On page 1 I described the phenomenon and causes of libration. Here I want to look at the complexities that influence the ability to observe the libration zones—those zones that are sometimes visible from the Earth and sometimes not. The causes of the latitudinal and longitudinal librations are quite different and change in different ways and it is the phase relationship between these two motions that dictate which areas of the Moon move into view.
Longitudinal libration is caused by the fact that the orbit of the Moon is an ellipse having the Earth at one focus*. Latitudinal libration is caused by the fact that the Moon's orbit is not in the plane of the Earth's equator and the Moon's axis of rotation is not at right angles to the plane of it's orbit. Both these causes are subject to precession—the gradual rotation of the axis of any rotating object, such as seen with a spinning gyroscope.
[*Actually the Earth and the Moon rotate about their common centre of gravity, called the barycentre, which lies about 4,600 km, or about ¾ of its radius, from the Earth's centre. This is a detail which does not affect the present discussion.]
Longitudinal libration is caused by the fact that the Moon's orbit is an ellipse and the major axis of that ellipse slowly rotates in a prograde (forward, west to east) direction within the plane of the orbit. The astronomical name for the major axis of an orbit is the line of the apsides, the apsides being the points closest and most remote from the orbited object—in this case the Earth—which, in this case, are the perigee and apogee of the Moon's orbit. This is known as apsidal precession and takes 8.84 years to complete one full rotation. Latitudinal libration is caused by the fact that the Moon's orbit is inclined to the Earth's equatorial plane and the Moon's equatorial plane is inclined to its orbit. These two planes precess together in what is called nodal precession (the nodes being the two points where the Moon's orbit intersects the plane of the Earth's equator) and is retrograde, that is in the opposite direction to the Moon's motion in its orbit, and takes 18.6 years to complete a full rotation.
The result of all this is that the Moon appears to rock north-south with a period that is slightly less than a sidereal month (the time it takes the Moon to revolve around the Earth relative to the stars) and rock east-west in slightly more than a sidereal month. The way the Moon actually appears to move depends on the phase relationship between these two motions. If they are in phase, that is they go positive together and negative together, the libration is north-east to south-west and the libration zones in the north-west and south-east are never well presented. If the two motions are 180° out of phase (that is one goes negative when the other goes positive) the libration zones in the north-west and south-east are well presented. If the situation is half-way in between (90° out of phase) then the Moon appears to wobble in a circle. These three situations are illustrated in Figures 1, 2, and 3 which show the librations each day for a period of 31 consecutive days in three different months when the librations are, roughly, in phase, 90° out of phase, and 180° out of phase. Obviously all combinations in between are possible.
| Plots of libration in latitude plotted against the libration in longitude for periods of 31 consecutive days | ||
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| Figure 1 Phase difference = ~0° |
Figure 2 Phase difference = ~90° |
Figure 3 Phase difference = ~180° |
The two precessions which give rise to these librations move around the orbit in opposite directions, and separate at about 0.34 days per sidereal month. This results in their coming together again in about 6 years.
| The latitudinal (in blue) and longitudinal (in red) librations plotted over a number of years. | |
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| Figure 4 The librations plotted through the 21st century. |
Figure 5 The librations and the phase angles (in green) between 2019 and 2045. Note that the scale for the phase angles has been divided by 18 to fit with the scale of the librations which run from about -10 to 10. |
These three cycles can be seen together in Figures 4 and 5, where the red line is the libration in latitude, having a period of 8.84 years; the blue line shows the libration in longitude having a period of 18.6 years; and the green line in Figure 5 shows the variation of the phase difference, rescaled to fit the graph, having a period of 6 years. The changes in libration are sinusoidal, but the phase angle changes in a linear fashion and changes direction at 180° giving rise to the triangular waveform shown. As time goes on from a phase difference of zero (when the two waveforms are exactly in phase) the phase difference increases to 360° and then jumps back to 0 because at that stage we have completed a whole circuit and a phase difference of, say, 370° is exactly the same as one of 10°. Equally a phase difference of 190° is exactly the same as a phase difference of -170°, but it is really only the absolute value of the phase difference which controls how the two librations combine to make the Moon appear to wobble in the sky. So to me it is clearer to think of the movements starting off exactly in phase, increasing until they are exactly out of phase, and then decreasing again until they come back into phase and the cycle starts again.
