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Douglas and I have had  some interesting and productive off-list
discussion, about lunars and other matters, and he sent me his latest
results before they were posted to Navlist, but I've been unable to comment
back, until now.

There are several things to say about those observations; a "text-book"
example, from which lessons can be drawn.

First, they are remarkably good, especially for an observer who is
relatively new to the game of lunars. The scatter about the fit-line in his
graphs would show a standard deviation of no more than 0.2' arc-minutes or
so, at a guess (without actually calculating) from his graphs, after
excluding that rogue point. I doubt if even the most seasoned observer
would hope to produce anything better than that.

Douglas' resulting deduction of GMT, to within a minute of time, is as good
as anyone would expect from a lunar. Taken with a time-sight, that would
give a longitude within 15 arc-minutes, which would content any mariner
from the18th-century, before the days of chronometers, when the only
alternative was dead-reckoning across an ocean. Those are observations that
Douglas should be proud of, which would qualify him to sail with Cook, or
for a post with the East India Company. The famous Longitude Prize asked
for 30 arc-minutes overall precision, some of the expected error arising
from deficiencies in the Moon position predictions of that era.

Douglas is keen on plotting the observed lunar distance against time, and I
agree with him about its usefulness in showing up anything going wrong,
such as the point he has taken to be some sort of blunder, and excluded.
Such a decision needs taking with some care, though. I think, in his
position, I would have excluded that point too. It may have resulted of a
misreading of the drum, by 1'. But it's a bit marginal; being less than 1'
out-of-line with the others. Anyway, because it's only one point out of 8,
the overall difference in the mean, between excluding and including it,
would only be 0.1 arc-minutes, or about 12 seconds in the GMT; neither here
nor there.

How have the slopes of the plotted lines been chosen? If that's been simply
done "by eye", to provide a fit that looks about right, the overall result
can not be stitistically better than by simply averaging angles and times
(after any exclusion has taken place). Lunar distance changes with time in
a way that's very close to a straight line, over the usual duration of an
observation, so simple averaging works well.  However, estimating the
approximate mid-time of the observation, the appropriate slope can be
pre-calculated, having taken account of all the clearing corrections in
reverse. Then, having plotted that known slope-line, it can be adjusted up
and down, strictly parallel with itself, to get the best fit to the points.
This has the advantage that there's only one parameter to fit, not two, and
the result should benefit from improved precision. In that way, only, can a
graphical procedure improve on simple averaging.

Squeezing all we can from those observations, it seems that two bodies, on
opposite sides of the Moon, gave rise to opposite (but unequal) time
errors, the interval between them being somewhat more than statistical
scatter would lead us to expect. So it looks as if a bit of systematic
error may be present. That's very likely due to the estimation of position
of the Moon's limb with respect to the centre of the body, affected as it
may well be by irradiation, due to brightness differences. This is where
the observer's skill, which comes from experience, matters. And it's where
modern observers have such an advantage, in being able to compare their
observations with a known time-and-longitude. In the 18th century, that
wasn't the case. Even when a known headland was sighted or harbour entered,
its charted position, sometimes in latitude but frequently in longitude,
would be imprecisely known and charted. Out at sea, there was nothing to
judge the quality of their lunars by.

I don't accept Douglas' conclusions about the impracticality of the lunar
distance method at sea. Thousands of mariners, without chronometers, relied
on it for nearly a century, having nothing better. Certainly, it was much
more of a challenge than he faced in has garden, compared with the motion
of a vessel and the sky obscured by sails.

George.

contact George Huxtable, at  george{at}hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
----- Original Message -----
From: "Douglas Denny" 
To: 
Sent: Thursday, September 23, 2010 12:16 AM
Subject: [NavList] Graphs of Lunar Distances.


I have been in correspondence with George Huxtable about Lunars and their
accuracy. I thought some input here might be of interest for the purposes
of discussion; if it is of interest.
-----
I have spent some time for my amusement programming my HP50G calculator for
almanac data and then moved on to Lunar distance clearing so naturally that
stimulated my getting the sextant out to try it out again.  I played around
with 'Lunars' many years ago and was not impressed then, getting poor
results so was interested to see it better could be achieved being a little
older and wiser :-) !

I relatively quickly obtained Moon/Jupiter shot one evening some weeks ago,
which gave a time result which was three minutes 'out' from true.  Last
Monday I tried again with more care, and found the Moon/Jupiter combination
was within six seconds of true, but the Moon/Vega combination was 51
seconds 'out'.

The main purpose of this short posting is to mention and promote the use of
simple graphs of the observations for obtaining better accuracy than would
otherwise be the case with simple averaging for the mean in a set of
observations.  I know that statistical calculations will give better
results too (at some considerably more trouble with calculations), but as I
have always tried to promote the practical side of navigation here, it
might be appropriate to mention it again with a recent example I have used
myself.

