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, define the quantity

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

and the force associated with this pressure is

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

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:

Numbers:

_{c} = 1.7x10^{11} N m^{-2}
= 1.7x10^{6} atmospheres

- G = 6.67x10
^{-11}N m^{2}kg^{-2} - average density of Earth = 5500 kg m
^{-3} - radius of Earth = 6400 km = 6.4x10
^{6}m