(4) The Moon: the Distant View

The Moon's the North Wind's cookie He bites it, day by day Until there's but a rim of scraps That crumble all away.	The South Wind is the baker He kneads clouds in his den, And bakes a crisp new moon that ... greedy.... North.... Wind ....eats....again!
"What the Little Girl Said" Vachel (Nicholas) Lindsay, 1879-1931.

Index

2a. The Sundial

3. The Seasons

3a. Angle
of sunlight

4. The Moon (1)

4a. The Moon (2)

4b. Moon Libration

5.Latitude and
Longitude

5a. Navigation

5b. Cross-Staff

5c. Coordinates

6. The Calendar

6a. Jewish Calendar

The Month

The monthly cycle of the moon (we won't capitalize the word here) must have mystified early humans--"waxing" from thin crescent ("new moon") to half-moon, then to a "gibbous" moon and a full one, and afterwards "waning" to a crescent again. That cycle, lasting about 29.5 days, gave us the word "month"--related to "moon," as is "Monday."

The civil year, January to December, no longer ties its months to the moon, but some traditions still do and their terms for "month" reflect the connection--in Arabic, "shahr", in biblical Hebrew "yerach" and also "chodesh" from "new," since it was reckoned from one new moon to the next. Jericho (pronounced Yericho), one of the oldest cities on Earth, took its name from "yerach," and of course, legends tell of many moon-gods and goddesses, e.g. Artemis and Diana.

Early astronomers understood the different shapes of the moon, noting that each was linked to a certain relative position between moon and Sun: for instance, full moon always occured when moon and Sun were at opposite ends of the sky. All this suggested that the moon was a sphere, illuminated by the Sun.

The moon's path across the sky was found to be close to the ecliptic, inclined to it by about 5 degrees. Eclipses of the Sun always occured when moon and Sun were due to occupy the same spot in the sky, suggesting that the moon was nearer to us and obscured the Sun. Eclipses of the moon, similarly, always occured at full moon, with the two on opposite sides of the Earth, and could be explained by the shadow of the Earth falling on the moon.

Lunar eclipses allowed the Greek astronomer Aristarchus, around 220 BC, to estimate the distance to the moon (see section #8c). If the moon and the Sun followed exactly the same path across the sky, eclipses of both kinds would happen each month. Actually they are relatively rare, because the 5-degree angle between the paths only allows eclipses when Sun and moon are near one of the points where the paths intersect.

. The cycle from each new moon to next one takes 29.5 days, but the actual orbital period of the moon is only 27.3217 days. That is the time it takes the moon to return to (approximately) the same position among the stars.

[IMAGE: Rotation of the Moon] Why the difference? Suppose we start counting from the moment when the moon in its motion across the sky is just overtaking the Sun; we will call this the "new moon," even though the thin crescent of the moon will only be visible some time later, and only shortly after sunset. Wait 27.3217 days: the moon has returned to approximately the same place in the sky, but the Sun has meanwhile moved away, on its annual journey around the heavens. It takes the moon about 2 more days to catch up with the Sun, to the position of the next "new moon," which is why times of the new moon are separated by 29.5 days.

The Face of the Moon

The visible face of the moon has light and dark patches, which people interpreted in different ways, depending on their culture. Europeans see a face and talk of "the man in the moon" while children in China and Thailand recognize "the rabbit in the moon." All agree, however, that the moon does not change, that it always presents the same face to Earth.

Does that mean the moon doesn't rotate? No, it does rotate--one rotation for each revolution around Earth! The drawings on the left, covering half an orbit, should make this clear. In them we look at the moon's orbit from high above the north pole, and imagine a clock dial around the moon, and a feature on it, marked by an arrow, which initially (bottom position in each drawing) points at 12 oclock.

In the top drawing the marked feature continues to point at Earth, and as the moon goes around the Earth, it points to the hours 10, 8 and 6 on the clock dial. As the moon goes through half a revolution, it also undergoes half a rotation If the moon did not rotate, the situation would be as in the bottom drawing. The arrow would continue to point in the 12-oclock direction, and after half an orbit, people on Earth would be able to see the other side of the moon. This does not happen.

We need to go aboard a spaceship and fly halfway around the Moon before we get a view of its other side--as did the Apollo astronauts who took the picture below.

The Gravity Gradient

This strange rotation of the moon is maintained because the moon is slightly elongated along the axis which points towards earth. To understand the effect we look at the motion of a body with a much more pronounced elongation--an artificial satellite with the shape of a symmetric dumbbell (drawing).

It can be shown that if the forces on the dumbbell (or indeed on a satellite of any shape) are unbalanced, it rotates around its center of gravity. That point will be defined later, but in a symmetric dumbbell with two equal masses marked A and B, the center of gravity is right in the middle between them.

Both masses A and B are attracted to the Earth, and if the attracting forces were equal, their tendencies to rotate the satellite ("rotation moments" or "torques") are equal and cancel each other, so that no rotation occurs. If however A starts closer to the center of Earth, the force on it is just a little stronger. Therefore the satellite will rotate until A is as close to Earth as it can be, which is a possible position of equilibrium. Of course, it may then overshoot its equilibrium position, and end up swinging back-and-forth like a pendulum, only slowly (like a pendulum) losing energy and settling down. The elongated moon acts like a dumbbell too.

The rotating force which lines up the moon or an orbiting dumbbell is the difference between the pull on A and on B. It depends not on how strongly gravity pulls these masses, but on how rapidly the pull of gravity changes with distance--on the "gravity gradient." Near Earth that is a gentle force, though still strong enough to line up elongated satellites. Among those was Triad, deliberately shaped like a long dumbbell with an additional payload in the middle, the first satellite to map the electrical currents associated with the polar aurora.

Near a black hole or pulsar, though, the gravity-gradient force can be fierce enough to rip a spacecraft apart.

Actually, the long axis of the Moon does not always point exactly to the center of the Earth, but swings back and forth around that direction, a motion known as libration. Most of this is caused because the Moon rotates around its axis with a fixed period, while its motion around its orbit slows down far from Earth and speeds up close to it. This speeding up and slowing down is the result of Kepler's 2nd law, discussed in section 12, and is a rather small effect, since the moon's orbit is very close to circular.

Because of libration, even though at any time only half the Moon is visible, over time 59% can be seen, since it lets astronomers look at the Moon from slightly different viewing directions. Librations are a rather specialized subject--but if you want to know more about it, click here.

Earthshine

At times when only a narrow crescent of the Moon is seen (e.g. a "new moon"), one can also see the rest of the Moon faintly outlined. The Sun now shines on almost all of the side of the moon turned away from Earth (those calling that "the dark side of the moon" are quite wrong!) and therefore it also illuminates most of the side of the Earth facing the moon. If you were standing on the moon at that time, a "full Earth" would shine brightly in your sky, and the faint "earthshine" of the darker part of the moon is just the reflection of some of that bright earthlight.

Earthshine is of interest to scientists, because its brightness is contributed by all the factors which turn back sunlight before it manages to heat the Earth--light reflected from the ground and from clouds, and light scattered back by dust and small particles ("aerosols") in the atmosphere. In a time when atmospheric scientists are trying to assess heating of the Earth by the greenhouse effect, earthshine measures a process which works in the opposite direction, reducing the heat our planet receives.

The fraction of light reflected is hard to estimate theoretically, but earthshine allows it to be measured. According to recent reports ("The Darkening Earth", Scientific American August 2004, p. 16), this fraction has been growing, reducing the amount of sunlight received by Earth and canceling about 1/3 of the greenhouse heating.