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Kent Nordstrom has provided a thoughtful contribution about the accuracy of 
lunar distance longitudes in the 19th century. However, he hasn't quoted any 
numbers for the various contributions, and it that way, some of the value 
has been lost. As we can see, some of the contributions he lists are 
negligible.

The bugbear of lunars is that dreadful factor of (about) 30. That is, every 
arc-minute of error in the lunar distance itself gives rise to an error in 
calculated Greenwich time of about 2 minutes, and thus an error of about 30 
arc-minutes in the longitude. That factor dominates the lunar problem. Other 
errors, particularly that in estimating local time, that are not subject to 
that factor of 30, so are much less important in assessing the longitude 
(though perhaps they should not be dismissed altogether). But we should 
concentrate on the errors in the lunar-distance itself.






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.
The Moon's altitude, in itself, is not a very important factor. It's 
importance is only in the fact that two corrections (important in 
themselves) used in the clearance process, depend on it. These are the 
parallax and the refraction. Moon parallax can never never more than about 
60', and as it varies with cos alt, to estimate that to, say 0.1' requires 
only that the altitude be known to within 6'; not at all demanding! 
Similarly to obtain a suitably precise figure for Moon refraction, which 
would then have been given in tables to the nearest second.. The Moon 
semidiameter is needed to a high precision, but is it's proportional to the 
horizontal parallax, and only about a quarter of it, that requirement is 
automatically met. Moon HP was quoted to the nearest arc-second, even in 
Maskelyne's first Almanac in 1767, even if he probably didn't known it then 
to that accuracy; but by the end of the century that Kent is considering, it 
certainly was well known.

There's a correction to the Moon's HP with observer's latitude, amounting to 
a few seconds, which is worth making if working to high accuracy, but that's 
a simple matter. There's a correction of similar size can be made to the 
Moon's semidiameter, depending on its altitude, but that's easy too. There's 
a tiny correction (always neglected in practice) might be made to the 
observed lunar distance, resulting from a tiny shift of the Moon's azimuth 
because of the Earth's non-spherical shape, but from what I remember it's no 
more than a very few arc-seconds.

So we don't need to worry, as Kent has been doing, about inaccuracies in the 
Moon's altitude itself; and similar arguments apply, but even more so for 
the other body, Sun or star or planet. It really boils down to the 
observer's precision in measuring the lunar distance, and in the Almanac's 
precision in predicting it. All those other errors become insignificant. If 
our navigator uses a scientific calculator to do his sums, or six-figure 
tables (or even an "approximate" method, which can in practice be remarkably 
precise), the additional errors intruduced are negligible, as long as he 
makes to mistakes.

Kent also worries about the errors in making a straight-line interpolation 
between the predictions, and suggests that second-differences need to be 
taken into account to allow for the curvature, depending on the interval. 
Well, Maskelyne settled on an interval of 3-hours, which seems remarkably 
prescient. It didn't ever need to be changed, and that interval of 3 hours 
prevailed over the whole history of lunars. Did any mariner need to do more 
than a straight-line interpolation between those predictions? Kent gives us 
no numbers, but I doubt it. Maskelyne did indeed need to use 
second-differences in CREATING these tables, but that was because he was 
calculating Moon positions at noon and midnight only (to save effort), so 
had to interpolate over that 12-hour interval to get his 3-hour tables.

In Maskelyne's day, few of his (or Mayer's) Moon predictions were more than 
an arc-minute out, and most were within half a minute. That very slowly 
improved over the next century, and by the 1860's, errors in Moon 
predictions would be rather negligible compared with errors in the lunar 
observation itself. And there the matter rested; sextants hardly changed.

Until the day that the Almanacs dropped their lunar predictions, a few years 
after 1900. Then things got a lot harder for the lunar navigator, if any of 
them remained. Instead of just looking up his lunar distance, to the second, 
he would have to obtain Moon and Sun positions individually, and calculate 
the included angle between them, to get lunar distance. And then all almanac 
corrections were provided only to the nearest 0.1', an error up to +/- .05. 
I reckon that a lotal of 14 such approximations are involved, which combine 
together to produce a standard deviation of 0.1'. (see Navlist 5282).But 
worse than that, the modern almanac states that the maximum error in its 
Moon GHA predictions might be as much as +/- 0.3'.

And more recently, things have got better again. With a computer on board, 
and appropriate software, you can predict lunar distances sufficiently 
accurately, for yourself. With internet access, you can download lunar 
predictions from Steven Wepster's website, to the second, at three-hour 
intervals, as for the old almanac.. If you want the whole thing done for 
you, can can rely of Frank Reed's lunar distance site. And now, once again, 
the overall error in the complete process is defined by the precision with 
which you can observe the lunar distance itself, and by little else.

