SD associated with that should be 15.9', as I
make it. To that has to be added a figure for "augmentation", which
varies with the sine of the altitude, and has a maximum value, at the
zenith, of about 0.3'. From his stated numbers, DW appears to have
applied an overall SD correction of 16.2', which may indeed be
correct, if the Moon altitude was high. But Moon altitude wasn't
given
(the horizon wasn't visible), and has to be calculated or estimated,
if only very roughly. That step (allowing for semidiameter,
augmented
suiably)  should really be part of the program, using a true Moon
altitude calculated within the program.

2 Refraction. Here, DW has called on the expression given in
Meeus
16.2, though neglecting a third-order term. The trouble with that, is
that Meeus states that below 15 degrees it will "give inaccurate, or
even completely meaningless, results". Yet it has been used to
correct
star altitudes, but really, the program should check first whether
those (calculated true) altitudes are within the usable range.  Or
better, perhaps, use some adaptation of Meeus 16.4, which is much
less
limited in angular range..

I am not saying here that using formula 16.2 gives rise to significant
error in this case; just that, under some circumstances, it could.

3. Parallax. DW uses an adaptation of the simple parallax formula
(HP
cos alt) which is rather more complex, and presumably this is to
make
the correction "backwards", on the basis of true calculated altitude
rather than observed altitude. I seem to remember that we have
discussed that matter, on Nav-L, in the past. Can he offer a
reference?

However, there's also a "standard" correction, which is often
neglected in navigation, but needs to be taken into account when
working to maximum precision (as we certainly are here). This is the
reduction in parallax on account of the Earth's ellipsoidal shape,
which increases from zero at the Equator to 0.15' at 60 deg latitude.
I can't find that in the program, but haven't delved into every
cranny. Is it hidden somewhere, implicitly, in one of the equations?

It may be that DW's program (perhaps with a bit of further
development) will indeed provide the guidance to Frank's method
that I
have been seeking, as an implementation of it, and I wonder
whether
Frank will give it his endorsement as such.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.




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