Lunar Distance by Calculator — The Letcher Method

The lunar distance method is a way to find UTC by measuring the angular distance between the moon and an adjacent body (sun, star or planet).  The measurements are called "lunars." And finding UTC is equivalent to finding your longitude. Recall we can find an accurate latitude without knowing the correct time, but standard procedures requires accurate time to find longitude. Lunars are thus a way to fill in your position if you have lost accurate time—or, starting from scratch, you never had accurate time.

It is an advanced skill in cel nav, because it requires very accurate sights and special analysis. Finding UTC to an accuracy of ± 30 sec or so is considered good work. We have notes on the history and practice of the method (starpath.com/lunars), which shows that there are essentially three approaches to the solution.

The first solution is the easy one. There are numerous apps online and for sale that solve lunars. Our own StarPilot was one of the first ones.  With these apps, you can just enter your DR position, the date and best guess of the right time, the name of the body you are using, and the lunar distance. There is usually an optional input for the measured altitudes of the moon and body, but these are not needed accurately, so they can be computed from the DR position. 

Everything else is computed and the output is the right time and right longitude, keeping in mind the uncertainty of the process: namely, every 0.1' of error in the  measured distance or clearing process leads to about 12 sec of time error, which corresponds to 3' for longitude error. 

The second solution is just the opposite: do it all by books and tables, like it was done in the late 1700s to early 1800s. The only math required is adding and subtracting longish numbers, but the tables and procedures can be complicated. With that said, this is the only logical solution, in that it is waterproof and no batteries are required.

Luckily, we have a modern all-paper solution created some years ago by Starpath friend and associate the late Bruce Stark. His book Stark Tables for Clearing the Lunar Distance — And Finding Universal Time by Sextant Observation has become a modern classic in cel nav studies. Experts consider his solution superior to some of those actually used historically (see discussion in the lunars link above.) But like the historic versions, doing this all with tables takes some practice before it becomes routine.

The third solution, which is the subject at hand, is a compromise of sorts between the first two. Namely we use a set of four trig equations that we can solve on any simple trig calculator along with standard sight reduction tables and a Nautical Almanac, and we compute the solution "by hand."

To my knowledge, this method was first put together by John S. Letcher, Jr in his excellent 1977 book on cel nav called Self-Contained Celestial Navigation with H.O. 208.  He was also a major technical authority and innovator of self steering devices, as presented in his 1974 book Self Steering for Sailing Craft. Both books are rare, but still found periodically in used book sites.

Below is an outline of his method. It has been elaborated upon in the 1980 book by Shufeldt and Newcomer called The Calculator Afloat, which is online in full. 

To help learn this process, we made an app (Letcher Lunar Distance Calculator)  and spreadsheet that solves the Letcher method that can be used to double check that you are computing the terms correctly. The app includes the examples presented in both books above. There is no logic to using this for actual solutions as there are other apps that are equally, if not a bit more, accurate that require much less input. The app and spreadsheet are purely training tools.

The idea here is you practice this a few times then write this prescription in your logbook to fall back on as needed, keeping in mind that the StarPilot, which covers all aspects of ocean and inland navigation computations, includes a lunar solution if needed.

It is important to skim over the discussion in the lunars link above. The summary is:

1) We measure the distance between moon and another body , which will be edge to edge. Then we use the Almanac to find the semi diameter of the moon (and sun if using it) so we can correct the edge to edge to get D, the lunar distance center to center, at time Ts which is our best guess of the UTC. In practice we need to take several sights and plot them D vs T then do a best fit to the line and find our best estimate of D and its corresponding Ts.

2) Then we "clear" that distance, which in this case means applying two corrections to D, refraction (R) and parallax (P). These corrections require us to compute the Hc of the moon (Hm) and of the body (Hb) at time Ts, which we do by looking up their GHAs and Decs, and then compute the distances with the formulas below. 

Note if you are underway, or have good horizon for the bodies on land, you can measure these heights in the normal way, and plot them to find a fix. The Lat of that fix will be correct, but the Lon will be off by the error in Ts, which we are finding from the lunars.

3) Once we have the cleared lunar (Dc) we need to see what time the two bodies where that far apart. We do this by computing the distance a the whole hour before Ts and at the whole hour after Ts.

Note that the lunar distance is just the zenith distance (z) to the moon assuming you are standing at the GP of the body being used. Thus D = z = 90º - Hc, which we can compute using the navigation triangle formula below (we just change our a-Lat to the dec of the body and our a-Lon to the GHA of the body).

Here is how the app input looks, showing also the Almanac data we need to enter.

Here is the sample solution from the Shufeldt book.

Once we have our measured Dc, we need to find out what time were these two bodies that far apart, keeping in mind the main premise of this measurement being that the moon moves eastward relative to the other bodies in the sky at a rate of about 12º per day, which is about 2' per minute. Not much, but right at the edge of our being able to measure this with standard sextant.

We figure this by interpolation as shown below:

Then within the accuracy of this method, Tc is he right time and Tc - Ts is our watch error.  The corresponding Lon error is 15'/1 min of time error.

The app is essentially two worked out examples that you can test with your own calculator.  The spreadsheets are there for those who want to look at that to see how these equations are entered into Excel.