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I know that helium is a very light and rare gas on Earth because Earths gravity is not strong enough to keep it. Instead, helium and hydrogen are rising through the atmosphere and escape into outer space.

My question is: How massive would Earth have to be so that it could keep helium in the atmosphere? 2, 5, or 10 times the actual mass? Could we, for example, compare it to Neptune or Saturn?

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    $\begingroup$ I believe it's more about percentage lost over time. Should be that the most energetic molecules at a given temperature of any gas sample can escape, it's just a matter of the cutoff line for how much energy is needed... and thus what percentage of gas escapes (over what amount of time). Indeed, it's probably almost unity for He/H. So the question becomes: over what time period do you want to keep the gases? $\endgroup$ – JeopardyTempest Feb 22 '17 at 22:14
  • $\begingroup$ Well, if we want a guarantee, we can go up to black hole level. If there is ever a chance for even one atom to escape, then eventually most, if not all of it will escape, though that may take forever, but it still counts. $\endgroup$ – Ryan May 1 '17 at 18:16
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    $\begingroup$ I don't know about all the gas giants, but Jupiter and Saturn both retain hydrogen and helium in their atmosphe re . So Saturn's mass or greater would do it. $\endgroup$ – 0tyranny 0poverty Jul 26 '17 at 21:52
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Atmospheric escape is the loss of planetary atmospheric gases to outer space. You'd never be able to contain ALL of any gas forever by gravity. Ultimately you end up in the rarefied atmosphere where there is some probability that a molecule of gas will reach escape velocity. The probability distribution is given by the Maxwell-Boltzmann distribution and the thermal escape mechanism is known as the Jeans escape.

On earth the problem is compounded by the fact that helium is lighter that the other gases. So (1) helium migrates to the upper atmosphere because of its density and (2) helium atoms on average have the same kinetic energy as oxygen and nitrogen molecules which means that helium atoms are on average traveling much faster than oxygen or nitrogen molecules.

All of this is made more complicated by needing a temperature profile of the atmosphere as a function of the height.

It doesn't help to assume temperature of background radiation because even at that temperature you can calculate the probability of a helium atom having 80% of light speed. This sort of like being able to calculate the probability of throwing $n$ heads in a row regardless of how big $n$ is.

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  • $\begingroup$ Perhaps it would be good for this question to give a rough reasonable estimate though... maybe something like the g needed to keep 99% of He\H for 10,000 years (assuming a uniform temperature... maybe 250 K) or something such. You choose the numbers, but it'd give some rough substance towards the basic idea Petr is really interested in, though obviously I'm certainly in total agreement that elucidating the real nature of the problem as you have is helpful too. :-) $\endgroup$ – JeopardyTempest Feb 23 '17 at 7:23
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    $\begingroup$ Wow, I see that it´s really complicated. I do not think I understand the mechanism why helium does not leave the atmosphere, but maybe we could have a relatively simple question with relatively simple answer ? Like.. what is the probability that there will be no helium atoms in 1 000 000 years ? (if we neglet the fact, that helium is still coming from Earth) Or probably a better question.. What percentage of helium (out of 0,000524..% we have today) would leave the atmosphere in the next 1 000 000 years ? Is that well asked question ? $\endgroup$ – Petr Hykš Feb 23 '17 at 22:28
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    $\begingroup$ The whole loss mechanism is even much more complicated than I outlined. Most of the loss of hydrogen and helium isn't because of the thermal loss mechanism. I had just edited the answer and added a link for "atmospheric escape." The wikipedia article mentions that overall earth loses "about three kilograms (3 kg) of hydrogen and 50 grams (50 g) of helium per second." $\endgroup$ – MaxW Feb 23 '17 at 22:35
  • $\begingroup$ A very simplified graph of escape velocity vs. surface temperature of a planet or moon is given here abyss.uoregon.edu/~js/ast121/lectures/lec14.html There are numerous escape mechanisms as mentioned in answers above, but if you go by this plot then the Earth's escape velocity would have to be raised from about 12 km/s to about 16 km/s (very rough approx). This would correspond to about 1.7 times increase in mass. However, escape velocity is also radius dependent, so if you maintain Earth's density then it's radius also increases and you need less of a mass increase say about 1.5 times. $\endgroup$ – Jack R. Woods Mar 3 '17 at 17:09

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