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The mass of the Moon can be found from a modified form of Kepler's third law of Planetary Motion.

You need to know the time taken for the Moon to orbit the Earth and the mass of the Earth.

Also you need the distance between the Moon and Earth.

Let the orbital period be P

Moon's mass = Mm

Earth's mass = ME

Distance apart = D

Gravitational Constant = G

Then: Mm + ME = (4pi2/G) x (D3/P2)

Note: That equation doesn't show up properly on some pages.

To clarify, it involves:

pi squared, D cubed and P squared.

Also, an artificial satellite orbiting the Moon can be used to find the Moon's mass by the same method:

mass of satellite (negligible) + mass of Moon = (4pi2/G) x (D3/P2)

D is now the distance of the satellite from the center of mass of the Moon and

P is the orbital period of the satellite.

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