Is the Moon moving away from the Earth?

Yes it is, but very slowly. The current rate at which the Earth day is increasing is 0.0018 seconds/century. The semi-major axis of the lunar orbit is increasing by 3.8 centimeters/year according to laser ranging measurements made since the 1970's using the Apollo 'corner cube reflectors' deposited on the surface by the astronauts.

It is expected that in 15 billion years, the orbit will stabilize at 1.6 times its present size, and the Earth day will be 55 days long equal to the time it will take the Moon to orbit the Earth. Of course, in less than 7 billion years, the Sun will have evolved into a red giant star and engulfed the Earth-Moon system, thereby incinerating it!

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