NavList:

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 “2^nd 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