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The top animation shows the Moon's orbit as it would be seen looking down on Earth from high above the North Pole, and it shows the Moon's phases as they would be seen from most places in the northern hemisphere.
The bottom animation shows the Moon's orbit as it would be seen looking down on Earth from high above the South Pole, and it shows the Moon's phases as they would be seen from most places in the southern hemisphere.
To see how the same motion can appear to go either clockwise or anti-clockwise depending on where it is viewed from, experiment with the animation below of the rotating Earth. Try viewing it from above the north pole and from above the south pole. (Click the up and down arrow buttons.)
THE MOON'S CYCLES AND ASSOCIATED TERMINOLOGY
Real motion and apparent motion
The Moon has two real motions and one apparent motion:
1) It rotates (spins) around its own axis, which is the line through the centre of the Moon from its north pole to its south pole. The Moon's north and south poles point toward (approximately) the same directions in space as the Earth's north and south poles, and, like the Earth, the Moon rotates from west to east. The Earth's and the Moon's rotation both obey the "right-hand rule," which is followed by most of the major bodies of our solar system (the planets and their Moons). The right-hand rule defines the relationship between the direction that we call "north" and the direction of the body's spin. If you make a fist of your right hand with the thumb pointing upward, your fist represents the body, your raised thumb represents the direction the north pole points toward and your curled fingers point in the direction of the body's spin.
2) The Moon also revolves around the Earth, meaning it orbits or circles the Earth. The direction of its orbital motion is the same as the direction of its rotation and that is also the direction of the Earth's rotational and orbital motions. If you were viewing the solar system from a point high above the Earth's north pole, those motions would appear anti-clockwise from your point of view.
3) Additionally, to us on Earth, the Moon appears to continuously move across our sky from east to west. At moonrise it appears above our eastern horizon, at moonset it descends below our western horizon and halfway between the two it reaches upper transit when it crosses our meridian heading westward. This westward motion is not real motion; it is apparent motion -- an optical illusion caused by our planet's rotation on its axis from west to east. Beause the Moon's real orbital motion eastward is in the opposite direction to its apparent westward motion, but much slower, it has the effect of making the period from one upper transit of the Moon to the next longer than 24 hours. That period (called a tidal day) is, on average, about 24 hours, 50 minutes.
The Moon's rotation is captured rotation, meaning that its rotational period is the same as its orbital period. In other words, in the same time that it takes the Moon to complete a single, 360° revolution around the Earth it also completes a single, 360° rotation around its own axis. The result of this is that the same side of the Moon is always turned toward the Earth. The side that constantly faces us is called the Moon's near side; it is the only side we see from Earth. The opposite side, called the Moon's far side, has only been seen from spacecraft in lunar orbit.
A lunation is a cycle associated with the Moon's orbital motion around the Earth. It commences when the Moon is at some notional starting point (P) in its orbit and it ends when the Moon next returns to P. Astronomy recognises several candidates for P. Consequently, there are different kinds of lunation, each with its own starting point, and each one has a different length. The length depends on whether P is stationary or moving as seen from our observation point (Earth), and, if moving, its rate and direction of motion relative to the Moon's motion.
Because the Moon's orbit is eliptical, not circular, it does not move at a uniform speed. Its orbital motion is fastest at perigee (the point in its orbit where it is closest to Earth) and slowest at apogee (the point furthest from Earth). For that reason, no lunation has a constant period; they all vary in length. But we are able to speak of their mean length.
Of the above five lunations, two are of interest to us here. The sidereal lunation, for which P is a fixed star, is the best measure of the Moon's "true" orbital period, i.e. the time it takes the Moon to complete a single, 360° orbit of the Earth, which we measure by observing how long it takes the Moon to return to the same position in our sky relative to a fixed star. (Sidereal comes from sider, the Latin word for star.) The Moon's sidereal period is about 27.32 days. From this, we derive the mean (angular) speed of the Moon's orbital motion: about 13.176° per day (360°/27.32 days).
Because we are primarily focussed here on the Moon's phases, we will be mainly concerned with the synodic lunation, which is a full cycle of lunar phases. The Moon's synodic period, the time taken to complete such a cycle, is about 29.53 days. This animation shows why the Moon's synodic period is about 2.21 days longer than its sidereal period.
For the synodic lunation, P is the Sun, and to us on Earth, the Sun appears to make a full circuit of the sky over the course of a year. This apparent motion is eastward and its rate is just under 1° per day (360°/365.242 days). So, although the Moon's mean orbital velocity is about 13.176° per day, eastward, its speed relative to the Sun's apparent eastward motion (as seen from Earth), is only about 12.19° per day.
The cycle of the Moon's phases notionally begins at astronomical New Moon. At that time, the Sun and the Moon both occupy the same celestial longitude as seen from Earth. Hence the name "synodic," which comes from synodos, the Greek word for meeting, because the cycle begins when the Sun and Moon appear to meet at the same celestial longitude.
