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Excuse the poor phrasing of the title, but it became apparent to me that the solar system has some degree of density (i.e it is not a perfect vacuum). So my question is if any of these particles (not bound with others like the moon, debris, dust etc) are orbiting the earth, and if so, the total mass of them. Note, I'm distinguishing this from gas in our atmosphere by defining them as particles that are solely at the height they are at due to their orbits.

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Very small.

While it's possible for an individual atom to orbit, it's tricky. Once a macroscopic object is in orbit, you have to apply a large force quickly, or a consistent small force over time to remove it. With unbound atoms, tiny forces would remove it.

Much of the interplanetary medium is plasma rather than neutral atoms. Those will be much more strongly affected by earth's magnetic field than they are by earth's gravity.

Britannica suggests a density for neutral hydrogen at 0.01 atom per $\text{cm}^3$. There might be some heavier atoms as well, but then again most of that material isn't in orbit. So I think it's conservative to assume they're all hydrogen and in orbit if close enough.

The Hill sphere for earth is about 100 million km in radius, or $4 \times 10^{39} \text{cm}^3$. Assuming one hydrogen atom for each 100, you get a mass of $7 \times 10^{10} \text{kg}$. That's about the same as asteroid Bennu.

Error bars here are huge, mainly from the difficulty of finding the percentage of that material that is truly orbiting the planet. It could be orders of magnitude less than that.

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