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For the delectations of the aficianados of Lunar distances, here are another set of results I took on the evening of Saturday 25th Sept 2010 here in Bosham on the South Coast of England. After such generous words from George who could resist posting another set?

The conditions were quite good with only slight Cirrus haze of the Moon but mostly very clear. Jupiter was very bright to the right of the Moon by 34 degrees and slightly lower in the SSW Az150. Capella was to the left of the Moon by 52 degrees and slightly lower in the NNw Az 48. Vega was to the right of the Moon by 94 degrees in the West Az 275. and slightly higher.

This time I used a tripod to steady the sextant to try to improve accuracy yet again. It is clear from the first graph of Moon / Jupiter where I used 'rested' and 'unrested' sights, that the results were not hugely different in either case - which surprised me.

Also, I found, again to my surprise, that having gone to more trouble yet again in steadying the sextant and trying hard, it made little difference to the overall accuracy as found before with the first set of lunars I shot, as the inherent variability seems to be around one minute of arc measuring no matter what, giving a result variability of two minutes of time. I cannot seem to better it. This means I have either reached my limits of measuring with the equipment used; or there are still systematic errors I am not aware of and eliminating.

The other thing I tested was George's hypothesis that a simple mathematical mean (average) of the results should be as good as my graphs. I found it appears he is right: with these results anyway.
For Jupiter, I tried a best fit line for all the points (lunar distance measurements) found; and for a second line which was only through the first points before changing from 'rested' to unrested' sextant; and finally with a mean result and time of all of them. It made little difference with any method, in fact the average came closer to the absolute than the graph - much to my chagrin - but still within the overall measurement accuracy so is still an open question - which is 'better'?

The main conclusion I have come to after this set confirm to me as found in my first experiments, that lunar distances are possible alright (ignoring completely the question of the profound difficulties in the calculations that sailors had in the past) but the system is inherently inaccurate (or more correctly: of strictly limited accuracy with practical measurement with a sextant) because of the slow rate of change of the Moon's position with the background of stars. That bringing with it a requirement for better than one minute of arc measurement accuracy absolute for only two minutes of time accuracy in the final result.

It would be very nice if the Moon only moved at twice the rate it does when Lunar distances would then be a piece of cake! Well almost. Chronometers would not have been needed to be invented :-)

For astronomers the slow rate of movement of the Moon is OK, where seconds of arc measurements can be achieved with land-based equipment (large telescopes with large divided circular scales) fairly easily, but is severely limiting for sea going navigators.

Nevertheless, I have to concede George Huxtable's correctly pointing out to me that Lunars were used as a successful method for a hundred years - by Cook even, in mapping New Zealand; so as a method in the right hands one cannot argue against it being practical and useful to sailors. I am just amazed they did it so well - better than I can achieve on dry land.

But then they did not have chronometers, so there was no alternative. If you wanted longitude you used Lunars or did without. From a practical navigating point of view I suspect even a 'rough' longitude obtained by lunar distance was of immense value and would make the difference between sailing one way to run down a latitude or the other way which was perhaps thousands of miles further; or make the difference as Sir Cloudisly Shovel found to his cost between sailing his ships onto the Scilly's rocks or not.

Anyway for those who have an interest here are more measurements and graphs:-

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

RESULTS:-

Lunars. Observations:-

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

Saturday 25th. IX. 2010.
All times GMT. (UTC)

MOON/JUPITER. (Jupiter to far limb)
(Tripod stand used for first six.
N.B I have just noticed the graph is incorrect slightly: it includes the seventh as marked).

22Hr-16'-50" 34deg-06'.5
19-17 8.2
20-55 8.3
23-49 9.4
26-15 10.4
28-33 11.0
30-14(hand held) 12.4
34-55 15.2
37-18 16.0
40-34 16.9
42-22 17.3
-----------------------------

MOON/CAPELLA. (Moon nearest limb to star)
(tripod stand for steadying sextant).

22Hr-51'-00" 52deg-06'.2
54-48 03.8
57-50 02.6
23Hr-01'-44" 01.7
03'-53" 00.1
05-19 52deg-00.0
07-52 51deg-58'.5
10-36 58.2
13-10 56.0
16-39 54.85
20-22 53.4
------------------------------

MOON/VEGA (Vega to far limb)
(tripod steady used)

23HR-34'-13" 94deg-24'.7
38-49 22.0
43-23 22.0
48-54 21.9
56-30 22.9
-------------------------------

The Moon/Vega sights were very difficult to achieve with a lunar distance of over 90 degrees even though the sextant was nearly vertical and canted to the right slightly by about 20 degrees. The swinging of the sextant to ensure correct alignment with the limb of the Moon was much more difficult than when the distance between Moon and star is only short, say 30 degrees, as the star at 90+ degrees has an unusual motion making eye/hand coordination exceedingly difficult. Another reason to avoid large lunar distances if at sea.

