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Since my last Navlist posting, I've had a short spell in hospital, and my 
pocket calculator came with me. Now I'm back out again, I can report back on 
some findings, to show my time there wasn't entirely wasted.

Jeremy wrote, in 5413-

"Here is my data around 0620 on 10 June 2008.

1) Sun LL: Hs 33deg 58.5� @ 06h 19m 13s UTC
2) Moon UL:  60deg 36.8� @ 06h 20m 44s
3) LD1: 86deg 10.3� @ 06h 21m 40s
4) LD2: 86deg 10.0� @06h 22m 20s
5) LD3: 86deg 10.2� @06h 22m 52s
6) LD4: 86deg 10.6� @06h 23m 24s
7) LD5: 86deg 10.6� @06h 24m 39s
8) Sun LL: 32deg 35.8� @ 06h 25m 19s
9) Moon UL: 61deg 49.2� @ 06h 26m 28s

DR (didn�t take a fix) Latitude was 15deg 14.0�N and Longitude was
144-04�E.  Ship was on course 270 at 12.0 knots.  IE was 0.0 T/P was
98deg F and 1010MB.  Height of eye is 106 feet.

I averaged the two sun lines to get 33deg 17.2� at 06h 22m 15s, then
the two moon altitudes to get 61deg 13.0� at 06h 23m 37s.  Finally the
5 LD�s averaged to 86deg 10.34� at 06h 22m 59s."

=======================
Those numbers didn't "add up", and Frank surmised (correctly, in my view) 
that overlapping, rather than the usual side-by-side, views of the Moon and 
Sun limbs had been taken.

I've had a go at working that lunar in my own way; which means that I like 
to get out of those damned sexagesimals and into decimals, so I can use my 
calculator properly. I will spell it all out in more detail than if I were 
just calculating for myself.

I agree with Jeremy and with Kent that the mean lunar distance is 86.1717� 
at a mean time of 6.3831hrs (give or take 2 or 3 arc-seconds, which between 
friends are not worth arguing about).

That needs correcting for the two semidiameters, in this (odd) case by 
adding the Moon's semidiameter, of 0.2606� (which includes "augmentation" of 
0.0035�), and (unusually) subtracting the Sun's, of 0.2625�.

so observed lunar distance between centres = 86.1698�   (this is d)

Then we have to take a weighted-mean of the two Sun observations so that 
it's the same as if it had been taken at that same moment, which gives-
Sun LL altitude at 6.3831 h = 33.1231�, which agrees pretty well with Kent's 
figure.
Similarly,
Moon UL altitude at 6.3831 h = 61.0879�, which disagrees a bit with Kent.

Now for some corrections for the Moon
61.0879 UL observed sextant alt Moon
 -.1667 dip
--------
60.9212 alt. above true horizon
 -.0083 refraction
-------
60.9129
 -.2606 less semidiameter Moon from UL obs.
-------
60.6523 alt Moon centre
 +.4622 parallax (=HP cos alt), where Moon HP = .9433�
-.0002 reduction of parallax
-------
61.1143 =M, corrected altitude of Moon centre
60.6606 =m, altitude of Moon centre before correction for parallax and 
refraction.
========================
Similarly for the Sun
33.1231 LL observed sextant alt Sun
 -.1667 dip
-------
32.9564 alt above true horizon
 -.0083 refraction
--------
32.9314
 +.2625 add semidiameter Sun to LL obs.
--------
33.1939
 +.0020 sun parallax based on Sun HP = .0024�
--------
33.1959 = S, corrected altitude of Sun centre.
33.2189 = s, altitude of Sun centre before correction for parallax and 
refraction.

Next we have to "clear" the observed lunar distance d of the effects of 
parallax and refraction, to arrive at the true lunar distance D.

s and m are the altitudes of Sun and Moon corrected for everything except 
parallax and refraction.
S and M are the altitudes of Sun and Moon corrected for everything including 
parallax and refraction.

I use the formula-
D = arc cos[(cos d - sin s sin m) cos S cos M /(cos s cos m) +Sin S sin M]
and I get D= 85.7384�. This is the corrected lunar distance, that has to be 
compared with prediction.

