In continuum mechanics, a fluid is said to be in hydrostatic equilibrium or
hydrostatic balance when it is at rest, or when the flow velocity at each point
is constant over time. This occurs when external forces such as gravity are
balanced by a pressure gradient force. For instance, the pressure-gradient
force prevents gravity from collapsing Earth's atmosphere into a thin, dense
shell, whereas gravity prevents the pressure gradient force from diffusing
the atmosphere into space. Hydrostatic equilibrium is the current
distinguishing criterion between dwarf planets and small Solar System bodies,
and has other roles in astrophysics and planetary geology. This qualification
typically means that the object is symmetrically rounded into a spheroid or
ellipsoid shape, where any irregular surface features are due to a relatively
thin solid crust. There are 31 observationally confirmed such objects,
sometimes called planemos, in the Solar System, seven more that are virtually
certain, and a hundred or so more that are likely.
Mathematical consideration = Derivation from force summation =
Newton's laws of motion state that a volume of a fluid that is not in motion
or that is in a state of constant velocity must have zero net force on it.
This means the sum of the forces in a given direction must be opposed by an
equal sum of forces in the opposite direction. This force balance is called
a hydrostatic equilibrium. The fluid can be split into a large
number of cuboid volume elements; by considering a single element, the action
of the fluid can be derived. There are 3 forces: the force downwards
onto the top of the cuboid from the pressure, P, of the fluid above it is,
from the definition of pressure, Similarly, the force on the volume
element from the pressure of the fluid below pushing upwards is
Finally, the weight of the volume element causes a force downwards. If the
density is ρ, the volume is V and g the standard gravity, then:
The volume of this cuboid is equal to the area of the top or bottom, times the
height — the formula for finding the volume of a cube.
By balancing these forces, the total force on the fluid is
This sum equals zero if the fluid's velocity is constant. Dividing by A,
Or, Ptop − Pbottom is a change in pressure,
and h is the height of the volume element—a change in the distance above
the ground. By saying these changes are infinitesimally small, the equation can
be written in differential form. Density changes with pressure, and
gravity changes with height, so the equation would be:
= Derivation from Navier–Stokes equations =
Note finally that this last equation can be derived by solving the
three-dimensional Navier–Stokes equations for the equilibrium situation
where Then the only non-trivial equation is
the -equation, which now reads Thus, hydrostatic balance can be
regarded as a particularly simple equilibrium solution of the
Navier–Stokes equations. = Derivation from General Relativity =
By plugging the energy momentum tensor for a perfect fluid
into the Einstein field equations and using the conservation condition
one can derive the Tolman-Oppenheimer-Volkoff equation for
relativistic stellar structure: In practice, Ρ and ρ are related by an
equation of state of the form f(Ρ,ρ)=0, with f specific to makeup of the star.
M(r) is a foliation of spheres weighted by the mass density ρ(r), with the
largest sphere having radius r: Per standard procedure in taking the
nonrelativistic limit, we let c→∞, so that the factor
Therefore, in the nonrelativistic limit the Tolman-Oppenheimer-Volkoff equation
reduces the Newton's hydrostatic equilibrium:
(we have made the trivial notation change h=r and have used f(Ρ,ρ)=0 to
express ρ in terms of P) Applications
= Fluids = The hydrostatic equilibrium pertains to
hydrostatics and the principles of equilibrium of fluids. A hydrostatic
balance is a particular balance for weighing substances in water.
Hydrostatic balance allows the discovery of their specific gravities.
= Astrophysics = In any given layer of a star, there is a
hydrostatic equilibrium between the outward thermal pressure from below and
the weight of the material above pressing inward. The isotropic
gravitational field compresses the star into the most compact shape possible. A
rotating star in hydrostatic equilibrium is an oblate spheroid up to a certain
angular velocity. An extreme example of this phenomenon is the star Vega, which
has a rotation period of 12.5 hours. Consequently, Vega is about 20% larger
at the equator than at the poles. A star with an angular velocity above the
critical angular velocity becomes a Jacobi ellipsoid, and at still faster
rotation it is no longer ellipsoidal but piriform or oviform, with yet other
shapes beyond that, though shapes beyond scalene are not stable.