I used altitudes obtained from computer almanac data (ICE programme) for the altitudes of the celestial bodies, for exact single time/lunar distance chosen from the graphs, or for the average, for use in clearing the distance.

Clearing the distance I used direct calculation with a programmed HP50g calculator using:-

D = arc cos ( sin S sin M + (cos d - sin s sin m ) (cos S cos M ) / (cos s
> cos m))

I used Cotter's book as a basis to start. Cotter has a long and very interesting chapter on the lunar distance methods and tables in 'A History of Nautical Astronomy' pps 205 to 243.

For the almanac Lunar distances, instead of calculating, as I do not yet have the planets almanac on my calculator I took the easy way out and used the nice ones produced by Oliv Soft in pdf format. These are to be found at:-

http://navigation.lediouris.net/

I used simple interpolation between the hourly lunar distances which is quite accceptable as the second differences, representing rate of change per hour of the moon's motion, is only 0.04 moa per hour, which in 30 moa per hour Moon's motion is only 4.7 seconds in time error per hour max.
(The early tables used three hourly Lunar Distances and the Moon's motion is not linear so prop logs were used to give greater interpolation accuracy - not needed here with hourly tabulated Lunar distances).

======================
CALCULATIONS.

MOON/JUPITER.

Moon/Jupiter. From Graph:
Observed distance to limb at 22HR-30'-00" was 34deg-12'.50
minus SD 14'.95 = 33deg-57'.55 observed LD.

Moon: SD 14.95 HP 54.90 Alt 39deg-33.2 plus refn 1'.2
gives observed alt = 39deg-34.4

Jupiter: true Alt 34deg-48'0 plus refn 1'.4
gives observed Alt = 34deg-49'.4

Cleared lunar distance = 33deg.897 470
= 33deg-53'.848
equiv = 22HR-31'-58".8

which is an error of + 1 minute 59 seconds. Which represents one minute of arc measurement accuracy.

This worked out for the lower line of the graph which only includes the first seven points (mostly steadied by tripod) = 22Hr-31'.1
i.e equiv = 1 minute 6 seconds in time error.
=========================

MOON / CAPELLA
from graph:

Using 23HR-00'-00" = Observed lunar distance to limb = 52deg-02'.0
plus SD 14'.95 gives Observed LD = 52deg-16'.95

Moon true Alt = 43deg-35'.7 plus refn 1'.1
gives Observed Alt = 43deg-36'.8 HP (Oliv Soft almanac) 54'.91

Capella true Alt = 33deg-35'.8 plus refn
gives observed Alt = 33deg-37'.3

Cleared Lunar distance = 52deg.136 969
=52deg-08'.21
equiv =23Hr-00'-32".1

i.e. an error of 32".1 in time equivalent to a quarter of a minute of arc measurement in the lunar distance.

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

MOON / VEGA

Using 'best fit' from graph (N.B. Which I made after I scanned the grpahs made for Jupter and Capella and is not included on enclosed picture of graphs. It was poor in measurement resolution of the five points compared to the other graphs - I can include it later if requested).

Observed Lunar distance to limb at 23Hr-40'-00" = 94deg-23'.25
minus SD 14'.95 gives observed Lunar distance = 94deg-08'.3

Moon true alt = 48deg-24'.4 plus refn 0'.9
gives observed Moon Alt = 48deg-25'.3

Vega true alt = 35deg-08'.1 plus refn
gives observed alt = 35deg-09'.5

Cleared Lunar distance = 93deg.574 628
= 93deg - 34'.478
equiv = 23Hr - 44'-29".7

An error of four minutes 29.7 seconds which is equivalent to two and a quarter minutes of arc measurement error.

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

USING AVERAGES

Moon / Jupiter

Mean time of sightings = 22hr-29'-36"
Mean Lunar distance to limb = 34deg-11'.96
Mean LD minus SD = 33deg-57'.01
Moon Observed Alt (true plus refn) = 39deg-31'.1
Jupiter observed Alt (true plus refn)= 34deg-48'.1

Cleared LD = 33deg.888 062
= 33deg-53'.28
equiv = 22Hr-30'-51".8

An error of 1 minute 15.8 seconds in time equivalent to about half a minute of arc measurement error.

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

Please excuse any obvious typos (no doubt I shall find them when I look through this again but cannot find any now as it was tiring doing all that)

Douglas Denny.
Chichester. England.
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