NavList:

Homer S., you wrote:
"I, too, am not clear on how to get a longitude after having determined GMT from a lunar distance measurement."

Part of the problem here is that we learn about this process under a title or description like "longitude by lunars". So where is the longitude equation?! Another problem is that students have a tendency to over-think the issue --it's really much simpler than it appears to be (unless you're going for the full historical treatment ...more in another paragraph below).

When you do normal modern celestial sights, there's no equation that says "longitude = [...]" and for that matter no corresponding equation that says "latitude = [...]". Instead, in the normal work, we get latitude and longitude by plotting and crossing lines of position. If you realize after you have worked up a multi-star fix that your GMT was wrong (maybe something simple, like mis-reading the minutes on a deck/hack watch), most navigators would not immediately know what to do to fix the "fix" except to re-work the whole thing with the corrected time. And, in fact, that is a viable, albeit lengthy, solution. The GMT from "lunars" can be treated the same way. The lunar observation fundamentally gives you Greenwich Time. That feeds into the solution for longitude the same way that a time from a chronometer or a radio time signal would feed into a position fix. There's no special procedure just because the GMT comes from a lunar.

So we just re-work the fix from beginning to end with the lunar GMT? Yes, that works just fine, but as I noted above, it's a bit of a "lengthy" solution. There is a simple shortcut valid for moderate changes in GMT: you shift the longitude of your fix by one minute of arc of longitude east of west for every four seconds of GMT difference. If your new or corrected GMT (from a lunar, a radio time signal, or other source) is later than the time standard you were using previously, then your fix moves west by one minute of longitude every four seconds (because all of the the subStar points that went into building your fix are moving in that direction with advancing time as the Earth itself turns east, towards the rising Sun side). [Note: if you have a Moon altitude in your fix, this is more complicated. So don't do that... Also, the exact ratio of four seconds per m.o.a. of longitude is slightly imperfect, missing some detail, but, as I say, it's fine for "moderate" changes in GMT]

Suppose you want to try for the more historical approach to this. In that case, you calculate the local time (equivalently the LHA) for the "other body" in the lunar observation [this is what Bob B. illustrated in his earlier post, and it was that "clue" that you mentioned from the very end of the last workshop]. For the most common historical case, that other body was the Sun so the local time here is the standard navigation calculation for "sundial time" or what we call "Local Apparent Time". You may recall from my workshop "Celestial Navigation in the Age of Sail" that this was the standard math for longitude for well over a century. It was known as a "time sight" (also known, in somewhat misleading terminology in British navigation as "longitude by chronometer"). You can work up a time sight for any body by turning around the usual equation for Hc that you know from the intercept method using the value for Ho as input and treating LHA as an unknown:
   cos Ho = sin Dec sin Lat + cos Dec cos Lat cos LHA   ...solve for LHA.
Historically it looked different on paper since the logarithmic solution benefits from a different form, but fundamentally this was the same math. Note that LHA would normally be treated as an angle in degrees, but it can be re-written very simply as an angle in hours (15°=1hr), and then LHA for the Sun is simply Local Apparent Time. When solved this way, the connection to longitude is much more obvious: the longitude (in time units) is just the difference in time from Greenwich to the ship: Longitude = Gr. Apparent Time - Local Apparent Time. There's an additional wrinkle to connect Apparent Time to Mean Time, but for most of the run of lunars (1760s-1834), the tables of lunar distances directly yielded G.A.T. so no extra step was needed.

It's worth remembering that all of this discussion (above) is long obsolete since no one —ever— loses GMT in any scenario where lunars might come to the rescue. Instead, in the modern world, we know the GMT (UT) when we take our lunars, and the result that matters, either for testing our observing skill or testing our sextants, is the error in the lunar observation itself. That's what my web app displays that observation error as the primary product of a lunar observation.

Frank Reed

PS: Celestial Navigation in the Age of Sail is now scheduled a month from now: Oct 18,19 at the Treworgy Planetarium at Mystic Seaport, and Oct 20,21,22 online.