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Re: Moon meridian passage and its Tilt
From: Frank Reed
Date: 2019 Sep 18, 11:10 -0700

This computation and process is identical to deciding whether a Moon Upper Limb shot is required instead of a Moon Lower Limb. It's also connected to a related observational issue in lunars. The math for this is slightly simpler if you already know the lunar distance (which can, of course, either be directly observed or calculated from the altitudes and the difference in azimuth). Then in the triangle ZMS (Zenith, Moon, Sun) you can easily set up the standard spherical "law of cosines" to solve for the angle M at the Moon corner. If that angle is less than 90°, the  central point on the Moon's bright edge is pointing up (UL sight required). If the angle is greater than 90°, the bright edge is pointing down (LL sight). And if the angle is 90° exactly, the Moon is "vertical" in the sense that the "shadow line" or terminator (or nearly enough, the N/S rotational axis of the Moon) is aligned vertically relative to the local horizon.

There's a useful visual way to think through the question of the Moon's orientation at meridian passage. Look at the ecliptic (Stellarium key shortcut: hit "comma" to show the ecliptic). It's the path of the Sun around the sky and also quite close to the paths of the major planets and the Moon. So the Sun is always on the ecliptic, and the Moon is always close to the ecliptic. The bright edge of the Moon always faces the Sun and the terminator is perpendicular to that direction. As an excellent first approximation the terminator is simply perpendicular to the ecliptic. That means that your question can be reduced to asking about the crossing angle between the ecliptic and the meridian. It's easy to see that this varies between 23.45° down and 23.45° up. The terminator is tilted in that same range to the left or right of vertical. The maximum "tilt" of the Moon occurs when the ecliptic has maximum angle relative to the meridian. This happens when the Moon is near 0h or 12h right ascension (0° or 180° SHA). And of course halfway in between those points (when the Moon is above Orion or passing through Saggitarius) the tilt will be nearly zero -- the terminator will be vertical.

We can do a little better. The Moon's orbit is inclined relative to the ecliptic at an angle or about 5.18°. The crossing point or "ascending node" cycles around the ecliptic in a period of about 18.6 years. In some years the Moon's orbital inclination enhances the range of angles for the tilt of the terminator to about +/-28.63°. In other years the range is reduced to +/-18.27°. The extremes still occur in roughly (?) the same RA/SHA directions. There's an easy way to decide what the extremes will be by looking at the Moon's passage through Sagittarius or above Orion in any given month. If the Moon is well south of the ecliptic as it passes through Sagittarius or well north of it when above Orion, then the maximum crossing angle (a week later or before) is enhanced. Otherwise it's reduced. This is a direct consequence of a simple property of great circles and their crossing angles (a simple case: the angle a great circle makes with the equator at its crossing is identical to the maximum latitude reached by that great circle).

By the way, there's another navigational question that's connected with this analysis. Sometimes we may want to use the Moon for a meridian passage latitude. Some English-speaking navigators like the pun name "LAM" or "Local Apparent Moon" for this sight. It's trickier than it sounds because the Moon is moving rather quickly on the celestial sphere. When that motion has a significant north/south component, it can throw off the latitude somewhat and ruin any attempt to get longitude from a series of sights around maximum altitude.  But this is only an issue when the Moon's motion is N/S and that happens exactly in those circumstances when the Moon is tilted relative to the vertical at LAM. But if we can see that the Moon's terminator is almost perfectly vertical at meridian passage, the problem goes away because the Moon's motion on the celestial sphere is then very nearly aligned E/W. We know that this happens when the Moon is above Orion or passing through/near Sagittarius.

Frank Reed
Conanicut Island USA

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