Fluid in hydrostatic equilibrium has a balanced downward force from gravity and upward force from fluid pressure. The hydrostatic equilibrium equation can calculate pressure in a planetary atmosphere. A volume of gas in space contracts under gravity until nuclear fusion produces a star in hydrostatic equilibrium. Stars eventually run out of hydrogen and collapse, leaving behind a solid iron core, neutron star, or black hole.
A volume of fluid, which may be a gas or a liquid, is said to be in hydrostatic equilibrium when the downward force exerted by gravity is balanced by an upward force exerted by fluid pressure. For example, the earth’s atmosphere is pulled down by gravity, but towards the surface the air is compressed by the weight of all the air above it, so the density of the air increases from the top of the atmosphere to the earth’s surface. This difference in density means that air pressure decreases with altitude so that the upward pressure from below is greater than the downward pressure from above, and this net upward force balances gravity downwards, keeping the atmosphere at a more or less constant height. When a volume of fluid is not in hydrostatic equilibrium it must contract if the gravitational force exceeds the pressure, or expand if the internal pressure is greater.
This concept can be expressed as the hydrostatic equilibrium equation. It is usually stated as dp/dz = −gρ and applies to a layer of fluid within a larger volume in hydrostatic equilibrium, where dp is the pressure change within the layer, dz is the layer thickness, g is the acceleration due to gravity and e is the density of the fluid. The equation can be used to calculate, for example, the pressure within a planetary atmosphere at a given height above the surface.
A volume of gas in space, such as a large hydrogen cloud, will initially contract under gravity, with increasing pressure toward the center. The contraction will continue until there is an outward force equal to the inward gravitational force. This is normally the point where the pressure at the center is so great that hydrogen nuclei fuse together to produce helium in a process called nuclear fusion which releases huge amounts of energy, resulting in a star. The resulting heat increases the pressure of the gas, producing an outward force to balance the inward gravitational force, so that the star is in hydrostatic equilibrium. As gravity increases, perhaps through more gas falling into the star, the density and temperature of the gas will also increase, providing more outward pressure and maintaining equilibrium.
Stars remain in hydrostatic equilibrium for long periods, typically several billion years, but will eventually run out of hydrogen and begin fusing progressively heavier elements. These changes temporarily throw the star out of balance, causing it to expand or contract until a new balance is established. Iron cannot be fused into heavier elements, as this would require more energy than the process would produce, so when all of the star’s nuclear fuel has eventually turned into iron, no further fusion can take place and the star will collapse. This could leave behind a solid iron core, a neutron star or a black hole, depending on the mass of the star. In the case of a black hole, no known physical process can generate enough internal pressure to stop the gravitational collapse, so hydrostatic equilibrium cannot be achieved and the star is thought to contract to a point of infinite density known as a singularity .
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