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Anonymous
The Moon will continue to move slowly away from the Earth until the tidal effects between the two are no longer of significance, whereupon the Moon's orbit will stabilize.
...the Moon is gradually receding from the Earth INTO A HIGHER ORBIT, and calculations (by e.g. NASA at the Jet Propulsion Laboratory) suggest that this will continue for about TWO BILLION YEARS. By that time, the Earth and Moon will become caught up in what are called a "spin-orbit resonance" in which the Moon will circle the Earth in about 47 days (currently 29 days) and both Moon and Earth will rotate around their axes in the same time, always facing each other with the same side.
TWO BILLION YEARS - considering this solar system is only expected to be around for about five billion years, the moon isn't going to get very far even if it was to somehow escape.
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