- To be at rest,
the net force
on every fluid parcel
must be zero.
- To see why a sphere is
in hydrostatic equilibrium
consider the identified layer in Image 1.
- The pressure in the layer is constant,
and so there is no tangential
pressure
gradient to push
a fluid parcel in the layer
tangentially.
- Acting on the layer,
we have 3
external forces:
pressure outward p_out*A,
pressure inward p_in*A,
and gravity mg,
where A is the surface area of the layer (which is the same to 1st order
on both inward and outward sides,
m is the layer mass,
and g is the
gravitational field
(which points inward).
For hydrostatic equilibrium
(i.e., balance of forces), we find that
p_out*A = mg + p_in*A
p_out = (m/A)g + p_in
p_out = ρg*dr + p_in ,
(dp/dr) = -ρg
where dr is the thinkness of the layer and ρ is the layer
density
and (dp/dr) is the pressure
gradient---which is
negative meaning that
pressure decreases going outward.
Note that the pressure outward
at any spherical shell
must support all the mass
above the spherical shell,
and so pressure increase going inward.
- Note gravity behaves very
simply for
spherically symmetric
mass distributions:
- It only pulls radially inward toward the center of the distribution.
- All the mass m(r) contained within
a sphere of
radius r centered on the
center of symmetry acts just as if it were
point mass
of mass m(r) at the
center of symmetry.
- All the mass outside of the
sphere of
radius r has NO
gravitational effect at all inside
the sphere.
- These great simplifications are due to the
inverse-square law
nature of
gravitational force.
The simplifications are summarized in the
shell theorem originally
derived by
Isaac Newton (1643--1727).
- The
hydrostatic equilibrium
spherically symmetric
sphere is
self-consistent solution for a clump of
of a self-gravitating
clump of
fluid.
It is self-consistent since nothing will move with all
the forces are balanced, and
so the structure is unchanging.
Actually, one has should
prove that the
hydrostatic equilibrium
spherically symmetric
sphere
is stable: i.e., that vanishingly small
perturbations damp out
and do NOT cause progressive change to some other structure.
We do prove this below actually.
- Now an arbitrary initial clump of fluid
(with initially zero
macroscopic
kinetic energy)
acting under only self-gravity
and pressure will arrange itself into
(i.e., relax to) a
hydrostatic equilibrium
spherically symmetric
sphere.
- Why?
Fluids have very low resistance
to shearing forces.
The ideal limit of a perfect fluid
has NO resistance at all.
A pair of shearing forces
are parallel,
but do NOT act along the same line.
Thus, they tend to make layers of a body slide over each other.
This is just what happens in fluids.
In the case of any initial clump, the
self-gravity
and pressure acting in combination
as shearing forces
will keep moving fluid parcels
around until they CANNOT anymore---which is when the clump has relaxed to the
self-consistent solution---where there are NO
shearing forces acting---which
is just the
hydrostatic equilibrium
spherically symmetric
sphere.
The fact the
self-gravity
and pressure
always move the clump toward
hydrostatic equilibrium
spherically symmetric
sphere
proves that that structure is stable.
- But you say why does fluid
NOT keep sloshing around the
hydrostatic equilibrium
spherically symmetric
sphere structure because
the
fluid parcels
still have kinetic energy
when they reach that structure.
What about conservation of energy?
The initial clump had more than the minimum possible
gravitational potential energy
and some
macroscopic
kinetic energy was
generated in the relaxaton process.
But during the relaxation process,
viscosity
dissipates
all the generated macroscopic
kinetic energy
into
waste heat.
In astrophysical contexts,
the waste heat will usually
be radiated away as
electromagnetic radiation (EMR).
Actually, a perfect fluid
has NO
viscosity, and
so can never relax to
hydrostatic equilibrium
unless there is some other way for it to lose macroscopic
kinetic energy.
- Now astro-bodies
can be made of all
fluids (e.g.,
stars), but they
can also be made or partially made of
solids like
most planets.
- But if the self-gravity
of a solid is sufficiently strong,
the resistance to shearing forces
will be so weak that
chemical bonds of the
solid will eventually break sufficiently and the
solid will act
as a physics plastic: i.e., it
will flow: layers will slide over layers.
- When will the flow stop?
Only the pressure force is strong enough
to resist sufficiently strong self-gravity.
Atoms strongly resist being compressed.
But note the pressure force does NOT
resist shearing forces.
So when the pressure force
and gravity balance on every
small bit of
matter flow will essentially stop
just described in general above.
Then one has
spherically symmetric
sphere self-consistent solution.
- If there is a centrifugal force
due to rotation
(relative to an inertial frame),
the self-consistent solution becomes approximately
an oblate spheroid.
- Since almost all
pressure-supported
astro-bodies
have rotation,
most pressure-supported
astro-bodies
with large enough self-gravity
to overcome shearing forces
are approximately oblate spheroids
with usually only a small amount of oblateness:
e.g., stars and
and planets.
- How large does a astro-body have
be to be pulled into nearly spherical shape?
Well this depends on
chemical composition,
heat energy content,
and rotation.
However, observations suggest the
empirical rule
that the size scale for a rocky astro-body
must be >∼ 600 km and
for a water ice
astro-body
must be >∼ 300 km
(see Wikipedia: Dwarf planet:
Hydrostatic equilibrium).
- Image 2 Caption: The Image 2 shows several structures relevant to our topic.
We explicate the structures as follows:
- A planetary system
is supported
against its host star's
gravity
by rotational kinetic energy
and angular momentum.
Without large
macroscopic kinetic energy,
astro-bodies
become pressure-supported
astro-bodies
(e.g., stars,
planets, and
asteroids).
- Most asteroids have
self-gravity too
small to pull them into
near spheres.
Their chemical bonds
(aided by pressure
and the centrifugal force)
allow them to have irregular shapes.
- The pressure force is strong
enough to hold up
mountains, but it is
isotropic.
It pushes equally upward and sideways.
The pressure force pushing
sideways acts itself as
shearing force which is
opposed by the
chemical bond forces
in the rock.
However, mountains will slump
if the combination of
pressure force
and gravity
can break the chemical bonds
of the rock.
The slumping stops when the
pressure force
and gravity are sufficiently
reduced that the chemical bonds
can resist their
shearing forces.
Note mountains on
Earth
are tiny compared to the
Earth because of the
Earth's relatively
high self-gravity.
- A typical planet
held up mainly by the
pressure force
with a little
centrifugal force
that causes
the equatorial bulge.
There can be little mountains
held from slumping by the
chemical bonds
of the rock.
- For a self-gravitating
fluid body of
UNIFORM density,
the exact shape solutions are
Maclaurin spheriods
and Jacobi ellipsoids.
- Can there be cases where all possible
pressure forces fail to
balance self-gravity?
Yes.
General relativity (GR)
predicts a sufficiently dense massive
object will collapse to being
black hole
with a
ring singularity
due to rotation
(i.e., a Kerr black hole)
or a point singularity
if there is no rotation
(i.e., a
Schwarzschild black hole).
The singularities have
finite mass and
zero
volume, and so have
infinite density.
However, maybe there some kind of finite
density structure inside a
black hole sustained
by something we don't know: an unknown
force,
motion (e.g.,
rotation
or oscillation),
or other physical effect.
But we don't know.
The inside of black holes is very uncertain.