jdc
Well-known member
Now that us astro-nav aficionados have been woken up by GHA's interesting post, here's something I've been meaning to seek opinions on for a while.
A proposed method for deriving time and longitude from lunar observations
There are some problems with using lunar distances to derive time and longitude:
(i) although lunar distance tables could be calculated from other ephemeris, they are not published any-more;
(ii) they require one to measure the angle between heavenly bodies (rather than between them and the horizon), which is difficult enough in calm conditions and a cloudless sky let alone on a small boat in any kind of sea;
(iii) they therefore require both bodies to be visible and 'in distance' simultaneously. This last is a really significant constraint.
I propose and describe here an alternative method which does not have these disadvantages. It does require a clock which can measure time intervals to a few tens of seconds over a some hours (12 say). But this is easily possible with the cheapest and most trivial clockwork, and would have been within the capability of Slocum's famous tin alarm clock. The clock does not need setting and can just be started from an arbitrary time. A further advantage is that the calculations are just the normal ones for the Marc St. Hilaire method and so should be familiar.
The method:
1.Start the clock. It will have an as yet undetermined error, call it To. You probably have some idea of the time, to within a couple of hours anyway (you maybe know which ocean you are on;-)
2. Determine your Latitude by measuring the altitude of the sun at local mid day (or from Polaris). Move this, if required, by DR to keep a reasonable estimate of Latitude at all times.
3. Sometime, could be a few hours earlier or later, find a star or two, and/or the sun, such that the azimuth is nearly at 90 or 270 degrees, ie more or less due E or W. Take sights of these stars and/or sun, recording time as displayed on the clock. These sights need not be simultaneous and could be spread over some hours. Record the times as T1, T2 etc. They will all be in error by the same amount, To.
4. Take a sight of the moon when it is near E or W, recording the time, Tm, which is still offset by To. This need not be done at the same time as the sun or star sights are taken, which is a key advantage.
5. In the normal way work out the longitude (long_star) which the (collection of) star and sun sights gives you. As normal one has to move PLs by allowing for DR if required. You have an assumed time and you know latitude, so longitude can be derived. This is the familiar 'longitude by chronometer' method.
6. Do the same calculation to derive the longitude but this time using the moon sight (long_moon). In general this will be different from that given by the star sights.
7. Difference these two longitudes to give delta_long (delta_long = long_star - long_moon). Write down the values of To and delta_long.
8. Guess a new value of To, and go back to step 5. Repeat until one has enough points to draw a graph of delta_long against To (or do it algebraically). The value of To when delta_long is zero is the true value of To and thus clock error, and the corresponding value of longitude is the true longitude. Hence a position fix is obtained without chronometer. Fast convergence can be achieved by knowing that the longitude of the moon advances against that of the sun at a rate of 360 degrees in a synodic month, ie ~29.5 days.
I see no reason that this method shouldn't be about the same accuracy as the classic method of lunar distances (which isn't all that good, maybe 15 miles). It does require more, but simpler, calculations than the classic method, but we have calculators nowadays...
Can anyone see a fatal flaw, and if not, has anybody used or seen this method described before?
A proposed method for deriving time and longitude from lunar observations
There are some problems with using lunar distances to derive time and longitude:
(i) although lunar distance tables could be calculated from other ephemeris, they are not published any-more;
(ii) they require one to measure the angle between heavenly bodies (rather than between them and the horizon), which is difficult enough in calm conditions and a cloudless sky let alone on a small boat in any kind of sea;
(iii) they therefore require both bodies to be visible and 'in distance' simultaneously. This last is a really significant constraint.
I propose and describe here an alternative method which does not have these disadvantages. It does require a clock which can measure time intervals to a few tens of seconds over a some hours (12 say). But this is easily possible with the cheapest and most trivial clockwork, and would have been within the capability of Slocum's famous tin alarm clock. The clock does not need setting and can just be started from an arbitrary time. A further advantage is that the calculations are just the normal ones for the Marc St. Hilaire method and so should be familiar.
The method:
1.Start the clock. It will have an as yet undetermined error, call it To. You probably have some idea of the time, to within a couple of hours anyway (you maybe know which ocean you are on;-)
2. Determine your Latitude by measuring the altitude of the sun at local mid day (or from Polaris). Move this, if required, by DR to keep a reasonable estimate of Latitude at all times.
3. Sometime, could be a few hours earlier or later, find a star or two, and/or the sun, such that the azimuth is nearly at 90 or 270 degrees, ie more or less due E or W. Take sights of these stars and/or sun, recording time as displayed on the clock. These sights need not be simultaneous and could be spread over some hours. Record the times as T1, T2 etc. They will all be in error by the same amount, To.
4. Take a sight of the moon when it is near E or W, recording the time, Tm, which is still offset by To. This need not be done at the same time as the sun or star sights are taken, which is a key advantage.
5. In the normal way work out the longitude (long_star) which the (collection of) star and sun sights gives you. As normal one has to move PLs by allowing for DR if required. You have an assumed time and you know latitude, so longitude can be derived. This is the familiar 'longitude by chronometer' method.
6. Do the same calculation to derive the longitude but this time using the moon sight (long_moon). In general this will be different from that given by the star sights.
7. Difference these two longitudes to give delta_long (delta_long = long_star - long_moon). Write down the values of To and delta_long.
8. Guess a new value of To, and go back to step 5. Repeat until one has enough points to draw a graph of delta_long against To (or do it algebraically). The value of To when delta_long is zero is the true value of To and thus clock error, and the corresponding value of longitude is the true longitude. Hence a position fix is obtained without chronometer. Fast convergence can be achieved by knowing that the longitude of the moon advances against that of the sun at a rate of 360 degrees in a synodic month, ie ~29.5 days.
I see no reason that this method shouldn't be about the same accuracy as the classic method of lunar distances (which isn't all that good, maybe 15 miles). It does require more, but simpler, calculations than the classic method, but we have calculators nowadays...
Can anyone see a fatal flaw, and if not, has anybody used or seen this method described before?
