Hydrostatic Equilibrium

Consider a cylindrical region (length dr, end area dA) at a distance r from the center of the Earth:


And the cylinder has properties

Now, what is the force of gravity acting on the cylinder?

Now, define the quantity

(what is g(r)?)

so that we have


What balances this gravitational force? Pressure. So let's calculate the pressure acting on the cylinder. (remember Pressure = Force/Area)

The cylinder feels the pressure from the stuff above it pushing down, plus the pressure from the stuff below pressing up. So the net pressure is


and the force associated with this pressure is



OK. These forces must balance for the sun to be in equilibrium:

A little algebra yields The Equation of Hydrostatic Equilibrium:

Estimating the central pressure of the Earth

First, let's assume the density of the Earth does not change with radius. Then we can express M(r) for the Earth as:

So that the equation of hydrostatic equilibrium becomes:

Now, let's integrate this from the center of the Earth to the surface:

To end up with an expression for the pressure at the center of the Earth:

  • G = 6.67x10-11 N m2 kg-2
  • average density of Earth = 5500 kg m-3
  • radius of Earth = 6400 km = 6.4x106 m
So plugging in, we get Pc = 1.7x1011 N m-2 = 1.7x106 atmospheres