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        riddles >> medium >> Air in Hole Through Earth
        
(Message started by: SWF on Dec 17th, 2003, 7:39pm)
Title: Air in Hole Through Earth
Post by SWF on Dec 17th, 2003, 7:39pm
A hole is drilled through the center of earth such that the atmosphere flows to the earth's center. What must the diameter of the hole be in order for essentially all of the earth's atmosphere to be in the hole?

I know this involves some dull calculation, but the answer should be surprising. The main thing to take into account is (hint): [hide]pressure varies with distance from center of earth[/hide].

Obviously, many unrealistic assumptions can be involved in this one. To simplify the problem and give a common ground for everyone, use standard physics problem assumptions and the following: Through the use of an elaborate climate control system, the inside of the hole is held at a habitable temperature of 300K. The ideal gas law applies with gas constant such that:  Pressure= Density * 86100 N-m/kg (about right for air at 300K).  "Essentially" all of the atmosphere in the hole means the pressure at sea level is one-millionth of normal atmospheric pressure of 1e5 N/m2 (it can't really reach zero).

Earth has uniform density = 5517 kg/m3;
Universal gravitation constant G=6.673e-11 N-m2/kg2;
Radius of earth = 6.37e6 m;  
Mass of earth's atmosphere = 5.1e18 kg.
Title: Re: Air in Hole Through Earth
Post by Eigenray on Dec 17th, 2003, 8:42pm
Sweet merciful crap! [hide]3.8e-70[/hide]m!
Title: Re: Air in Hole Through Earth
Post by Eigenray on Dec 17th, 2003, 11:30pm
Assuming a linear temperature change from T=6000K at the center to T=300K at the surface, I get an answer of about [hide]83 Angstrom[/hide], which is much more reasonable.
Title: Re: Air in Hole Through Earth
Post by SWF on Jan 12th, 2004, 8:19pm
Excellent, Eigenray.  Your first answer is the same thing I get for the diameter of the hole, although I haven't tried the linear temperature variation, since I want the hole to be habitable. However, being less than 1e-50 times the diameter of an electron means this hole would be difficult to climb into.

The important point to remember is: be careful when making deep holes in the earth or the surface will run out air.
Title: Re: Air in Hole Through Earth
Post by Speaker on Jan 12th, 2004, 8:30pm
That is an incredible fun fact. I just have to find some way to work it into polite conversation.
;D
Title: Re: Air in Hole Through Earth
Post by Eigenray on Jan 12th, 2004, 11:14pm

on 01/12/04 at 20:19:24, SWF wrote:
Excellent, Eigenray.  Your first answer is the same thing I get for the diameter of the hole

Oh good.  I had to correct it twice cause I kept losing constants.


Quote:
The important point to remember is: be careful when making deep holes in the earth or the surface will run out air.

Words to live by.
Title: Re: Air in Hole Through Earth
Post by John_Gaughan on Jan 14th, 2004, 8:06am

on 01/12/04 at 20:19:24, SWF wrote:
The important point to remember is: be careful when making deep holes in the earth or the surface will run out air.

Given the thickness of the Earth's crust and the feasibility of drilling into the mantle, I am more concerned about magma pouring out of the mantle than air pouring in.
Title: Re: Air in Hole Through Earth
Post by Sameer on Feb 2nd, 2004, 12:41pm
Assuming there was no earthly distortion until man arrived and a footstep by a very heavy person caused total annihilation!!! I wonder if this made dinosaurs extint!!  ;D
Title: Re: Air in Hole Through Earth
Post by John_Gaughan on Feb 2nd, 2004, 9:57pm

on 02/02/04 at 12:41:27, Sameer wrote:
Assuming there was no earthly distortion until man arrived and a footstep by a very heavy person caused total annihilation!!! I wonder if this made dinosaurs extint!!  ;D

Smoking caused the dinosaur extinction. Everyone knows that!
Title: Re: Air in Hole Through Earth
Post by fil on Apr 24th, 2004, 8:13pm
Just stumbled across this discussion, and I'm deffinately lost with yaalls equations.  

Are you saying that a small hole drilled through the earth would require more air to fill it than the atmosphere has to provide?  

Kindly,
Fil
Title: Re: Air in Hole Through Earth
Post by SWF on Apr 27th, 2004, 7:31pm

on 04/24/04 at 20:13:43, fil wrote:
Are you saying that a small hole drilled through the earth would require more air to fill it than the atmosphere has to provide?
Yes, but only under the unrealistic assumptions stipulated by the problem. Using the Ideal Gas Law, means doubling the pressure of the gas doubles the density. The increase in density allows more gas fit in the hole, which further raises the pressure and density.

It turns out pressure as a functions of distance from center of earth, r, is something like

P0*exp(C*(R*R-r*r))

where R is radius of earth and P0 the pressure at earth's surface. C is a constant involving G, density of earth, T, [pi], R, and temperature.

Eventually pressure computed from Ideal Gas Law reaches a level where the substance assumed to be a gas, has a higher density than any normal solid. Well before that point, the approximation of density doubling with a doubling in pressure is incorrect.

I haven't tried to figure out how big the hole would need to be to consume the atmosphere assuming some maximum permissible density.


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