What would happen to the Moon if the Earth changed its mass by 1/2 or 1/4?

The escape velocity from any orbit about the Earth is given by


  2         2 G  M            
V    =   ----------- 
              R


But the circular velocity of a body in a stable orbit is just

 2           G  M
V     =  ------------
               R

This means that if the Earth suddenly lost 1/2 of its mass, the circular velocity it once had BEFORE the change, would now equal the escape velocity, and so the satellite would escape. If the Earth only lost 1/4 of its mass, the Moon would go into a highly elliptical orbit whose perigee distance would be about equal to its current average velocity in its orbit. Its apogee distance would probably increase by a factor of two or so from its current mean distance. But you would have to consult a textbook on celestial mechanics to get the correct new orbit parameters.
Copyright 1997 Dr. Sten Odenwald
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