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Subject: Towards a basis for Bruce Stark's Tables
From: Fred Hebard (Fred@XXX.XXX)
Date: Fri Jan 03 2003 - 02:35:10 EST
OK, some history of this major obsession of mine,
I used to sail and do coastal piloting, and always wanted to do
celestial, but have lived inland for 30 years. Then I noticed the
price of a Davis Mark 3 on the Celestaire site and bought it. Then
of course spent hundreds of dollars on sight reduction books when I
couldn't get my position down to better than 20 miles, trying to
figure out what was going on. Thought maybe elevation above sea
level was involved, but in my case that's a half mile difference in a
6000 mile radius of the earth; elevation affects lunar parallax and
refraction a bit, but not much more. Finally got my position to
about 2 miles once. Decided I needed a better sextant. Got a 1948
Husun Mate on Ebay for a little more than $200 because the batteries
had burst the handle; but the enamel was not noticeably chipped in
any of the photos and all the parts were there. Found I would have
to remove the index arm to remove the handle for repair. Delayed
doing that. Ran a lunar and got to within 12 seconds of GMT! Maybe
that Husun was OK, but handle needed fixing. So removed index and
fixed handle, but feared destruction of the calibration due to a
small blunder while removing index. To check calibration, I would
like some accurate interstellar distances, thus need to apply Borda's
method to correct for refraction. But how do all those tables in
Bruce Stark's book work? A far more abstruse question.
Table K, one of the big ones, is log(haversine()). Found a reference
to Gauss' formulas at
http://mathworld.wolfram.com/SphericalTrigonometry.html . They are
similar to Napier's formulas. So one of those must be the formula
for the Gaussian table. It clearly is not the normal approximation
to the binomial.
I have gotten about as far as this as I have time, and sure would
like to see Bruce Stark's method laid out in detail. It appears to
be a fairly straight-forward application of log haversines to
spherical triangle trigonometry. He appears to be using Borda's
method for clearing the distance, and the standard method for
computing intercelestial distances (with no correction for
refraction).
Here is as far as I've gotten with his method for computing
intercelestial distances. Let del(GHA) be the absolute value of the
difference in GHA between the two bodies, del(dec) be the
corresponding value for declinations. ~ is the operator for finding
the absolute value of a difference. Mdec is the declination of one
body and Sdec the declination of the second; logs are to the base
10; hav is haversine; archav its inverse; and Gauss are values from
his Gaussian table:
=archav{-Gauss[(log(hav(del(GHA))) + log(Mdec) + log(Sdec))
~log(hav(del(dec)))]
+ [log(have(del(GHA))) + log(Mdec) + log(Sdec)) ___or___ log(hav(del(dec)))]}.
The ___or____ function chooses the lesser of the two values.
So what is the Gaussian, and how do you hook all these guys together
in standard mathematical notation? Hopefully, this will help and
inspire somebody, maybe Bruce, to lay this out!
Thanks,
Fred
-- -------------------------------------------------------------------------- Frederick V. Hebard, PhD Email: mailto:Fred@XXX.XXX Staff Pathologist, Meadowview Research Farms Web: http://www.acf.org American Chestnut Foundation Phone: (276) 944-4631 14005 Glenbrook Ave. Fax: (276) 944-0934 Meadowview, VA 24361
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