[Don Joe]

Well, at the core, there'd be zero effective gravity, so I'd imagine that the air pressure would be rather low, since for some distance above the core, I suspect the effective "source" of gravity would be upward, reducing any downward pressure at the air at the bottom of the hole.

[boris]

"Well, at the core, there'd be zero effective gravity, so I'd imagine that the air pressure would be rather low, since for some distance above the core, I suspect the effective "source" of gravity would be upward, reducing any downward pressure at the air at the bottom of the hole."

Wrong.

The equation of hydrostatics says P = Integral from 0 to radius of earth of rho(r)*g(r)dr. G(r) is easy; it is zero at the center but non zero at all other radii. I believe it is a linear function with radius.

Rho(r) is the density of the air which is stronly non-linear and requires a very robust equation of state.

One hundred miles of air over our heads produces 14.7 psi. Earth's radius is 6378136 meters or 3963 miles.

There is the solution. All you need is a (non-ideal gas law, eg a "real gas" law) for the equation of state of air.

Go to it, genius.

--Boris