# Hydrostatic equilibrium

Hydrostatic equilibrium or hydrostatic balance is the condition in fluid mechanics where a volume of a fluid is at rest or at constant velocity. This occurs when compression due to gravity is balanced by a pressure gradient force.[1] For instance, the pressure gradient force prevents gravity from collapsing the Earth's atmosphere into a thin, dense shell, while 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 (apart from the Sun), sometimes called planemos,[2] in the Solar System, seven more[3] which are virtually certain, and a hundred or so more which are likely.[3]

## Mathematical consideration

If the highlighted volume of gas is not moving, the forces on it upwards must equal the forces downwards.

Newton's laws of motion state that a volume of a fluid which is not in motion or which 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 the hydrostatic balance.

We can split the fluid into a large number of cuboid volume elements. By considering just one element, we can work out what happens to the fluid as a whole.

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,

$F_{top} = - P_{top} \cdot A.$

Similarly, the force on the volume element from the pressure of the fluid below pushing upwards is

$F_{bottom} = P_{bottom} \cdot A.$

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:

$F_{weight} = -\rho \cdot g \cdot V.$

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.

$F_{weight} = -\rho \cdot g \cdot A \cdot h$

By balancing these forces, the total force on the fluid is

$\sum F = F_{bottom} + F_{top} + F_{weight} = P_{bottom} \cdot A - P_{top} \cdot A - \rho \cdot g \cdot A \cdot h.$

This sum equals zero if the fluid's velocity is constant. Dividing by A,

$0 = P_{bottom} - P_{top} - \rho \cdot g \cdot h.$

Or,

$P_{top} - P_{bottom} = - \rho \cdot g \cdot h.$

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.

$dP = - \rho \cdot g \cdot dh.$

Density changes with pressure, and gravity changes with height, so the equation would be:

$dP = - \rho(P) \cdot g(h) \cdot dh.$

Note finally that this last equation can be derived by solving the three-dimensional Navier-Stokes equations for the equilibrium situation where

$u=v=\frac{\partial p}{\partial x}=\frac{\partial p}{\partial y}=0.$

Then the only non-trivial equation is the $z$-equation, which now reads

$\frac{\partial p}{\partial z}+\rho g=0.$

Thus, hydrostatic balance can be regarded as a particularly simple equilibrium solution of the Navier-Stokes equations.

## 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 (critical) 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% fatter at the equator than at the poles. A star with an angular velocity above the critical angular velocity becomes a Jacobi (scalene) 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.[4]

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.

### 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 (a planemo), 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 the 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.

In the Solar System, it appears that icy objects with a diameter larger than ca. 400 km are usually in hydrostatic equilibrium, while those smaller than that are not. Icy objects, though, can achieve hydrostatic equilibrium at a smaller size than rocky objects. The smallest object known to be in hydrostatic equilibrium is the icy moon Mimas at 397 km, while the largest object known not to be is the rocky asteroid Pallas at 532 km.

Since the terrestrial planets and dwarf planets (and likewise the larger satellites, like the Moon and Io) have irregular surfaces, this definition evidently has some flexibility, but as of yet a specific means of quantifying an object's shape by this standard has not 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.[5]

### 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 the earth and maintaining pressure differences with altitude.