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Just what the title states.

Any body, as it travels through space, probably sweeps meteoric/general dust from the vicinity into itself. How much (in tons/kilograms) dust does the moon accumulate daily?

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The amount of dust swept up is proportional to the surface area, and since the moon is about 27% the size of the Earth is sweeps up about 7.5% as much dust.

The amount of dust swept up by the Earth is estimated as anything from 30,000 to 80,000 tons per year. The estimates are only approximate because they're based on measuring the dust content of the upper atmosphere by flying planes through it. Anyhow, assuming you believe these figures that means the moon sweeps up 3000 to 6000 tons.

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