A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2020 Feb 1, 18:08 -0800
Paul Hirose, you wrote:
"My software says the Moon to Venus rate in your case was +0.271' per minute time. That's 67% of the total angular velocity of both bodies (100% means the bodies are moving directly toward or away from each other)."
That isn't right. First the rate is way off. It should about 0.43' per minute of time. Why so far out? I suspect you're using the topocentric rate which folds in changes from the Moon's varying parallax in altitude. But we clear the parallax from the sights, so what counts is the rate of change of the corresponding cleared (or geocentric) distances. You may remember some 15 or 16 years ago (yes, that long ago!) when George Huxtable got caught in a trap which he thought was an important discovery and labeled "parallactic retardation". George later tried to erase all traces of his "parallactic retardation" but alas, that's not how the Internet works. The rate of change of the topocentric distance does have some relevance in narrow cases, but it's really not the right way to look at the sensitivity to errors in observed lunars in most cases.
The Moon-Venus rate for Jim's lunar was over 95% of the Sun-Moon rate at the same date and time. So quite respectable, hardly reduced at all. You suggested that this is connected with whether the bodies were moving directly toward or away from each other. If you had seen the Moon and Venus that evening, or if you had simulated the circumstances in a simple app, like Stellarium, you would have seen that the ray from the Moon to Venus was very nearly aligned with the center of the Moon's limb (very nearly perpendicular to the line through the horns) so this geometric factor was not especially important. It is worth noting that Venus was chasing the Moon on this date which reduced the rate relative to other celestial bodies, including the Sun. But this is down in the noise.
Finally, it's important to notice that this was the same date as lunar apogee. The Moon's angular rate moving against the celestial sphere is noticeably slower at apogee than at perigee --for distances measured from any other celestial body. The Moon's angular rate is reduced because it's physically moving slower (in "miles per hour" relative to the Earth) and because an angular rate, even at the same speed, is lower when the object is further away.
You suggested a multiplier of "3.7" to get the error in GMT from the calculated error in the observed distance. And that's way off (presumably for the same reason, as above). In fact, the multiplier, if properly calculated in detail for this case, looks to be about 2.3. For a Sun-Moon lunar on this date, the multiplier would have been about 2.2. That is, an error of 1 minute of arc in the observed Venus-Moon lunar distance would produce an error of 2.3 minutes (2m18s) in the resulting GMT. The usual, recommended, quick rule is a simple multiplier of 2.0 (equivalent to a multiplier of 30 for the error in longitude). The primary reason the multiplier is 2.3 for this lunar and not 2.0 is because it was lunar apogee.