A planet is shaped like thin donut or torus and has atmosphere made of pure helium μ = 4 g/mole?

Question

The radius of the girth of the planet is a=1000km, and the larger radius c of the donut itself is unknown, but very large c>>a. I use notation a and c as can be seen here:
http://mathworld.wolfram.com/Torus.html

Due to global warming the temperature of the atmosphere is gradually increasing. At what temperature will the atmosphere completely 'evaporate' and dissapear?

The acceleration of gravity on the surface is g=1m/s².

First note to Dr. R:
spherical planets do have finite potential at infinity and their atmospheres evaprate at any temperature. Cylindrical and planar 'planets' have inifinite potentials at infinity and do not have to evaporate.

Now:
acceleration of gravity is
g(r) = Go (a/r)
potential of gravitation field:
P(r) = ∫Go (a/r) dr = Go a ln(r/a)
Density of states:
dV/dE = dV/dr dr/dE =
2πr x 1/μ 1/g(r) =
2πr x 1/(μ Go) (r/a) =
2π/(μ Go a) r²

dV/dE = 2π/(μ Go) exp(2E/Eo), where Eo = μ Go a.

Note that critical exponent exp(2E/Eo) popped up. Wow.
Between energies E and E+ΔE there are ~ ΔEexp(2E/Eo) populated according to Boltzman distribution
exp(-E/RT) and hence there are
dN = ΔE exp(E(2/Eo - 1/RT)) of gas.

Consequently at critiacal temperature
Eo = 2RTc, the population of switches from predominently ground states to predominantly high-energy states, that is gas escapes to infinity.

Answer:
Tc = μaGo/(2R) = 240K

Answer 1

This is a tough question that I can't answer without learning too much that I've forgotten.

Tricky part is to find the escape velocity of the torus. Initially, because of the large c >> a ratio, it will act like any infinity long cylinder. This means that g falls off at 1/(a+h). At some point, I'm guessing until h =1/8 c, g will start falling off rapidly until it reaches 1/(c+r)^2 at about h = c.

[No, I don't want to do the integration to figure this answer out, and no, I could not find it on the internet, and no, all my physics books burned up in a fire many years ago so I have no references]

Once you get the escape velocity, you probably want your temperature such that rms velocity for the gas molecules is the escape velocity. This will allow the Helium to literally boil away.

But let's do the back of the envelope calculation (I'm only attempting this because it is a good question, but very tough, and you scared everyone else off):

Solve for Oribital V.torus (around a)
g.torus=v^2 /r
v=sqr (g.torus * r)
= 1000 m/sec = 1 km/sec

Assume the escape velocity for the torus ~ 2 * Orbit velocity = 2 km/sec (educated guess)

Velocity of sound on earth ~ 330 m/sec at 290 K. My recollection velocity of gas molecules is about 4/3 the velocity of sound or 440 m/sec

Molar weight air ~ 28 g/mole

Air/Helium = 28/4 = 7

Velocity Helium/Air ration = Sqr 7= 2.64

Velocity Helium at 290 K = V air * 2.64 = 1160 m/s

Escape V.torus / V.He.290K = 2000 / 1160 = 1.72

T.torus = 290 K * 1.72^2

~ 862 K

Was I close? Just used everything I could remember. Probably the biggest source of error would be the escape velocity from the torus.

Answer 2

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Answer 3

The question is clever, but ill posed. Technically, the atmosphere of any planet is evaporating continuously due to the tail end of the Boltzman distribution extending to infinity. It's just a question of time.

Answer 4

Wow! What a cool question! Congrats to the TA who came up with it.

Ahhhh... I wish I could be back in school and I would get to solve these again.

Too bad. Now I get to solve the real ones... like how to write a pipelined accumulator of arbitrary width in VHDL that can run at 250MHz in an FPGA. Wanna swap?

:-)