Assuming that the Sun wouldn't evolve into a red giant, how long would the Earth's atmosphere last?

Question

It's not going to last forever - Jeans Escape is going to eventually act on the atmosphere after trillions of trillions of years: see http://faculty.washington.edu/dcatling/Catling2009_SciAm.pdf

Answer 1

This is a quick answer, but the last paragraph of the article you linked says that the process will be important within about 1 billion years. I'm not sure if this is distinct from the fact that the Sun will be significantly hotter anyway because it is gradually expanding during its main-sequence life (i.e. before it even starts to become a giant) which will heat the Earth and thus increase the Jeans escape rate.

I think what the question is really asking is: how rapidly is the atmosphere escaping now, and how long would it take the atmosphere to escape? I tried briefly to dig around for an appropriate equation. The relevant Wiki section is mostly tagged "citation needed", which didn't really help, but claims that $O_2$ is sufficiently bound that it would take more than a trillion years to escape.

For what it's worth, one can compute the fraction of gas particles that has a vertical velocity greater than the local escape velocity from the Maxwell distribution, which is basically a normal distribution.

$\int_{v_\text{esc}}^\infty f_vdv=\int_{v_\text{esc}}^\infty\sqrt{\frac{m}{2kT}}\exp\left(-\frac{mv^2}{2kT}\right)dv$

$=\int_{v_\text{esc}}^\infty\sqrt{\frac{1}{2c_s^2}}\exp\left(-\frac{v^2}{2c_s^2}\right)$

$=\frac{\sqrt{\pi}}{2}\text{erfc}\left(\frac{v_\text{esc}}{\sqrt{2}c_s}\right)$

At the surface, you can use $v_\text{esc}\approx11.2$km.s$^{-1}$ and $c_\text{s}\approx331$m.s$^{-1}$ which, according to my computer, gives $5.02\times10^{-251}$. This is a very crude approximation. Ideally you'd average over the total mass of the atmosphere with the appropriate distributions. To get a rate you'd need the acceleration, since this just gives an unbound fraction rather than how rapidly that fraction is lost. But I hope this gives some solidity to the fact that it'd take a very long time.

Answer 2

What other assumptions are you going to make? Once you make one non-physical assumption, every further extrapolation is self-inconsistent. Comets are at least speculated if not accepted to make a decent contribution to atmospheric volatiles. Does comet bombardment continue at the rate it would if the Sun stayed happily main-sequence? At the presumably higher rate as if the Sun did go red giant and perturbed every orbit in the Solar System? Life continues to survive and maintains near-present-day atmospheric composition? Life gets wiped out some five billion years from now for no reason?

I don't think this question is answerable, unless you just do

logbase2(present atmospheric pressure/criterion pressure for calling the atmosphere "gone")*(Jeans escape characteristic time scale for N2), which I believe will be the shortest of the critical components of the atmosphere.