# How much mass must Earth lose to lose its Moon?

Maybe I am wrong about the basic concept here, but for my curiosity: how much mass would need to leave the Earth (like satellites and rockets) to create an imbalance between the Earth and Moon so that the Moon can escape the Earth's gravitational pull and leave its orbit?

Half.

The escape velocity for an object at a distance $D$ from an object of mass $M$ is $\sqrt{2GM/D}$. The circular orbital velocity (the Moon is on an orbit that's close enough to circular that I'll just assume this) at the same distance is $\sqrt{GM/D}$. Setting the escape velocity from the Earth with it's new reduced mass $M_{\rm new}$ equal to the orbital velocity around the Earth with it's usual mass $M_{\rm old}$ gives:

$$\sqrt{\frac{2GM_{\rm new}}{D_{\rm Moon}}} = \sqrt{\frac{GM_{\rm old}}{D_{\rm Moon}}}$$

Which immediately gives:

$$\frac{M_{\rm new}}{M_{\rm old}} = \frac{1}{2}$$

Note that you'd need to remove all this mass to well outside the Moon's orbit, probably well away in the direction opposite where you're going to send the Moon. Otherwise, you'll expand the orbit of the Moon, but once it gets back outside the distribution of expelled mass, it will remain bound to the system.

This will obviously not be achieved with rockets launching satellites, chiefly because rockets work by throwing mass out of the back end... mass that would get left behind. It's pretty obvious that $0.5\,{\rm M}_{\oplus}$ in rocket fuel (even if half the Earth's mass could be made into rocket fuel!) is not enough rocket fuel to launch $0.5\,{\rm M}_{\oplus}$ in payload to escape velocity. Even if it could, you wouldn't be left with half the Earth, but instead $0.5\,{\rm M}_{\oplus}$ in rocket exhaust...

• Man, the virial theorem is magic. – rob Jun 24 '16 at 16:13
• That was a GOOD, short, and concise answer! – David White Jun 24 '16 at 16:14
• @rob Indeed, pure sorcery. – Kyle Oman Jun 24 '16 at 16:15

Assuming that we are making a certain amount of the earth's mass disappear instantly somehow, doing this will decrease the escape velocity of the earth. The moon is already moving at its orbital velocity in an orbit centered on the current center of mass of the earth-moon system.

If we took away so much mass (instantaneously, somehow) that the escape velocity of the less-massive version of earth was equal to the lunar orbital velocity, then the moon would escape the earth's pull.