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Somewhere I've read it is in the order of some tens of billions of years, but unfortunately I can't find any reference with it in Google.

I suspect, the losses are coming mainly from analyzing the statistical distribution of the upper atmosphere, extending with the interaction of the Solar Wind. The result is probably a finite element model and not an analytical solution. Am I right? What is the result?

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Even in the exosphere, at 1800 deg centigrade, the only components of the Earth's atmosphere that can currently attain escape velocity are hydrogen (3 kg per sec), and and helium (next to nothing). The notion that the Earth's atmosphere would be stripped away by solar wind but for the protective magnetic shield, is a myth. So there is no 'characteristic time of loss of the Earth's atmosphere' in the sense of slow decay. There is certainty that, as the sun approaches the red giant stage, several billion years from now, there will be catastrophic heating of the atmosphere such that oxygen and nitrogen will be stripped away, leaving Earth to overheat in runaway carbon-dioxide. Unless, that is, if gravitational mayhem, caused by galactic collision with Andromeda, flings Earth substantially further from the sun.

Oxygen, nitrogen and inert gasses can escape from bodies the size of the moon, Venus and Mars because the thermal process, 'Jeans Escape', in less massive bodies incurs substantially lower escape velocities.

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    $\begingroup$ Exosphere? 1800 C? Huh? $\endgroup$
    – Gimelist
    Oct 8, 2016 at 7:56
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    $\begingroup$ @Michael It is very rare gas, it doesn't mean that the satellites would evaporate :-) Protons coming from the Sun can be considered as 1million K hidrogen gas. The physics of the rare gases is very different. They are rarer as the best vacuum what can be produced on Earth. $\endgroup$
    – peterh
    Oct 8, 2016 at 10:18
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    $\begingroup$ Thank you very much the answer! I suspect, the "there is no characteristic time of loss of the Earth's atmosphere", means here essentially, that it would be much too long for a meaningful calculation (compared to the other significant processes, for example the Sun's depletion of Hydrogen)? $\endgroup$
    – peterh
    Oct 8, 2016 at 11:47
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    $\begingroup$ Just to add to the explanation of the ultra-hot exosphere, the gas at this altitude (which extends beyond the International Space Station) is so thin it is almost not there. Individual atoms or molecules can be in the order of a metre apart, and very seldom physically interact. So although the temperature is so high, the amount of heat is not far from zero. Think of spark from an angle grinder on iron - then think of of sparks millions of times smaller. They would land on your skin and you wouldn't even notice them. $\endgroup$
    – Gordon Stanger
    Oct 9, 2016 at 12:27

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