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Re: GMT of lunar altitudes when possible?
From: Frank Reed
Date: 2019 Jul 9, 20:16 -0700

Herman Dekker, you wrote:
"When I read about GMT of lunar altitudes. (I know not the most accurate method for determining GMT) there is mentioned the moon must be near E or W bearing."

Not so much! That's not the critical test. The key with GMT/UT by lunar altitude is that the Moon's altitude should be maximally impacted by the Moon's motion relative to the celestial sphere (background stars). For that to happen, the horns of the Moon, or equivalently the Moon's north-south axis, should be nearly horizontal. If that's the case, then the Moon's motion on the celestial sphere of about 0.5 minutes of arc per minute of time has maximum impact on the Moon's altitude. If the Moon is tilted as much as 30°, even 45°, the reduction in accuracy is not all that bad (about 13% less accurate at 30° tilt). Note, too, that the Moon can be quite close to due E or W and yet its horns might be nearly vertical in higher latitude at the right point in the Moon's orbit --directly contradicting the prime vertical rule.

Getting UT by lunar altitude is less accurate than UT by lunar distances fundamentally because the horizon is uncertain. This problem can be reduced by shooting a second altitude (star or planet) that is as nearly aligned in the same azimuth as possible. Uncertainty in horizon refraction (and resulting anomalous dip) can then cancel out. In fact, if the Moon and the second body are on identical azimuths, within some limits, then there is a direct and simple correspondence between UT by lunar altitudes and UT by lunar distances. The latter will still almost always be more accurate since we can count on Moon-star angle observations being typically three or four times more accurate than altitudes of the bodies separately.

So what about these sources that say the Moon should be near the prime vertical? They're wrong. It's not terribly important, from a certain point of view, because in fact when the horns of the Moon are nearly horizontal, the Moon will generally bear either east or west within +/-30° or so. It will also usually be relatively low in the sky when this condition is met. But from another point of view, it's clearly not right at all: seeing the Moon nearly due east or due west does not imply that the horns will be horizontal.

Note that some sources will tell you that you need to shoot a third star. This is not entirely true. If your latitude is known from any source, including moderately good dead reckoning, then you don't need that third star. Fundamentally for this method, we need a Moon altitude when the horns are nearly horizontal and one altitude of another body, preferably on similar azimuth to reduce the impact of anomalous refraction.

Let's think this through visually, in terms of the appearance of the sky. Suppose you go outside in twilight some night next week and you see several bright stars in the eastern sky. They all have altitudes which you could measure and record. Nestled among those bright stars, you see the Moon, and you record its altitude, too. Now suppose we wait around for five minutes. Those stars and the Moon will all climb in altitude by as much as 1.25°, dependent on latitude, etc., and the sky will darken as twilight deepens. But if we wait five minutes and simultaneously jet off to the west by 1.25° of longitude (not quite exact since we have to deal with the sidereal rate, but nearly so), then the stars will stay at the same altitudes, and the twilight will stay at almost exactly the same relative brightness. The Moon, however, will move. It will shift among the stars "on the celestial sphere" by about 2.5' of arc. Its altitude will change by that amount multiplied by the cosine of the angle of the Moon's motion across the celestial sphere relative to the perpendicular to the local horizon. If the horns are horizontal, all of that motion impacts the altitude. If the horns are vertical, then the motion is horizontal, and the altitude doesn't change. It's the tilt of the horns that determines the effectiveness of this method of getting absolute time by lunar altitude.

Note: I haven't completely convinced myself that I have thought this through completely, so I'm willing to be convinced otherwise. But I wouldn't post this without some solid confidence that this is right. I rarely have opportunities to shoot lunar altitudes for UT so practical tests have not come up.

Frank Reed
Conanicut Island USA

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