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- What would be air pressure in a borehole through the center of a cool Earth?

The basic barometric formula for calculating atmospheric pressure is

I was arguing with a friend about the atmospheric pressure at the center of a cool Earth (no heat at the core) if a hole were drilled through it along the axis of rotation and open so air in the tube is connected to air above the surface. The acceleration of gravity decreases to zero at the center, since only the sphere below a random point (located between surface and center) contributes to the gravitational force. With g = 0 at the center, the pressure would be small. The friend says that's nonsense, that the pressure only a couple of hundred kilometers down would be super high and that would keep the pressure at the center really high. Neither of us has the mathematical acumen necessary to get beyond hand-waving.

Has anyone seen, or is brilliant enough to derive, a function that would give us an idea of what the approximate pressure would be at the center?

**(Ph: pressure at height h) = (Pzero: pressure at height zero) x exp(-mgh/kT)**, where the height**h**unit is meters,**P**unit is pascals,**m**is the mass of an average "air" molecule,**g**is the acceleration of gravity,**k**is the Boltzmann constant, and**T**is the temperature in kelvins. Acceleration of gravity**g**drops off with altitude, but not by much, so treating it as constant is a good approximation.I was arguing with a friend about the atmospheric pressure at the center of a cool Earth (no heat at the core) if a hole were drilled through it along the axis of rotation and open so air in the tube is connected to air above the surface. The acceleration of gravity decreases to zero at the center, since only the sphere below a random point (located between surface and center) contributes to the gravitational force. With g = 0 at the center, the pressure would be small. The friend says that's nonsense, that the pressure only a couple of hundred kilometers down would be super high and that would keep the pressure at the center really high. Neither of us has the mathematical acumen necessary to get beyond hand-waving.

Has anyone seen, or is brilliant enough to derive, a function that would give us an idea of what the approximate pressure would be at the center?