I've taken the predictions quoted by Kent (who may have "rolled his own", 
and they agree, within an arc-second with those you can get from Steven 
Wepster's website, at http://www.math.uu.nl/people/wepster/tables.html) and 
converted to decimal degrees, these are, for 10 June 08-

GMT 0600, D = 85.5331�
GMT 0900, D = 87.0181�
and to deduce the time of the observation, we have to do a 
reverse-interpolation between them.

The result I get is 06h 24m 53s as the final result for the GMT of the 
observation.

This has to be compared with Jeremy's averaged chronometer time of 06h 22m 
59s for his lunar distances. So my conclusion is that his chronometer was 
slow by 1 minute 54 seconds. Or, perhaps more realistically, that his 
chronometer was correct and his lunar-derived GMT was in error by 1 minute 
54 seconds. That isn't a bad result, for a lunar. It would have put his 
deduced longitudes out by about 28.5 arc-minutes, which would have (but only 
just) qualified Jeremy for the Longitude Prize, if he had been observing 
around 240 years ago. The total error in angle in the whole process was less 
than an arc-minute, which for a first-shot at a lunar, is creditable work 
indeed.

If anyone is bothered to check my numbers, and finds anything to question, 
or any step hard to follow, I hope he will let me know.

That compares pretty well with what Kent Nordstrom deduced, in [5476], when 
he wrote-

"By using my average time for the LD observations 06-22-59 the corresponding 
true LD from a NA is 85d 23m 43s. If I ,without paying attention to the GMT 
06-22-59 but only use this distance in my calculations I get a GMT of 
06-23-43."  I disagree (but only marginally), with that GMT, but it comes 
even closer to the time of Jeremy's observation than does my calculation; 
within a minute, indeed.

However, I do NOT understand Kent when he continues-

"At this time the sun�s apparent time is 16-59-47.5. The TE is +31.5 s, 
which gives the MT of 15-59-47.5 + 00-00-31,5=16-00-19.


The difference between MT and GMT, that is 16-00-19 � 06-23-43=09-36-36 or 
corresponding to the long. E 144d 09m.



So it seems to me that Jeremy�s distances are not correct."



I don't see what Sun's apparent time has to do with the matter, or MT, or 
longitude. The aim of a lunar is to establish GMT, to set a local clock 
right; nothing more than that. Steven's web page of lunar predictions is 
marked as being for UT (= GMT), unlike the early almanacs, which until 1834 
were based on Greenwich Apparent Time. Kent's predictions must also be for 
GMT, so I suggest he is bringing in complications that don't exist.



Armed, now, with a figure for GMT (if we ignore what the chronometer was 
saying), it's easy to deduce, from the Almanac, a position for the Sun (or 
the Moon, for that matter, or both) and from the observed altitude, generate 
position lines in the familiar way, which should provide a good longitude, 
but (from the way they cut) not a good latitude, which deserves another 
observation at a different time.



==========================================



Jeremy wrote, in [5469]-



"I will also say that getting lunars at sea will be inherently more 
inaccurate then land.  Roll alone can change my height of eye several feet 
and ship's vibration also make it more difficult to get a steady platform 
(it pretty much forces you to shoot while standing.)  I am actually quite 
happy with about 1' of error for sea lunars.  Sitting on a beach, I'd expect 
a bit better however...."

Changing height of eye will have little affect on a lunar, being no more 
than a correction to a correction.

 Adding "... I don't have logs, but I was wondering how accurate old time 
sailors got with their lunars at sea?"

I doubt if that's an easy question to answer. Trouble is, only recently have 
navigators known REALLY where they were when out in the ocean. Lunar 
distance navigation loses its point once land is in sight. So it's only when 
taking lunars for training and for practice that they would be taken from 
well-known positions, and then the results would seldom be logged. 
Comparisons of lunars with chronometers could be informative, if at the end 
of a voyage a chronometer had been shown to have kept good time overall, but 
I don't know of any retrospective study that has ever been made.

Mariners usually had log trig tables available with sufficient decimal 
places (5 or 6) so as not to contribute significant error, together with 
familiarity in using them (though without much understanding of what they 
were doing, in general, I suspect).

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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