Hydrostatic equilibrium

Lore & Background

Hydrostatic equilibrium, also known as hydrostatic balance or hydrostasy, describes the state of a fluid or plastic solid at rest, where external forces like gravity are exactly counteracted by a pressure-gradient force. In Earth’s planetary physics, this balance prevents gravity from compressing the atmosphere into a thin, dense shell while simultaneously stopping the pressure-gradient force from dispersing the atmosphere into space. More broadly, this equilibrium is the fundamental reason objects in space take on a spherical shape. In astrophysics and planetary geology, it serves as the defining criterion that distinguishes dwarf planets from small solar system bodies. An object in this state has a symmetrically rounded shape, typically an ellipsoid due to rotation, with any irregular surface features resulting from a relatively thin solid crust. Beyond the Sun, about a dozen such equilibrium objects are confirmed to exist in the Solar System. The mathematical foundation rests on Newton’s laws: for a fluid volume with constant velocity, the net force must be zero. By analyzing a cuboid volume element, the downward force from pressure above, the upward force from pressure below, and the downward weight of the element are balanced. This force balance yields the differential equation dP = -ρ g dh, where density (ρ) and gravity (g) can vary with height and pressure. This equation can also be derived as a simple equilibrium solution of the three-dimensional Navier–Stokes equations. In general relativity, the Tolman–Oppenheimer–Volkoff equation describes the structure of a static, spherically symmetric relativistic star, which in the nonrelativistic limit reduces to Newton’s hydrostatic equilibrium. The principle applies to stars and planets, which may have been fluid in the past or whose solid material deforms like fluid under high stress. A rotating star or planet in this equilibrium typically forms an oblate spheroid; for example, the star Vega, with a 12.5-hour rotation period, is about 20% larger at its equator than from pole to pole.

Reader's Guide

The concept of hydrostatic equilibrium is fundamental in astrophysics and planetary geology, serving as the distinguishing criterion between dwarf planets and small solar system bodies. It explains why objects in space are spherical: the balance of gravity and pressure-gradient force shapes them into symmetrically rounded ellipsoids, with any irregular surface features resulting from 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. The mathematical derivation from force summation shows that for a hydrostatic fluid, the change in pressure with height is proportional to density and gravity. This equilibrium can also be derived from the Navier–Stokes equations as a simple equilibrium solution, and from general relativity via the Tolman–Oppenheimer–Volkoff equation for static, spherically symmetric relativistic stars.

Did You Know?

• Hydrostatic equilibrium is the condition of a fluid or plastic solid at rest when external forces are balanced by a pressure-gradient force.
• It prevents gravity from collapsing Earth's atmosphere into a thin, dense shell and prevents the pressure-gradient force from diffusing the atmosphere into outer space.
• It is the distinguishing criterion between dwarf planets and small solar system bodies.
• In addition to the Sun, there are a dozen or so equilibrium objects confirmed to exist in the Solar System.