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I understand that moon does not have an atmosphere because it's escape velocity is low and atmosphere gases if it had would escaped into vacuum.

I know Earth has a relatively larger value of escape velocity. But I wonder if Global Warming plays any significant role in this.

I mean when the temperature of gases increases as a measure of how much its average kinetic energy (thus velocity) increases.

I'm curious to know whether atmospheric gas's velocity will rise significantly upto escape velocity due to Global Warming in the future or not.

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Atmospheric escape occurs on Earth via several distinct processes. The dominant process is called charge exchange escape, and accounts for 60-90% of escaped particles. This process is caused by collisions of high energy cosmic rays with atoms and molecules in the upper atmosphere. This process is mostly independent of the energy of the atmospheric molecule and therefore will not be affected by an increase in the atmospheric temperature.

The second most dominant process is Jean's escape which accounts for 10-40% of escape. This occurs when an atmospheric atom or molecule is at the extreme high end of its species's velocity distribution and actually has a velocity greater than earth's escape velocity. Particles in the atmosphere have velocities according to a Maxwell-Boltzmann probability distribution: $$ f(v) = \big( \frac{m}{2 \pi k_B T } \big )^{3/2} 4 \pi v^2 \exp \big (-\frac{m v^2}{2 k_B T} \big )$$

Note that this distribution is dependent on the temperature of the atmosphere, $T$, and the mass of a gas molecule, $m$. Increasing $T$ will technically broaden the distribution and shift more particles in the tail to velocities greater than the escape velocity. But, the important point is that an extremely small fraction of particles are actually above $v_{esc}$, and a shift of a few degrees will have a negligible effect on the fraction with $v > v_{esc}$. The Earth's escape velocity is ~11 km/s

To get a sense of the orders of magnitude involved: the Earth's atmosphere is about $5.5 \times 10^{18}$kg, and it loses 3 kg of hydrogen per second. Hydrogen is the lightest element, so heavier species like oxygen have significantly smaller losses.

Therefore, to answer your question, if we assume that the entire atmosphere heats by a few degrees due to global warming, there will be an extremely small but negligible increase in Jean's escape. Therefore, this is probably the least severe consequence of global warming.

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