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.

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

----- Original Message ----- 
From: "KENT AE NORDSTR�M" 
To: 
Sent: Sunday, June 08, 2008 7:32 PM
Subject: [NavList 5380] Accuracies in LD's


In an earlier e-mail I promised to provide my owm views on accuracies for 
LD's. Unfortunatley I have not been succesful to connect to NavList 
probably beause of my bad e-mail. Frank Reed has kindly helped to setup  my 
new account. Anyway here are my comments. If you find my writing below 
clumsy pls accept that I am neither English nor American:

Firstly: My comments are all related to the way old navigators did the 
lunars, that is roughly in the time span l760-1860.

Secondly: In order to smooth measurement errors the method I have applied 
requires

measurements of altitudes of the body, then the moon, then three distances, 
then altitude of the moon and finally altitude of the body. Hence all 
measurements will be normalized to the mean time of distance measurements.

Thirdly: I have assumed that there is no GPS onboard and no 
chronometer/other accurate time piece available.

Forthly: The reckoned position is not very accurate as it was in the old 
days.



When the altitudes of the moon and the other body are reduced some 
uncertainties are introduced even if corection tables are used:

- the moons altitude has be corrected for augmentation

- altitudes of the moon and the sun (if used) have to be corrected for the 
differencies in refraction between the measured egde and the geocentre of 
the body.

Corrections taken from tables provide average values, hence by using table 
values some small errors are introduced.



Refraction, in particular for low altitudes, are always uncertain. 
Refraction data and corrections for deviation in air pressure and 
temperature from normal taken from most tables are coarse and may (if you 
are lucky) represent the actual situation.



The moon's true local altitude has to compensated for earth flatness. The 
size and sign of this correction is dependant of the azimuth to the moon. If 
the azimuth is less than 90 degrees than a negative correction must be 
introduced, otherwise the correction is positive. The difficulty here is 
that azimuth must be estimated (not calculated because no other ephemeris 
data are available this early in the calculation).



The moon's HP has to be corrected for earth flatness before calculating the 
local parallax.



Correct reduced altitudes have implications on the calculation of the LD and 
consequently on the GMT, however errors in altitudes have much less 
implication on the GMT than an erronous  LD.



The observed distance has to compensated for the moon's augmentation. This 
is tricky because the angle between the distance line and the vertical line 
through the moon (and sun if used) has to be observed (which in old times 
was the normal way to do). This estimation will in practice be rather 
un-precise (if ephemeris data are available these angles can of course be 
calculated).



Finally. the earth flatness will also have implications on the final LD. A 
correction for the moon's parallax in azimuth due to earth flatness is 
needed before reaching the calculated true distance.



As you can see from above there are many sources of errors in each step of 
the calculation of the LD.



To this we have the classical difficulties for navigators:

-   how accurate are measurements of altitudes and distance(s)?

-   How accurate are the readouts of the sextant?

-   How accurate is the index error?



But for finding what we are searching for, i.e. the longitude, there are 
more difficulties.



Assuming that LD's are tabulated for the navigator (as in the old NA's) and 
that the calculated true distance will provide a first GMT then the next to 
consider is the correction for "2nd differencies", which is a way to 
compensate for un-regular movements of the moon. The longer interval between 
the tbulated true LD's the more need there is for this correction. Then a 
final GMT is obtained.



The local time (Mean Time=MT) has to be found as well. When the sun is used 
obtaining MT is rather trivial because the suns's LHA represents apparent 
time, which corrected with the time equation, gives the local time. If the 
other body is a star or a planet it is a little more complicated to find the 
MT (the moon is not recommended due to un-regular movements). What is needed 
here is to find the difference in sideral time between the reference point 
in time=Aries passage of the upper meridian (where an opinion of the 
longitude is needed, we have here some kind of Catch 22 situation) and the 
calculated sideral time at the observation, which is =LHA+Ascensio Recta. 
In this case accuracies in altitude, declination and latitude as well as the 
GHA of Aries will all have implications on the accuracy because they all 
depend on the final GMT.



The difference in sideral time has to be converted to MT and thereafter we 
can do the longitude calculation as  MT-GMT or vice verse.



It is of course possible to try to quantify errors as per above and 
calculate a final error using the method of Gaussian normal distribution. I 
am not going to do that but instead stating my personal opinion based on my 
own knowledge and the brief analysis above: Longitude can be found typically 
within 15-20 nautical miles from correct longitude. A very well trained 
observer may sometimes obtain 5-10 nautical miles. All measured on the 
equator.



By this answer I have hopefully answered Franks question about LD accuracies 
some weeks back.



Kent N





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