(Celestial longitude is how we measure east-west separation of solar system objects in our sky. It is used here in the same sense as ecliptic longitude, which is the correct astronomical term.)
At New Moon, the Moon is on the Earth-Sun line between the Sun and the Earth. Because the Moon is then at the same celestial longitude as the Sun, a New Moon is also called a lunar conjunction.
At Full Moon, the Moon is again on the Earth-Sun line, but on the opposite side of Earth from the Sun, so that the three bodies lie on that line in the order Sun, Earth, Moon. This means that the Moon is at the opposite celestial longitude to the Sun as seen from Earth. Therefore a Full Moon is also called a lunar opposition.
At New Moon, lunar elongation is 0° and at Full Moon it is 180°. As shown graphically in the above animations, lunar elongation is the angle Sun, Earth, Moon, i.e. the angle at the vertex of the Earth-Sun line and the Earth-Moon line. It is an angular measure of the east-west separation in our sky between the Sun and the Moon as seen from Earth.
More generally in astronomy, elongation refers to the angle body, Earth, parent, where body and parent are a solar sytem body revolving around a parent body, usually a planet and the Sun.
It is important to remember that exactly half of the Moon's surface is lit by the Sun at all times (except during a lunar eclipse, when the Moon passes through the Earth's shadow). Like the Earth, the Moon always has a sunlit side and a night side. The changes in illumination of the Moon that we see from Earth are changes in the amount of the Moon's near side that is lit. This occurs because, as the Moon orbits the Earth, different parts of it face the Sun, so, as on Earth, the sunlit side keeps moving around the Moon's surface and the amount of the Moon's near side that it covers is continuously changing, ranging from none of it to all of it. This directly affects the shape of our view of of the Moon from Earth. Those different shapes are called lunar phases.
At New Moon, when lunar elongation is 0°, only the Moon's far side is lit up by the Sun; its near side is turned fully away from the Sun, so none of that side is visible to us at that time.
During the first half of the lunation (between New Moon and Full Moon), while lunar elongation increases from 0° to 180°, increasingly more of the Moon's near side becomes illuminated. The sunlit portion that we see grows. During this half of the lunation the Moon is said to be waxing. This is an old English word for growing. It is allied to the German word wachsen.
Halfway through the synodic lunation, when lunar elongation is 180°, the Moon's near side is turned fully toward the Sun and all of that side is lit up by the Sun, so we see a Full Moon.
During the second half of the lunation (between Full Moon and New Moon), while lunar elongation decreases from 180° to 0°, the sunlit portion that we see diminishes. During this half of the lunation the Moon is said to be waning, meaning shrinking in size and brightness.
Lunar elongation also governs the times of day and night during which the Moon is "up" (i.e. above our horizon).
At New Moon, when lunar elongation is 0°, the Moon rises at sunrise and sets at sunset. It is up during the day but is not visible in the sky for the reason explained above.
During the first half of the synodic lunation, when the lunar elongation angle is increasing, elongation is said to be eastern elongation because the Moon is to the east of the Sun in our sky during all of that half of the lunation. It rises after sunrise and sets after sunset.
At Full Moon, when lunar elongation is 180°, the Moon rises at sunset, and sets at sunrise. It is up all night and it reaches upper transit about 25 minutes after true local midnight.
During the second half of the synodic lunation, when the lunar elongation angle is decreasing, it is called western elongation, because the Moon is to the west of the Sun in our sky during all of that half of the lunation. It rises before sunrise and sets before sunset.
Here are the names of the Moon's phases in the order that they occur during the synodic cycle, and the lunar elongation angles corresponding to each phase.
First half of lunation
During the two crescent phases, the shape of the phase is like a banana, concave on one side and convex at the opposite side. The convex side is the side that is growing outward or shrinking inward. For southern hemisphere observers, the waxing crescent is shaped like a "C" and the waning crescent is shaped like the rounded portion of a "D", so the letters "C-O-D" can be used as a mnemomic for the full cycle. For northern hemisphere observers, the reverse is true.
At 90° elongation, the Moon is at its First Quarter or Last Quarter phase. Unlike the other phases, these are NOT descriptions of the shape of that phase, which is a half moon at that time. Since both of those phases have a half-moon shape, to call them Half Moon would not be specific enough. So we call those two phases First and Last Quarter, because at those two times the Moon has completed one quarter and three quarters of its synodic cycle. For southern hemisphere observers, the First Quarter phase is shaped like a "C" and the Last Quarter phase is shaped like a "D". For northern hemisphere observers, the reverse is true.
Gibbous means a shape similar to (but not quite) a disk, i.e. it is not perfectly circular, but it is concave (bulges) on both sides. While the Moon is in its two gibbous phases, one side has a semi-circular shape, and the opposite side (the side that is growing or shrinking) bulges outward, but is not a perfect semi-circle.