Final answer:
The view of the moon changes from a full moon to a third quarter moon about one week after the full moon. During the third quarter moon, only half of the Moon's visible side is illuminated, and the Moon rises around midnight and sets around noon, providing good conditions for observing surface details because of the oblique sunlight casting sharp shadows.
Step-by-step explanation:
The change from a full moon to a third quarter moon occurs as the Moon goes through its phases in a cyclical pattern. About one week after the full moon, the Moon reaches the third quarter phase. During this phase, half of the Moon's visible side is in darkness. This is because the Moon has moved three-quarters of the way around its orbit relative to Earth, and the side facing the Sun is partially facing away from the Earth. The illumination of the Moon decreases from the full moon, where it is 100% illuminated, to the third quarter moon, where only 50% of the side facing Earth is illuminated.
For observers on Earth, it is significant that the moonrise and moonset times change as well. After the full moon phase, the Moon rises approximately at sunset and sets at sunrise. However, at the third quarter phase, the Moon rises around midnight and sets around noon. The transition between these two moon phases provides an excellent opportunity to study the Moon's surface relief, as oblique lighting during the third quarter casts sharp shadows on the Moon's surface features, revealing more topographical detail than the flat lighting provided by sunlight during the full moon.