NavList:

DW, you wrote:

"I believe it possible to calculate the partial of LD with respect to
Long analytically."

This came up in the context of the longitude error that results from a given error in the measured lunar distance. I think you're thinking in terms of a lunar line of position (as we discussed in the past three months) in which a lunar distance observation at known GMT yields a line of position, and two lines of position yields the observer's latitude and longitude. There may well be a partial derivative analysis that might possibly be of use in that case. But it's important to distinguish these lunar lines of position from a traditional 18th/19th century lunar distance analysis in which the cleared lunar distance yields Greenwich Time. That Greenwich Time combined with a value of Local Time from a time sight gives longitude. There's no particular complexity in this case in calculating the rate of change of the true lunar distance and multiplying it by the error in observed distance resulting in the error in the longitude. For example, the rate of change of the geocentric lunar distance might be 1.2 minutes of arc in two minutes of time. If the error in the observed lunar is 0.6 minutes of arc, then the error in the time is 60 seconds. And of course, one minute error in time implies 15 minutes of arc error in longitude. The rate of change of the geocentric lunar distance varies up and down a little bit around 1.0 depending on how close the Moon is to perigee or apogee and depending on whether the Moon is moving more or less directly toward/away from the star. This variation is rather small in the great majority of cases, so if you want a quick estimate of the longitude error from a given error in the lunar distance, just multiply by 30. That is, for a traditional lunar distance observation, an error of 0.1 minutes of arc in the observed lunar distance implies an error of 3 minutes of arc in the resulting longitude, assuming no errors in the clearing calculation or the almanac data.