A new moon is technically just a moment in time and is invisible from earth because it is between earth and the sun. It is only illuminated by the sun reflected from the earth. Unless the earth sun and moon line up perfectly in 3D and we get a solar eclipse it is not visible.
Most new moon are not solar eclipses because the moon is slightly above or below the sun. So they line up in 2D but not in 3D
When we start to see the moon it is a waxing crescent. When it begins to be visible it is still very close to the sun in the sky and the result is that moonrise and sunrise are at the same time.
In the same way is a full moon only a moment in time when all of the moons we see is illuminated by the sun. Earth is in between the moon and the sun. If they line up in 3D we get a lunar eclipse.
Because it it the opposite side of the earth the moonrise is at sunset. You can technically see both because the atmosphere bends light.
The lunar cycle of around 29.5 days is the time it take to move once around earth relative to the sun. If you just look at the period it take to move around earth relative to stars far away it is 27.3 days
In 29.5 days earth moves 29.5/365.25 = 8 % of its orbit around the sun. So the moon needs to move 8% more than a single lap to line up the same with the sun. 29.5/27.3 = 1.08
Because a new moon and full moon depend on the position of the moon relative to the sun the time it rises will always be the same relative to the sun
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