Could Pluto support an atmosphere of methane?

[PeterPoPS]

Homework Statement

Pluto is believed to have a radius of 1500 km, a mass of 1.5x10^22 kg, and a surface temperature of 55 K. Could Pluto support an atmosphere of methane, CH4?

Homework Equations

The Attempt at a Solution

First some information:
The problem is from a book "Introductory Statistical Mechanics", so far topics like basic thermodynamics, some probability / statistics and the canonical ensemble together with the equipartion theorem have been brought up.
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My guess to solve the problem was to get the energy of the molecule via the equipartion theorem ($$E=\frac{3k_{B}T}{2}$$), only having 3 degrees of freedom since the rest (vibrational and rotational) are "frozen out". And somehow relate the energy to some centripetal force pushing the molecule towards the planet. The problem with this idea is that I need the rotational velocity i think? And none is given in the problem. Also, the centripetal force will be calculated macroscopically giving a very large force and the energy from the equipartion theorem is very small so they are not comparable.
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The answer to the problem is: Yes (Pluto can support an atmosphere of methane)

Best regards Peter

 

[Borek]

Think about the escape velocity.

 

[PeterPoPS]

Think about the escape velocity.

Thank you! But do I not need the Gravitational constant for Pluto to do that calculation?

If found an approximation on Wikipedia "Escape velocity"
"For a body with a spherically-symmetric distribution of mass, the barycentric escape velocity ve from the surface (in m/s) is approximately $$2.364×10^{-5} m^{1.5}kg^{-0.5}s^{-1}$$ times the radius r (in meters) times the square root of the average density ρ (in kg/m³), or:"
$$v_e \approx 2.364\times10^{-5}r\sqrt{\rho}$$

Calculating with this equations i get that the speed of the molecule is around 280m/s and that the escape velocity is around 1160m/s, indicating that the molecules does not have enough energy to leave the planet.

Is this the way you were thinking of? I'm just having doubts because I had to look up the equation for escape velocity and the molar mass for CH4 (which maybe I should have known)

 

[Borek]

That's more or less approach I was thinking about.

There are some details here that I am not sure about (there is always fraction of the gas that is fast enough to escape, so in fact atmosphere can be "bleeding" all the time, just very slowly), so second opinion won't hurt. But from what I know what you did is the most important step.

 

[PeterPoPS]

That's more or less approach I was thinking about.

There are some details here that I am not sure about (there is always fraction of the gas that is fast enough to escape, so in fact atmosphere can be "bleeding" all the time, just very slowly), so second opinion won't hurt. But from what I know what you did is the most important step.

Thank you very much!

About the atmosphere "bleeding", I think that could be neglected because as you say it is not the most important part of the problem