- Hydrostatic equilibrium
is when a fluid
is at rest everywhere because
everywhere the forces are balanced.
- Gravity
and the pressure force are usually the
forces.
- For the diagram,
Newton's 2nd law of motion
(AKA F=ma) specializes to
      ma = 0 = -mg + (p_1-p_2)*A ,
where acceleration a = 0
because of the constraint of
hydrostatic equilibrium,
A is the area of planar layer,
m is the mass of the planar layer,
-g is the gravitational field
and g is positive,
p_1 is the pressure below the layer,
p_2 is the pressure above the layer,
and we take up as positive.
- Note p_1 > p_2 for
hydrostatic equilibrium.
- Another perspective is that p_1*A must equal the
gravitational force
on all the mass above the lower layer.
To generalize, in all
spherically-symmetric
hydrostatic
self-gravitating
pressure-supported body,
the upward pressure force
of every layer must balance the
gravitational force
on all the mass above
that layer.
For example, this is true for the
Earth's atmosphere and
the Earth's interior.
To generalize, it is true for
moons,
planets,
and stars.
- Now let Δp = p_2 - p_1 .
Let Δr be the thickness of the layer and let its
density ρ be
asymptotically constant as
Δr→0.
Then in limit
as Δr→0, we obtain
      lim_(Δr→0)(Δp/Δr)
= lim_(Δr→0)[(-m/(Δr*A)]g = -ρg ,
or finally
      (dp/dr) = -ρg ,
where (dp/dr) is the
derivative of
pressure p with respect to height r.
- For a star at
radius r from the center,
g=Gm(r)/r**2, where m(r) is the
mass enclosed within the
spherical shell of radius r.
This formula follows from
the shell theorem originally
derived by Isaac Newton (1643--1727).
Our planar layer is asymptotically
equivalent to a small part of a spherical shell,
and so we can substitute for g in the
formula above to get
      (dp/dr) = -Gm(r)ρ(r)/r**2 ,
where ρ(r) is the density
of a star
as a function
of stellar radius.
This equation is one of the
4 equations of stellar structure:
the
stellar hydrostatic
equilibrium equation.
The 4 equations of stellar structure
are the equations that are solved
to obtain a
stellar structure model,
a hydrostatic that is.
Stars undergoing large-scale motions
require hydrodynamic modeling
which is much more difficult and less certain.
- The
stellar hydrostatic
equilibrium equation,
as the derivation implies, apples to all
spherically-symmetric
hydrostatic
self-gravitating
pressure-supported bodies.
So it applies
moons,
planets,
and stars.
Corrections do have to be made for deviations from
spherical symmetry,
of course.
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