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