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The moon phase is the shape of the directly lit portion of the Moon by the Sun. The moon phase is dictated by the angle between the Moon, the Earth, and the Sun. More specifically, the moon phase is dictated by the ecliptic longitude, the longitude of the orbit of the Earth around the Sun, of the Moon compared to the ecliptic longitude of the Sun.
A new moon occurs when the ecliptic longitude of the Moon is the same as the ecliptic longitude of the Sun. At this point, the disk of the Moon will appear to not be illuminated.
The first quarter moon occurs when the ecliptic longitude of the moon is 90 degrees away from the ecliptic longitude of the Sun and occurs after the new moon but before the full moon. The moon will appear to be half illuminated at this point. In the Northern hemisphere of the Earth, the third quarter moon will appear to be illuminated on the right side and in the Southern hemisphere of the Earth, the third quarter moon will appear to be illuminated on the left side.
A full moon occurs when the ecliptic longitude of the Moon is 180 degrees away from the ecliptic longitude of the Sun. At this point, the disk of the Moon will appear completely illuminated.
The first quarter moon occurs when the ecliptic longitude of the moon is 90 degrees away from the ecliptic longitude of the Sun and occurs before the new moon but after the full moon. The moon will appear to be half illuminated at this point. In the Northern hemisphere of the Earth, the third quarter moon will appear to be illuminated on the left side and in the Southern hemisphere of the Earth, the third quarter moon will appear to be illuminated on the right side.