- #1

Juanda

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- TL;DR Summary
- Helium leaks into space continuously. Big things don't. What is the threshold?

While reading a similar and deservedly closed post a contradiction came to my mind. The supposed contradiction is related to Statistical Physics where my understanding is only conceptual so correct me where I might be wrong.

I remember reading that lightweight gasses can escape Earth's gravitational field because, due to the statistical nature of reality, some particles at the top of the atmosphere may get enough kinetic energy from heat (random kinetic energy of nearby particles) to reach the escape velocity. If there's nothing around to collide with it (and there isn't because we said it's already at the top of the atmosphere) then it'll just keep going and leave the gravitational field from Earth. I'd love to add a graph showing how the chances of this event happening are related to temperature. I recall they exist but I can't find a proper source.

As far as I know, this doesn't happen to hydrogen because it's such a reactive element that it'll bond with heavier elements making that event impossible to happen (I guess you'd always say there's a small chance in statistical physics but it's near zero). Helium on the other hand, being the noble gas it is, only bonds with itself so it doesn't get to be heavy enough to be retained by Earth in the long term so we're constantly losing helium to space. Helium is also being created on Earth by some unrelated processes but that's not the point of the post.

So that's the introduction. Let's now go with the post. There are many details that make the real-world scenario complicated such as the influence of the sun for example in the form of day and night, heat and solar winds so let's try to simplify it by imagining a starless perfectly round and solid planet similar in mass to Earth with a heat source within so the crust is uniformly at 20ºC (68F). Its atmosphere is composed of only CO2 (something I presume is heavy enough to make a stable atmosphere).

From that, we'd be able to establish the density and temperature distribution of the atmosphere as a function of height. I couldn't in my tries but I believe we'd have all the information necessary. If there is any prerequisite missing to derive the density function feel free to add it.

Now let's take an infinitely rigid spherical balloon 1m in diameter and 1g in mass. Then, it'll keep raising until its density equalizes with the density of the CO2 outside which as we said is a function of height. Anyways, at least in classical mechanics, this is not a especially hard problem.

Is there a point in statistical mechanics where we'd have to worry about the balloon escaping Earth's gravity? Is it a matter of shrinking the balloon? If it were the mass and size of a molecule it'd be able to escape. If it's much bigger we know the chances are near zero and the behavior can be described with classical mechanics.

I believe there's a point where we could say something like:

What do you think?

I remember reading that lightweight gasses can escape Earth's gravitational field because, due to the statistical nature of reality, some particles at the top of the atmosphere may get enough kinetic energy from heat (random kinetic energy of nearby particles) to reach the escape velocity. If there's nothing around to collide with it (and there isn't because we said it's already at the top of the atmosphere) then it'll just keep going and leave the gravitational field from Earth. I'd love to add a graph showing how the chances of this event happening are related to temperature. I recall they exist but I can't find a proper source.

As far as I know, this doesn't happen to hydrogen because it's such a reactive element that it'll bond with heavier elements making that event impossible to happen (I guess you'd always say there's a small chance in statistical physics but it's near zero). Helium on the other hand, being the noble gas it is, only bonds with itself so it doesn't get to be heavy enough to be retained by Earth in the long term so we're constantly losing helium to space. Helium is also being created on Earth by some unrelated processes but that's not the point of the post.

So that's the introduction. Let's now go with the post. There are many details that make the real-world scenario complicated such as the influence of the sun for example in the form of day and night, heat and solar winds so let's try to simplify it by imagining a starless perfectly round and solid planet similar in mass to Earth with a heat source within so the crust is uniformly at 20ºC (68F). Its atmosphere is composed of only CO2 (something I presume is heavy enough to make a stable atmosphere).

From that, we'd be able to establish the density and temperature distribution of the atmosphere as a function of height. I couldn't in my tries but I believe we'd have all the information necessary. If there is any prerequisite missing to derive the density function feel free to add it.

Now let's take an infinitely rigid spherical balloon 1m in diameter and 1g in mass. Then, it'll keep raising until its density equalizes with the density of the CO2 outside which as we said is a function of height. Anyways, at least in classical mechanics, this is not a especially hard problem.

Is there a point in statistical mechanics where we'd have to worry about the balloon escaping Earth's gravity? Is it a matter of shrinking the balloon? If it were the mass and size of a molecule it'd be able to escape. If it's much bigger we know the chances are near zero and the behavior can be described with classical mechanics.

I believe there's a point where we could say something like:

*For a spheric rigid balloon X in volume and Y in mass there is a Z% chance that it will escape Earth's atmosphere within 1 day.*What do you think?