## Lunar Libration and its Relationship to Phases and Distances

On average, the Moon rotates at the same rate as it revolves around the Earth, so it keeps the same side pointed towards us. However, sometimes we are able to see a bit more around the edges, an effect called libration. This webpage describes this phenomonon, and you can use the ideas here to figure out times when you can look to see parts of the Moon that are normally invisible to us.

The primary cause of libration in longitude (E to W) is the Moon's eccentric orbit. When the Moon is closer to the Earth it moves faster, and so the lunar rotation no longer keeps up with the orbital motion. The Moon moves right to left from day to day as it revolves around the Earth (observer facing south), so during perigee when the Moon is closest we are seeing the greatest increase of visibility on the right side. This side is west in our sky, but is defined as the eastern side of the Moon (there's a reason, see below). After about a week, the orbital motion is back to average and the extent to which we see the eastern side of the Moon, which has been increasing until this point, is maximized. Similarly, the visibility of the western limb of the Moon is maximized about a week after apogee. Another way to put it - when the first quarter Moon is at apogee, a week later during full Moon we have a good western libration; when the first quarter Moon is at perigee, the subsequent full Moon will have a good eastern libration. This behavior is shown in the following plot for the year 2019. The total effect is about +/- 8 degrees.

A plot of the longitudinal (east-west) libration of the Moon vs. distance between the Earth and the Moon for the year 2019. The arrows show how the Moon moves in the diagram. The visibility of the eastern limb of the Moon is maximized about a week after perigee, and the visibility of the western limb of the Moon is maximized a week after apogee.

What about latitudinal (north-south) librations and lunar phases? How these effects intertwine with longitudinal (east-west) libration is shown in the plot below. The blue areas denote full Moon, times when the illumination is greater than 99%. Because full Moons are not synchronized with perigees, and longitude librations are determined by the time of perigee, it follows that longitude librations do not line up with lunar phases. Note that the blue part of the curves (full Moon) sometimes have eastern librations, sometimes western librations, and sometimes effectively no librations. What matters for longitude librations is the time of apogee and perigee, not the phase.

Plot of the lunar libration (black line: longitude [east-west]; red line: latitude [north-south]) vs. day of the year for 2019. Time of apogee and perigee are marked with small and large dots, respectively. Longitude librations are synchronized with perigees and apogees, but latitude librations drift relative to perigees and apogees by several days over the course of the year.

Just around the left side of the Moon there is a huge basin, Mare Orientale (for the 'Eastern Sea') that just starts to come into view during the most extreme western librations. It is called the `Eastern Sea' because back when it was named the left side of the Moon was called the eastern limb, as it is in the east as we view it. However, we now call that limb the western limb because it would be to the west for an astronaut walking on the Moon. These east/west choices can be confusing, which is why I included a picture of the Moon in the figure.

If you wanted to see Mare Orientale, the best view is with the most western libration. You'd also want it illuminated, and maybe at sunrise, which would be a full Moon. The other time with long shadows would be sunset, but that's basically a new Moon and would be too close to the Sun to view well. From the graph, in 2019 you can see that the full Moon with the most western elongation occurred late in the year in December. The Moon goes through the extremes of eastern and western libration every month, so you could see Mare Orientale as long as the Moon is past full, so the region is illuminated, and has a large western libration. For 2019 those two conditions were satisfied during the second half of the year.

What about latitudinal librations? These motions mainly result from the 7-degree tilt of the Moon's rotational axis relative to its orbital plane (like seasons on the Earth). The period of latitudinal librations is mainly set by the orbital period of the Moon, known as a sideral month. This period (27.32 days) is slightly shorter than the longitudinal libration period set by the time between perigee passages (27.55 days, known as an anomalistic month). Hence, variations in latitude librations occur a bit faster than those for longitude. There is also a small component caused by precession of the lunar rotation, something that happens over an 18.6 year period. The result is that the peaks of the latitude librations move ahead relative to the longitudinal ones by about 4.6 days per year.

For example, early in 2019 the Moon's western libration (minimum on black curve) lined up reasonably well with its northern libration (maximum on red curve). Hence, the Moon was basically nodding in a line along the NE->SW->NE direction as we view it (NW->SE->NW libration). But by later in the year the latitude libration curve had shifted earlier relative to the longitude one by about 4 days. By December, the Moon's maximum eastern libration was followed 7 days later by a maximum northern libration, then a maximum western libration in another 6 days, and a maximum southern libration in another 8 days, returning to the next maximum eastern libration in 6 more days. This motion is like a circle. Hence, the Man in the Moon went from nodding to lolling his head about in a counterclockwise direction as we look at him. He's like a celestial bobblehead.

If you wanted to insist upon the best view of Mare Orientale, you might want both maximum western and southern librations. These synchronized in September of 2015, and do so again six years later in September of 2021. The six years comes from a combination of the 18.6 period of precession, and the 8.8 year cycle from the difference between anomalistic and sidereal months.

It is fun to think about being able to see around to part of the back side of the Moon that we normally never see. Sometimes the press calls this the `dark' side of the Moon, though the Sun rises and sets there just like it does on the side that faces the Earth.