Sunday, September 25, 2005

Lunar Librations

After my recent post on the Harvest Moon, a friend emailed me with this:

I'm somehow disappointed with your picture of the moon at apogee and perigee. The non-circular orbit also has the effect of producing lunar librations so we can see more than 50% of the moon. I can't help but think this is in sync with the apogee and perigee of the moon so the two images should look different (the two extremes) instead of being a 30% reduction of one another. But I'm just guessing...

The animation at right (shamelessly ripped from Astronomy Picture of the Day, but modified to slow it down) shows the lunar libration my friend speaks of. I, too, knew of the librations without really understanding the mechanism behind them. My friend's email nudged me into a little research.

My initial guess about lunar libration was that these two things caused it:

1: The tilt of the lunar orbit (5.2 degrees) with respect to the earth's rotation, coupled with the latitude (on the Earth) of the observer. This slightly "tilts" the view an observer has, to see slightly past the north or south pole.

2: The displacement of an observer viewing the moon at moonrise versus an observer at moonset. This would allow an observer to see slightly past the nominal east and west sides of the moon.

It turns out that both of the mechanisms I described above do contribute to libration (both latitudnal and longitudinal), but I completely missed two other mechanisms, the first of which causes the most libration:

1: The moon's rotational rate is constant, while its speed in its orbit about the Earth is variable. More on this below.

2: The moon's rotation is tiled about 1.5 degrees to the plane of its orbit. This adds another 1.5 degrees to the first effect I had guessed, for a total of 6.7 degrees.

For me the most interesting of these four mechanisms is the one that has the biggest effect, and that I had completely missed. It depends on a marvelously subtle interplay between the moon's constant rotational speed and variable orbital speed. As my friend guessed, the ellipical (non-circular) orbit of the moon is the cause of this — but the effect is the opposite of what he guessed: at apogee and perigee, there is no libration at all!

This unexpected outcome derives from the characteristics of an elliptical orbit. At perigee (the point of the moon's closest approach to the Earth) the moon is moving faster than at apogee (the point of the moon's farthest distance from Earth). The way I puzzled this out was to divide the moon's orbit into quarters, with one dividing line from perigee to apogee, and the other at 90 degrees to that. Starting at the perigee, let's call the quarter-orbits I, II, III, and IV. The moon is observably at dead center (with respect to longitudinal libration) at both perigee and apogee. To explain the longitudinal libration, let's follow the moon through the four quarters of its orbit:

Quarter I: The moon starts at perigee, where it is moving faster than at any other point in its orbit. During this quarter orbit, the moon will slow down some, but it's still moving faster than the average speed over the entire orbit. This means that the moon reachs the end of the first quarter of its orbit in slighly less than one quarter of the time it takes for an entire orbit — so the moon will have rotated slightly less than one quarter of a rotation. In fact, it turns out that the moon at this point has only rotated about 82 degrees — thus providing a view for an Earth-bound observer that is 8 degrees (90 - 82 = 8) of longitude shifted.

Quarter II: The moon is moving further from Earth, and is slowing down. At the end of this quarter, it has moved through exactly half of its orbit, and it has rotated through exactly half its rotation — so no longitudinal libration.

Quarter III: The moon is moving closer to Earth, and is speeding up. However, its speed during this quarter is lower than the average speed over the entire orbit. That means that the moon takes slightly more than one quarter of an orbital period to reach the end of this quarter orbit. Because of that, the moon has rotated slighly more than one quarter of a rotation — about 98 degrees instead of 90. That means that an Earth-bound observer has a view that is shifted 8 degrees (90 - 98 = -8) longitudinally, in the opposite direction as occurred in quarter I.

Quarter IV: The moon is moving closer to Earth, and is speeding up. At the end of this quarter, it has moved through its entire orbit, and it has rotated exactly once — so no longitudinal libration.

I'll bet that was more than most of you ever wanted to know about lunar libration! But if you're anything like as loony as I am, you can find more information about lunar libration here, here, here, especially here, and here.