[net.physics] Explanation of pressure equation

lew (04/14/83)

I posted an equation for the pressure of an isothermal atmosphere in
a linearly increasing gravitational field:

p(z) = p1 * exp( -(mg/kT) * 1/2 * z^2/R )

where p1 is the pressure at z=0 and g is the g-field value at z=R .
The formula applies to the pressure in a deep shaft, drilled from the
surface. It is obtained by integrating the differential equation for the
pressure:

dp = -q * g(z) * dz = -(p*m/kT) * (g*z/R) * dz

dp/p = -(mg/kT) * z/R * dz

log(p) = -(mg/kT) * 1/2 * z^2/R + constant


Here, q is the mass density and it goes to (p*m/kT) by the ideal gas law.
g(z)=g*z/R is an expression of the linear increase of gravity from 0
at the center (z=0) to g at the surface (z=R).

If you adjust p1 to give p(R)=p0, and then compare the values obtained
for z<R with the values obtained from a downward extrapolation of the
exponential atmosphere, you will find that the former are less than the
later. The values diverge slowly as z decreases from z=R.

The value at z=0 is the square root of the value obtained from a downward
extrapolation of the exponential atmosphere, in units of p0.

	Lew Mammel, Jr. ihuxr!lew

warren (04/14/83)

I believe the pressure calculations assume that there is an infinite
amount of air available to fill up the whole after it is drilled. 
With such large pressures, there may not be enough air in the earths
atmosphere to ignore the effect of the loss of air from above ground
into the hole.  As long as we are having fun calculating, how big
(diameter) does the whole through the earth have to be in order to a
"significant" amount of the atmosphere into the center of the earth?
 This begins to sound like a good plot for a bad sci-fi movie.
-- 

	Warren Montgomery
	ihnss!warren
	IH x2494

soreff (04/15/83)

Is the assumption of an isothermal compression really valid?  I was under the
impression that the temperature variation in the atmosphere was better
modeled with an *adiabatic* compression.  This gives a power law, rather than
an exponential law for pressure or density increase with depth.
	-Jeffrey Soreff (hplabs!soreff)

Schauble.HIS_Guest@MIT-MULTICS (04/15/83)

Unfortunately, I don't have time this week to check your equation,
but...

You make several references to a linearly INCREASING gravitational
field. In the case we are considering, the field is DECREASING, ranging
from 1G at the surface to 0 G at the center. Is this the case your
equation is drawn for??

          Paul