# What “g” would be needed to keep helium on Earth?

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?

• 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? – JeopardyTempest Feb 22 '17 at 22:14
• 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. – Ryan May 1 '17 at 18:16
• 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. – 0tyranny0poverty Jul 26 '17 at 21:52

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.