Hydrostatic equilibrium

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Hydrostatic equilibrium or hydrostatic balance occurs when compression due to gravity is balanced by a pressure gradient force in the opposite direction. 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 other small solar system bodies, and has other roles in astrophysics and planetary geology.

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Mathematical consideration

For a volume of a fluid which is not in motion or is in a state of constant motion, Newton's Laws state that it must have zero net force on it – the forces up must equal the forces down. This force balance is called the hydrostatic balance.

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

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

In this equation, the minus sign comes from the direction – this force supports the volume element, rather than pulls it down (We are presuming that positive force acts down, if you read "down" as "up" the results are the same for equilibrium).

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 gas is

This is zero if the gas is not moving. If we divide 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:

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

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