Crash into the Sun, orbit the Sun, or fly off into space?
The centre of mass of the Earth and the Moon now orbits the Sun. If the Earth disappeared, the Moon would continue to orbit the Sun. If the Earth diasappeared there could be no full moon because that is what we see from Earth and relates to the moon's orbit about the Earth relative to the direction to the Sun.
The speed of the moon around the sun varies by a few percent through its orbit, so while it will continue to orbit the sun, the eccentricity of the orbit will be different (likely larger) and the average distance will change slightly.
Would the Moon then be tidally locked with the Sun?
I hope not. The angular momentum of the Moon about it's axis will not change suddenly. It would take a very, very long time for a solid moon to tidally lock to the Sun. So the length of a day on the Moon would remain at about 29½ days as it is now.
The gravitational attraction between the Earth and Moon keeps them together in the same orbit about the Sun. Up until the instant that the Earth disappears, gravity has kept the Moon orbiting the Moon-Earth barycentre.
The gravitational “tension” between the Moon and Earth act at 90° to the Moon's orbit and operate to pull the Moon from a straight line passing the Earth, into an orbit about the Earth.
But, when there is a full moon, the Moon is on the opposite side of the Earth from the Sun, so it is then travelling close to parallel with the Earth's orbit around the Sun.
Removing the Earth, (at full Moon), would not let the Moon fly directly away from where the Earth was, but the Moon would continue to travel parallel with it's original orbit about the Sun.
If instead, the Earth disappeared when there was a half moon, it would introduce a greater disturbance to the orbit of the Moon about the Sun.
Not really greater.
The speed of Moon relative to Earth is about 1 km/s.
The speed of Earth relative to Sun is about 30 km/s.
In order for Moon to crash into Sun, Moon would have to be propelled at 30 km/s, in a specific direction.
In order for Moon to fly off into space, Moon would have to be propelled at at least 12 km/s.
Since the orbital speed of Moon is about 1 km/s, Moon would continue to orbit Sun.
Treating Earth's orbit as exact circle - if Earth vanished at full moon, then Moon would be travelling faster than Earth, and therefore would be at an elliptical orbit bigger than Earth orbit, whose perihelion touches Earth orbit.
If Earth vanished at new moon, moon would be travelling slower than Earth, and therefore would be at an elliptical orbit smaller than Earth orbit, whose aphelion touches Earth orbit.
If Earth vanished at half moon, moon would be travelling across Earth orbit, and therefore would be at an elliptical orbit that intersects Earth orbit.
I agree, it would not make much difference. From your figures when the moon is furthest from the Sun (at full moon) it would be travelling at about 31 km/s at a greater radius from the Sun, but at new moon, speed would be 29 km/s on a shorter radius from the Sun. Since orbital velocity is not proportional to r, but to √(1/r), I agree. If the orbits were circular at the start of the experiment, then after the deletion of Mother Earth, the Moon's orbit would be elliptical about the Sun.
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