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Hydrogen and helium are quite rare in Earth's atmosphere, despite being the most abundant elements in the universe. Planets such as Mars have even lost almost all their atmosphere.

The usual explanation is that in a gas of a certain temperature you have a normal distribution of kinetic energy around that temperature. A few particles will have enough energy to reach escape velocity and therefore escape the planet’s gravitational influence. But where do they go? Wouldn’t this mean that there should be a dust cloud all throughout the solar system, mostly in each planet’s orbit? Why does it not coalesce into small clumps and eventually asteroids?

And what about all the other velocities between low earth orbital velocity and escape velocity? Shouldn’t we have gasses (especially the lighter ones) in orbits all the way to the Moon and beyond?

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    $\begingroup$ Different question, but the many answers make for good reading: Does “What happens beyond Kármán, stay beyond Kármán”? $\endgroup$ – uhoh Dec 26 '19 at 2:34
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    $\begingroup$ What is your basis for saying hydrogen is quite rare on Earth? $\endgroup$ – Bob516 Dec 27 '19 at 1:58
  • $\begingroup$ Space is quite big, you know? ^^ $\endgroup$ – Zaibis Dec 27 '19 at 6:39
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    $\begingroup$ @Bob516: Hydrogen as a free gas. Of course as an element (e.g. in water molecules) it’s quite abundant. To quote Wikipedia: “hydrogen gas is very rare in the Earth's atmosphere (1 ppm by volume) because of its light weight, which enables it to escape from Earth's gravity more easily than heavier gases” $\endgroup$ – Michael Dec 27 '19 at 8:56
  • $\begingroup$ Thanks for the edit, shorter and better. $\endgroup$ – Michael Dec 27 '19 at 15:02
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Escape from planetary atmospheres of the terrestrial planets in our solar system is dominated by ions in absolute numbers, as opposed to neutral particle species. Particles can be any type of molecule or atom here, mostly $\mathrm O^{+}$ and $\mathrm N^{+}$ for Earth.

For the case of Earth, a particle, once ionised in the upper thermosphere, can couple to the terrestrial magnetic field. From there, it will start orbiting in the magnetosphere, and it can be picked up by the solar wind at the magnetospheric bow-shock. Once transported by the solar wind, most of the species will be transported into interstellar space.

Your argument with the rings of material (not dust, see below) would be valid, if the velocity of the escaping particles would somehow be fine-tuned in a way as to stay in orbit. However, escape velocity from Earth is $14\ \mathrm{km/s}$, the orbital speed at $1\ \mathrm{AU}$ is $30\ \mathrm{km/s}$, so most particles will escape from Earth, not staying in orbit around the sun.
Add the solar wind on top of that, having typical speeds of $100{-}400\ \mathrm{km/s}$ and you get blown away into space pretty good.

There is of course a small chance that some of those escaped particles will be picked up on their way out by the other planets. This fraction is a function of the geometric cross-section of Hill-sphere of the encountered planets, but is still very small compared to the total escaping flux. I remember having read an article about a fraction of the atoms being lost by Venus, being later picked up by Earth, but can't remember the source. This results in most gas ending up in interstellar space, as previously stated.

Assuming a constant average velocity of the solar wind of $\sim 100\ \mathrm{km/s}$, the solar wind can traverse the $\sim 150\ \mathrm{AU}$ until interstellar space in just about 6 years. From there, the lost particles contribute to the interstellar medium, although in negligible contributions.

Note that a 'dust cloud' is different from escaping atmospheric gas. 'Dust' in space are usually minerals. The most prominent ones are olivines, pyroxenes, forsterites, etc. that form macroscopic crystalline structures and could never escape by Jeans or hydrodynamic escape from an atmosphere, except via meteorite impacts.

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  • $\begingroup$ Some atmosphere is also lost to reactions with the planet's surface. CO2 for example is readily bound up in Calcium Carbonate. $\endgroup$ – Ryan_L Dec 26 '19 at 6:11
  • $\begingroup$ @Ryan_L Yet OP's question is about particles attaining escape velocity, hence I addressed those. Atmosphere can be released again through volcanism and plate tectonics, but we're not opening this can of worms here. $\endgroup$ – AtmosphericPrisonEscape Dec 26 '19 at 11:29
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    $\begingroup$ Isn't 400 km/s faster than light speed? $\endgroup$ – Jost Dec 26 '19 at 12:03
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    $\begingroup$ @Jost google.com/search?client=firefox-b-d&q=speed+of+light as you can convince yourself easily, it is not. It's about 0.1% of the speed of light. For gaseous winds in space a quite typical speed. For a very massive object, this would be much harder to achieve, but is also possible, the keyword here is 'hypervelocity stars'. $\endgroup$ – AtmosphericPrisonEscape Dec 26 '19 at 12:37
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    $\begingroup$ @AtmosphericPrisonEscape Oh, yeah - Speed of light is about 300'000km/s - 3 orders of magnitude higher. Sorry :/ $\endgroup$ – Jost Dec 26 '19 at 12:45
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Basically it is blown away by the solar wind, headed to interstellar space. A light particle in orbit around the Sun will tend to be pushed further out with time because of both solar wind and photonic pressure.

Note there are a few pockets of dust around, but they are very hard to see. It takes being in a relatively stable point, usually the L4/ L5 points of a system (Earth-Moon, Earth-Sun) for it to stick around. And that dust is heavier stuff which will stick around easier than a gas.

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