A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Wendel Brunner
Date: 2019 Feb 25, 19:32 -0800
The Latitude is known from meridian passage of the sun or some other body. The iteration involves assuming a time, and determining the corresponding assumed longitude for the assumed time from an observation of the altitude of a body and the known Latitude. Calculate the Observed Lunar Distance at the assumed position and time, and compare that with the actual Observed Lunar Distance. The difference provides a correction for the assumed time, and thus a corrosponding correction for the assumed Longitude. I use a time change of 115 seconds/minute of arc of LD, a middle geocentric rate of LD change, to make that time correction, but any value around that will do. Then, with the new assumed time and Longitude, do it again. This converges to less than 0.1' of difference in Lunar Distance - better than your measurements- in two iterations in almost all cases. At that point, you are done.
You can start with errors in assumed time of hours, with a corresponding error in assumed Longitude of 15 degrees for each hour, and it still converges in two iterations.
This is a lot of calculation, considerably more than with more traditional methods of clearing the Lunar distance and getting time. But if you have a position from a uncertain clock, entering the position and clock time along with an Observed Lunar Distance into a calculator programmed for Lunars, like the StarPath calculator (or the Casio one I programmed), gives the correct time and Longitude in seconds. Or you can do it with a hand calculator, or even with logarithms if you are a masochist. This simplifies the observations, and requires only the measurment of the Lunar Distance and the altitude of any one other celestial body. These observations do not need to be simultaneous; just both timed by the uncertain clock. The observations required for more traditional methods of Lunar Distance, however, are (generaly) not that difficult, so the trade-off of increased calculation effort may definitly not be worth the effort. But if you can't get the altitudes of the sun or moon because of clouds or fog, this may be the only way to go.
This is a link to a complete article describing the process in detail.