Part of my text to George (to save my writing it again):-

I took Moon/Jupiter and Moon/Vega sights.  Jupiter was 28 degrees to the
left of the Moon, same level;  and Vega was 67 to the right and higher.
I took several shots over some minutes each.   I dispensed with the
altitude
shots by bubble sextant as I decided I am interested only in the accuracy
considerations of the lunar distance itself,  and decided to use calculated
(almanac) altitude values for the 'clearing' using my calculator programme
(based on the rigorous formula mentioned in Cotter's book 'History of
Nautical Astronomy').

I then made graphs of the time / lunar distance observations.
Graphing observations. I think, gives better accuracy than by just
averaging shots. The change in lunar distance over say 15 or 20 minutes is
indeed quite enough to be 'graphable'.

Graphs are very powerful,  as they allow instantly to see the trend with
the
eye and have an enormous advantage in being able to disallow any obvious
rogue shots immediately. I think it is probably a better method than a
'proper' statistical calculation (eg. least squares) because you can
eliminate shots which would otherwise be included;   it is also simple, can
be done quickly, possibly faster than a calculation procedure (unless
programmed), and is a kind of 'analogue', visual, integration method for
just
a few points (observations) -obtaining much better accuracy than any other
method.  You will note in the graph of Vega/Moon this is shown quite well
with one shot being 'way out' which is immediately discarded.

The best result was the Moon/Jupiter combination giving a six second error
only.  The Moon/Vega set was 51 seconds 'out' from absolute.

This was probably because Jupiter is very bright, the LDist was smaller and
the sextant easier to handle and adjust for co-incidence of object/Moon's
limb.
This is an important note for practical lunar distances in my opinion,  as
it is quite clear to me that stars are difficult to use unless they really
are very bright. I had to use shades for the Moon for both Jupiter and
Vega, but especially so for Vega, it being much less bright than Jupiter.
---------

My original negative conclusion about 'lunars' was confirmed with my recent
attempt, where I obtained time within only three minutes for a single shot
is therefore still valid in my mind;  i.e. 'lunars' are very difficult
indeed to get anything like real accuracy for practical use of obtaining
time down to seconds.

The Moon only moves 30 minutes of arc in one hour against the background of
stars;  i.e.  one minute of arc in two minutes of time.  That implies a
necessary measurement accuracy of one second of arc measurement for a
theoretical three seconds of time accuracy in the final result... and that
ignoring all the errors and inaccuracies within the method... A very tall
order indeed when at sea ! Impossible in fact.

I cannot understand (and still do not believe) how it was possible at all
at sea to use it as a practical method,  and still think it was most likely
force-fed to mariners by the astronomers in the eighteenth century for the
simple reason there was no other practical method at all available until
Harrison. Also, because the astronomers were used to dry land and
terra-firma (the more firma - the less terror), and they were used to using
instruments which they knew were good for seconds of arc accuracy at any
time.

For today's use Lunars are great fun on dry land;  but I remain deeply
sceptical about their use in any way as a practical method of navigation
expertise when at sea.

Douglas Denny.
Chichester. England.

P.S.
I do hope there are no terrible typos in this as it is a bit long. Sorry if
there are.

=============================
RESULTS:-

Lunars. Observations:-

Observer's position:
(WSG84)
Lat:  North 50deg-49.910
Long: West 000deg-51.300

Monday 20th IX 2010. All times GMT. (UTC)

MOON/JUPITER.
21Hr-50'-10" ....... 28deg-33'
     53-55                 30.5
     57-28                 31
     59-44                 29.8
22Hr-01'-46"               28.7
     03-39                 27.9
     04-52                 27.3
     07-29                 27.1

Later (not used):-
22Hr-44'-36"         28deg-14.5
     46-28                 13.5
     48-10                 12.8
     49-16                 12.2
     50-50                 12.1
===============================

MOON/VEGA

22Hr-25'-15"         67deg-13'.0
     28-05                 13.5
     31-29                 13.7
     33-01                 14.5
     35-05                 13.1
     36-24                 13.8
     37-48                 14.1
     41-15                 14.5
===============================

I used altitudes obtained from almanac data for the exact single time/lunar
distance chosen from the graphs, for use in clearing the distance.

I used Moon/Jupiter from the graph at 22Hr-01'-00" with observed LD to limb
of 28deg-29'
this, minus 14.7 Semi-Diam giving 28-14.3 observed LD. A calculated HP was
53.999'
The 'cleared' LD was:-  28deg-08.'756  giving an interpolated time using
the Oliv Soft Lunar distance almanac of 22Hr-01'-6.4"
i.e.  an error of 6.4"


=============
for the Moon/Vega combination:_

I used 22Hr-40'-00" from the graph with LD Vega to limb of 67deg-14.4
plus SD this time gives observed LD of 67deg-29.'1
The 'cleared' LD was 66deg-58.'97
which interpolated for time gives 22Hr-39'-8.1"
i.e. an error of 51.9"

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

   

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