Hydrostatic equilibrium


In fluid mechanics, hydrostatic equilibrium, also called hydrostatic balance and hydrostasy, is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary physics of Earth, the pressure-gradient force prevents gravity from collapsing the atmosphere of Earth into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into outer space. In general, it is what causes objects in space to be spherical.
Hydrostatic equilibrium is the distinguishing criterion between dwarf planets and small solar system bodies, and features in astrophysics and planetary geology. Said qualification of equilibrium indicates that the shape of the object is symmetrically rounded, mostly due to rotation, into an ellipsoid, where any irregular surface features are consequent to a relatively thin solid crust. In addition to the Sun, there are a dozen or so equilibrium objects confirmed to exist in the Solar System.

Mathematical consideration

For a hydrostatic fluid on Earth:

Derivation from force summation

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 three 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 the structure of a static, spherically symmetric relativistic star in isotropic coordinates:
In practice, Ρ and ρ are related by an equation of state of the form f = 0, with f specific to makeup of the star. M is a foliation of spheres weighted by the mass density ρ, with the largest sphere having radius r:
Per standard procedure in taking the nonrelativistic limit, we let, so that the factor
Therefore, in the nonrelativistic limit the Tolman–Oppenheimer–Volkoff equation reduces to Newton's hydrostatic equilibrium:
. A similar equation can be computed for rotating, axially symmetric stars, which in its gauge independent form reads:
Unlike the TOV equilibrium equation, these are two 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. This equilibrium is strictly applicable when an ideal fluid is in steady horizontal laminar flow, and when any fluid is at rest or in vertical motion at constant speed. It can also be a satisfactory approximation when flow speeds are low enough that acceleration is negligible.

Astrophysics and planetary science

From the time of Isaac Newton much work has been done on the subject of the equilibrium attained when a fluid rotates in space. This has application to both stars and objects like planets, which may have been fluid in the past or in which the solid material deforms like a fluid when subjected to very high stresses.
In any given layer of a star, there is a hydrostatic equilibrium between the outward-pushing pressure gradient and the weight of the material above pressing inward. One can also study planets under the assumption of hydrostatic equilibrium. A rotating star or planet in hydrostatic equilibrium is usually an oblate spheroid, an ellipsoid in which two of the principal axes are equal and longer than the third.
An 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 from pole to pole.
In his 1687 Philosophiæ Naturalis Principia Mathematica Newton correctly stated that a rotating fluid of uniform density under the influence of gravity would take the form of a spheroid and that the gravity would be weaker at the equator than at the poles by an amount equal to five fourths the centrifugal force at the equator. In 1742, Colin Maclaurin published his treatise on fluxions in which he showed that the spheroid was an exact solution. If we designate the equatorial radius by the polar radius by and the eccentricity by with
he found that the gravity at the poles is
where is the gravitational constant, is the density, and is the total mass. The ratio of this to the gravity if the fluid is not rotating, is asymptotic to
as goes to zero, where is the flattening:
The gravitational attraction on the equator is
Asymptotically, we have:
Maclaurin showed that the component of gravity toward the axis of rotation depended only on the distance from the axis and was proportional to that distance, and the component in the direction toward the plane of the equator depended only on the distance from that plane and was proportional to that distance. Newton had already pointed out that the gravity felt on the equator has to be in order to have the same pressure at the bottom of channels from the pole or from the equator to the centre, so the centrifugal force at the equator must be
Defining the latitude to be the angle between a tangent to the meridian and axis of rotation, the total gravity felt at latitude is
This spheroid solution is stable up to a certain angular momentum, but in 1834, Carl Jacobi showed that it becomes unstable once the eccentricity reaches 0.81267.
Above the critical value, the solution becomes a Jacobi, or scalene, ellipsoid. Henri Poincaré in 1885 found that at still higher angular momentum it will no longer be ellipsoidal but piriform or oviform. The symmetry drops from the 8-fold D point group to the 4-fold C, with its axis perpendicular to the axis of rotation. Other shapes satisfy the equations beyond that, but are not stable, at least not near the point of bifurcation. Poincaré was unsure what would happen at higher angular momentum but concluded that eventually the blob would split into two.
The assumption of uniform density may apply more or less to a molten planet or a rocky planet but does not apply to a star or to a planet like the earth which has a dense metallic core. In 1737, Alexis Clairaut studied the case of density varying with depth. Clairaut's theorem states that the variation of the gravity is proportional to the square of the sine of the latitude, with the proportionality depending linearly on the flattening and the ratio at the equator of centrifugal force to gravitational attraction. Clairaut's theorem is a special case for an oblate spheroid of a connexion found later by Pierre-Simon Laplace between the shape and the variation of gravity.
If the star has a massive nearby companion object, tidal forces come into play as well, which distort the star into a scalene shape if 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, 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 0.11 to 0.14 for various surveys.