If the star has a massive nearby companion object then tidal forces come
into play as well, distorting the star into a scalene shape when rotation alone
would make it a spheroid. An example of this is Beta Lyrae.
Hydrostatic equilibrium is also important for the intracluster medium,
where it restricts the amount of fluid that can be present in the core of a
cluster of galaxies. We can also use the principle of
hydrostatic equilibrium to estimate the velocity dispersion of dark matter in
clusters of galaxies. Only baryonic matter emits X-ray radiation. The
absolute X-ray luminosity per unit volume takes the form where and are
the temperature and density of the baryonic matter, and is some function
of temperature and fundamental constants. The baryonic density
satisfies the above equation : The integral is a measure of the total
mass of the cluster, with being the proper distance to the center of the
cluster. Using the ideal gas law and rearranging, we arrive at
Multiplying by and differentiating with respect to yields
If we make the assumption that cold dark matter particles have an isotropic
velocity distribution, then the same derivation applies to these particles,
and their density satisfies the non-linear differential equation
With perfect X-ray and distance data, we could calculate the baryon density at
each point in the cluster and thus the dark matter density. We could then
calculate the velocity dispersion of the dark matter, which is given by
The central density ratio is dependent on the redshift of the cluster and is
given by where is the angular width of the
cluster and the proper distance to the cluster. Values for the ratio range from
.11 to .14 for various surveys. = Planetary geology =
The concept of hydrostatic equilibrium has also become important in determining
whether an astronomical object is a planet, dwarf planet, or small Solar
System body. According to the definition of planet adopted by the International
Astronomical Union in 2006, planets and dwarf planets are objects that have
sufficient gravity to overcome their own rigidity and assume hydrostatic
equilibrium. Such a body will normally have the differentiated interior and
geology of a world, though near-hydrostatic bodies such as the
proto-planet 4 Vesta may also be differentiated. Sometimes the
equilibrium shape is an oblate spheroid, as is the case with Earth. However, in
the cases of moons in synchronous orbit, near unidirectional tidal forces create
a scalene ellipsoid, and the dwarf planet Haumea appears to be scalene due
to its rapid rotation. It had been thought that icy objects
with a diameter larger than roughly 400 km are usually in hydrostatic
equilibrium, whereas those smaller than that are not. Icy objects can achieve
hydrostatic equilibrium at a smaller size than rocky objects. The smallest
object that appears to have an equilibrium shape is the icy moon Mimas
at 397 km, whereas the largest object known to have an obviously
non-equilibrium shape is the rocky asteroid Pallas at 532 km. However,
Mimas is not actually in hydrostatic equilibrium for its current rotation.
The smallest body confirmed to be in hydrostatic equilibrium is the icy moon
Rhea, at 1,528 km, whereas the largest body known to not be in hydrostatic
equilibrium is the icy moon Iapetus, at 1,470 km.
Because the terrestrial planets and dwarf planets have irregular surfaces,
this definition evidently has some flexibility, but a specific means of
quantifying an object's shape by this standard has not yet been announced.
Local irregularities may be consistent with global equilibrium. For example,
the massive base of the tallest mountain on Earth, Mauna Kea, has deformed and
depressed the level of the surrounding crust, so that the overall distribution
of mass approaches equilibrium. The amount of leeway afforded the definition
could affect the classification of the asteroid Vesta, which may have
solidified while in hydrostatic equilibrium but was subsequently
significantly deformed by large impacts. = Atmospherics =
In the atmosphere, the pressure of the air decreases with increasing altitude.
This pressure difference causes an upward force called the pressure
gradient force. The force of gravity balances this out, keeping the
atmosphere bound to Earth and maintaining pressure differences with
altitude. See also
List of Solar System objects in hydrostatic equilibrium
Statics Two-balloon experiment
Notes References
White, Frank M.. "Pressure Distribution in a Fluid". Fluid Mechanics. New York:
McGraw-Hill. pp. 63–107. ISBN 978-0-07-128645-9.
External links Strobel, Nick.. Nick Strobel's Astronomy
Notes.