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I recently studied that NASA has planned to tow and place it in the orbit of the moon. My doubt is when asteroid is placed in the orbit near moon.since the gravitational field of earth is very high.what will it revolve around the moon or the earth. Can anyone clarify my doubt ??

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The real threat is not the Earth's gravity, but the gravitational anomalies (mascons - for "mass concentrations") on the Moon. They have significant impact on the lifetime of Moon's satellites and the choice of orbits. – Deer Hunter Apr 11 '13 at 10:21
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There are stable orbits around the Moon, also Nasa, Esa, and many more have satellits orbiting the Moon, and they don't "decay" towards the Earth.

To look for the physics and no only examples search for Hill sphere:

Acording to the last link, the Moon's sphere of influence is 38 Moon radii or $66.1\cdot 10^{3}km$ plenty of space to host an asteroid.

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The setup is just like this: Earth around Sun, Moon around Earth and finally asteroid around Moon. Here's a phrase I've got from Nat Geo (assuming this is the news you've studied)

NASA wants to identify an interesting asteroid flying around deep space, figure out a way to capture the spinning and hard-to-grab orb, nudge it into our planetary region and then set it into orbit around the moon, the agency announced Wednesday.

So, NASA really plans to make the orbit of asteroid around Moon. There's nothing complicated in it. If you're thinking that the orbits depend on gravitational field only, then you can also ask, "Why the moon orbits around Earth and not the sun which has high gravitational field?" And you can expect an answer that the Earth is nearer relative to Sun.

The same goes to the asteroid. It's much nearer to the moon. Gravitational field being an inverse-square law, says that the Earth's field is larger - but the effective flux on the asteroid is small when compared to that of the moon.

But, it confused me when you asked - "what will it revolve around - the moon or the earth?" while NASA already declared that it's gonna place the asteroid around the Moon..!

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