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The American Ephemeris and Nautical Almanac for the Year 1855 contains
Tables for Correcting Lunar Distances by William Chauvenet. He provides
an explanation and two examples.

https://books.google.com/books?id=_yUOAAAAQAAJ&pg=RA1-PA20&focus=viewport

Compare his Sun lunar solution with modern solutions:

08:09:01  Chauvenet
08:11:46  Antoine Couëtte
08:08:30  me (full solution)
08:11:55  me (lunar time sight)

Those times are not all directly comparable. Chauvenet and I assume the
position is approximate. The altitudes of both bodies affect the
solution. Antoine's computation assumes position is known. I call that a
"lunar time sight." My software allows that also, and its result is on
the last line.

However, I don't think Chauvenet's lunars are a good test for modern
software. The main problem is his ephemeris, which differs from the JPL
DE406 by 18 arc seconds. Therefore, I have re-computed both examples
with JPL HORIZONS, USNO MICA, the Bennett refraction formula, the
Nautical Almanac dip formula, and Chauvenet's own formula for refracted
semidiameter. Those formulas are different from my software in order to
provide an independent check.

Sun lunar: 1855 September 7 08:09:01 UT1. Delta T = 7.5 seconds.
Position N 35.5 W 030.0. Height of eye 20 feet (6.1 m), temperature 75 F
(23.9 C), and station pressure 29.1 inches Hg (985.4 mb). Lower limb
altitudes above the sea horizon: Sun 5.78908°, Moon 49.90027°. Lunar
distance, near limb to near limb: 43.87438°.

The solution from my lunar distance program is 08:09:10 UT1, 9 seconds
after the correct time.

Fomalhaut lunar: 1855 Aug 30 05:42:02 UT1 (delta T = 7.3). Position S
55°20′ W 120°25′. (NOTE SOUTH LATITUDE) Height of eye 18 feet (5.5
meters), station pressure 31 inches Hg (1049.8 mb), temperature 20 F
(-6.7 C). Moon lower limb 6.58649 above sea horizon. Fomalhaut altitude
52.71724. Moon far limb to Fomalhaut 46.50609.

The solution from my program is 05:41:51, 11 seconds before the correct
time.

I'm satisfied with that accuracy. Altitude of the lower body in each
case is about 6 degrees, so the solution is exceedingly sensitive to
refraction. To create the synthetic observations I was careful to use a
refraction formula different from my program, so a discrepancy is
unavoidable.

Of course in the real world you wouldn't shoot lunars at such low
altitude. I suspect Chauvenet created those examples to make his method
look good, and simpler solutions (specifically Bowditch) look bad. Even
the atmosphere conditions are obviously non-standard: high temperature
and low barometer in one example, opposite conditions in the other.
That's why I call the lunars "difficult."

More details on how I constructed the synthetic observations is on my
page of lunar software tests.

http://home.earthlink.net/~s543t-24dst/lunar3/tests.html

   

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