Hydrostatic equilibrium
In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, 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.
[1] In the planetary physics of Earth, the pressure-gradient force prevents gravity from collapsing the planetary atmosphere into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into outer space.
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.
Similarly, the force on the volume element from the pressure of the fluid below pushing upwards is
one can derive the Tolman–Oppenheimer–Volkoff equation for the structure of a static, spherically symmetric relativistic star in isotropic coordinates:
[4] A similar equation can be computed for rotating, axially symmetric stars, which in its gauge independent form reads:
From the time of Isaac Newton much work has been done on the subject of the equilibrium attained when a fluid rotates in space.
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.
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.
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 (including the effect of centrifugal force) would be weaker at the equator than at the poles by an amount equal (at least asymptotically) to five fourths the centrifugal force at the equator.
[5] In 1742, Colin Maclaurin published his treatise on fluxions in which he showed that the spheroid was an exact solution.
Newton had already pointed out that the gravity felt on the equator (including the lightening due to centrifugal force) 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
(including the effect of centrifugal force) is This spheroid solution is stable up to a certain (critical) angular momentum (normalized by
Above the critical value, the solution becomes a Jacobi, or scalene, ellipsoid (one with all three axes different).
Henri Poincaré in 1885 found that at still higher angular momentum it will no longer be ellipsoidal but piriform or oviform.
[7][8] Poincaré was unsure what would happen at higher angular momentum but concluded that eventually the blob would split into two.
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.
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.
However, in the cases of moons in synchronous orbit, nearly unidirectional tidal forces create a scalene ellipsoid.
Also, the purported dwarf planet Haumea is scalene because of its rapid rotation though it may not currently be in equilibrium.
The smallest object that appears to have an equilibrium shape is the icy moon Mimas at 396 km, but the largest icy object known to have an obviously non-equilibrium shape is the icy moon Proteus at 420 km, and the largest rocky bodies in an obviously non-equilibrium shape are the asteroids Pallas and Vesta at about 520 km.
The smallest body confirmed to be in hydrostatic equilibrium is the dwarf planet Ceres, which is icy, at 945 km, and the largest known body to have a noticeable deviation from hydrostatic equilibrium is Iapetus being made of mostly permeable ice and almost no rock.
[15] In 2024, Kiss et al. found that Quaoar has an ellipsoidal shape incompatible with hydrostatic equilibrium for its current spin.
They hypothesised that Quaoar originally had a rapid rotation and was in hydrostatic equilibrium but that its shape became "frozen in" and did not change as it spun down because of tidal forces from its moon Weywot.
For example, the massive base of the tallest mountain on Earth, Mauna Kea, has deformed and depressed the level of the surrounding crust and so the overall distribution of mass approaches equilibrium.
The force of gravity balances this out, keeps the atmosphere bound to Earth and maintains pressure differences with altitude.