If the previous paragraph is too confusing, I am sorry, but I have been as confused as you may have been. The problem is that there are three different processes going on here. (1) The motion of the Moon around its orbit. (2) The precession of the apsides which controls longitudinal libration. (3) The precession of the nodes which controls the latitudinal libration. Figures 1 to 3 show the libration day by day through a period of 31 days (slightly more than the period of the Moon's phases), whereas Figure 4 shows the librations at intervals of a calendar year, and Figure 5 at intervals of a sidereal lunar month (27 days), through the years in an attempt to show how the phase relationships change as the two precessions progress.
As a consequence of how the curves in Figures 4, particularly, and 5 have been calculated, care has to be taken inferring when the cycles come into and out of phase. The graphs in Figure 4 were synchronised with the true libration for 26th June 2008 which is when I took my pictures of Mare Orientale so the other points are the libration the Moon would have had had it reached the same point in its orbit on 26th June in successive years. There is one other matter to be taken into consideration—the phase of the Moon when the libration occurs. There is no point in a favourable libration in the north-west, for example, if it occurs at around first quarter. The zone will be in darkness; try again in six months.
As an example of the practical consequences of all this, consider my pictures of Mare Orientale. These were taken in June 2008 when the longitudinal libration was about as far west as it ever is and the latitudinal libration was favourable but not at its maximum. But the latitude of M. Orientale is at only 20° south so the less-than-maximum, latitudinal libration was actually slightly beneficial. In 2019, I considered repeating these observations, but the librations by then were almost 180° out of phase and so good libration to the south-west was never occurring. Theoretically the librations should come into phase again in 2022 and indeed they do, being very favourable on 9th June 2022. The only problem is that the moon on that date is at first quarter. The best time will probably be on 14th January 2023 when the libration both in latitude and longitude will not be as good as it was in June 2008 but is quite favourable nonetheless. The next best opportunity appears to be around 15th August 2028 when again the librations will not be as good as they were in 2008. It seems that June 2008 was a particularly good time to observe Mare Orientale. The reasons are to do with the eccentricity of the Moon's orbit.
The eccentricity of the Moon's orbit is subject to continual change. Indeed it changes during a single orbit. These changes cause the distance at apogee and perigee to change and this effects the longitudinal libration. During the period 2008 to 2011 the distance of apogee varied over a range of only about 0.7% whereas the perigee varied over a range of 3.8%. Of course this also results in a change in the Moon's orbital velocity and hence the degree of libration. These changes result from a periodic variation in the length of the orbit's major axis (the line of the apsides) and this is a maximum when this axis points towards the Sun. In order to observe the extreme librations zone in the west, we need to look at some time between full moon and new moon, so third quarter is a sort of average time to look. To get maximal libration we need that to be at either perigee or apogee which means that the orbit will be orientated with its major axis at right angles to the direction of the Sun. This is when the orbital eccentricity is at its minumum and the perigee and apogee distances are at a minimum. It seems we cannot win....
To sum all this up, the complexities of the Moon's motion is so complex that only sophisticated software can really keep track of it. The fact that it can do so is evidenced by the fact that the time and track of solar eclipses can be predicted years in advance and are accurate—else why would any one travel halfway around the world to view an eclipse? I use the data given in the Virtual Moon Atlas and have no reason to suppose it is not good.
Finally is you want a good explanation of all the Moon's orbital motions I can recommend an article in NASA's archives called Eclipses and the Moon's Orbit by Fred Espenak of NASA's Goddard Space Flight Centre.
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