Atmospherics of a Space Station

Date: 11/26/2000 at 02:54:36
From: Matthew H.
Subject: Simulated gravity and air pressure.

I understand that this question has as much to do with physics as it 
does mathematics. I have asked it of both math and physics instructors 
at my college, and have not gotten useful results. I appreciate any 
help, or pointers to useful resources, that you can offer.

I wish to have some understanding of the atmospheric dynamics of a 
space habitat of a given size.

Take a closed cylinder of a radius of r meters that is rotating at a 
rate of w radians per second (to simulate a desired gravity.) I wish 
to fill this cylinder with a gas of density p such that when it 
reaches equilibrium (that is, through friction all or part of it 
begins to rotate with the cylinder) it will be at a desired pressure 
at r meters from the cylinder's axis.

My question is this: Is there a formula to determine what the gas 
pressure is a given number of meters from the cylinder's axis?

Thank you for your time.

Date: 11/28/2000 at 04:05:56
From: Doctor Mitteldorf
Subject: Re: Simulated gravity and air pressure.

Matthew:

There are two equations you need. First, I'm sure you're familiar with 
the formula for centrifugal force, the surrogate for gravity in this 
situation: rw^2 is the equivalent of our "g" here on earth. But g is 
nearly constant throughout the earth's atmosphere, whereas rw^2 
increases linearly across the radius of the cylinder.  

Translated into terms of energy: the potential energy of a particle on 
earth is given by mgh, but in your space cylinder there's a 
pseudo-potential that can be found by integrating rw^2 to give 
-(1/2)mr^2w^2.

The second equation you need is less familiar: At a constant 
temperature T, the density of molecules in a region where their energy 
is E is proportional to the Boltzmann factor exp(-E/kT). In this 
situation, I believe you can get away with using our pseudopotential 
formula for E, and substituting 

     E = -(1/2)mr^2w^2 

into the Boltzmann factor.

So the pressure as a function of radius within the cylinder is 
proportional to 

     exp((mr^2w^2)/(2kT)) 

where m is the mass of a molecule of gas and kT is the Boltzmann 
constant times temperature.

For practical values of r and w, the difference in pressure from axis 
to rim of the cylinder is negligible. But if r and w are very large, 
you can use the above to predict pressure as a function of radius. (To 
actually get numbers out of the formula, you will need to normalize 
your values with an integral of the function exp((mr^2w^2)/(2kT)) over 
a finite range of radii; it looks like an un-doable integral, akin to 
the standard error function, but in fact there's a geometric factor of 
r in the integral that makes it easy.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/   

Date: 11/29/2000 at 19:22:06
From: Matthew Hauge
Subject: Re: Simulated gravity and air pressure.

Why does integrating rw^2 yield the pseudo-potential of a particle of 
mass m? And how exactly are you integrating rw^2 to get -0.5mr^2w^2? 
Second, how terribly does it complicate matters when the temperature 
is not constant? Third, I intend to work with large values of r, and 
small values of w. I'm not sure what you mean by "normalize your 
values..." and "it looks like an un-doable integral, akin to the 
standard error function, but in fact there's a geometric factor of r 
in the integral that makes it easy."

I have not had a great deal of calculus, so I appreciate your 
patience.

What I want to do here is understand what kind of weather could be 
expected in a large, rotating cylinder given a heat source along part 
of the axis, and a heat sink at the ends of the cylinder. I'm not sure 
what the right questions to ask are, so I tend to start with something 
basic like air pressure with altitude.

Matthew

Date: 11/30/2000 at 13:24:04
From: Doctor Mitteldorf
Subject: Re: Simulated gravity and air pressure.

Dear Matthew,

You're asking good questions, but I can't answer them in a short space 
in a language that you are comfortable with.  

I think you have a subject area that you're highly motivated to 
investigate, and that's a great start. But it's my guess that you'll 
have to pay your dues and put in some time studying the fundamentals 
of physics and calculus before you can answer them, or even understand 
other people's answers.

The question that you asked originally can be answered with a solid 
knowledge of first-year college calculus and physics; but these new 
questions require modeling and numerical analysis, with specialized 
techniques that only a graduate student in fluid dynamics would be 
comfortable with.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/