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, 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 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:
 

     

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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:

Numbers:

• 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 P_c = 1.7x10¹¹ N m^-2 = 1.7x10⁶ atmospheres