Hydrostatic equilibrium occurs when compression due to gravity is balanced by a pressure gradient which creates a pressure gradient force in the opposite direction. The balance of these two forces is known as the hydrostatic balance.

Mathematical Consideration

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For a volume of a fluid which is not in 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,

$F\_\{top\}\; =\; P\_\{top\}\; \backslash cdot\; A$

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

$F\_\{bottom\}\; =\; -\; P\_\{bottom\}\; \backslash cdot\; A$

In this equation, the minus sign comes from the directon - this force supports the volume element, rather than pull it down (I'm saying positive force act's down, but this doesn't matter).

Finally, the weight of the volume element causes a force downwards. If the density is ρ, the volume is V and the acceleration due to gravity is g, then:

$F\_\{weight\}\; =\; \backslash rho\; \backslash cdot\; g\; \backslash cdot\; V$

We can split volume into the area of the top or bottom, times the height.

$F\_\{weight\}\; =\; \backslash rho\; \backslash cdot\; g\; \backslash cdot\; A\; \backslash cdot\; h$

By balancing these forces, the total force on the gas is

$F\_\{total\}\; =\; F\_\{top\}\; +\; F\_\{bottom\}\; +\; F\_\{weight\}\; =\; P\_\{top\}\; \backslash cdot\; A\; -\; P\_\{bottom\}\; \backslash cdot\; A\; +\; \backslash rho\; \backslash cdot\; g\; \backslash cdot\; A\; \backslash cdot\; h$

This is zero if the gas isn't moving. If we divide by A,

$0\; =\; P\_\{top\}\; -\; P\_\{bottom\}\; +\; \backslash rho\; \backslash cdot\; g\; \backslash cdot\; h$

Or,

$P\_\{top\}\; -\; P\_\{bottom\}\; =\; -\; \backslash rho\; \backslash cdot\; g\; \backslash cdot\; h$

P_top-P_bottom 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.

$dP\; =\; -\; \backslash rho\; \backslash cdot\; g\; \backslash cdot\; dh$

Density changes with pressure, and gravity changes with height, so the equation would be:

$dP\; =\; -\; \backslash rho(P)\; \backslash cdot\; g(h)\; \backslash cdot\; dh$

Applications

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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.

Astrophysics

In astrophysics, in any given layer of a star, there is a balance between the thermal pressure (outward) and the weight of the material above pressing downward (inward). This balance is called hydrostatic equilibrium. A star is like a balloon. In a balloon, the gas inside the balloon pushes outward and the elastic material supplies just enough inward compression to balance the gas pressure. In the case of a star, the star's internal gravity supplies the inward compression. The isotropic gravitational field compresses the star into the most compact shape possible: a sphere.

Atmospherics

Hydrostatic equilibrium can explain why the Earth's atmosphere does not collapse to a very thin layer on the ground. In the atmosphere, the pressure of air decreases with increasing altitude. This causes an upward force, called the pressure gradient force, which tries to smooth over pressure differences. The force of gravity, on the other hand, almost exactly balances this out, keeping the atmosphere bound to the earth and maintaining pressure differences with altitude. Without the pressure gradient force, the atmosphere would collapse to a much thinner shell around the earth, and without the force of gravity, the pressure gradient force would diffuse the atmosphere into space, leaving earth with hardly any atmosphere.

See also

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• Statics

Reference

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Strobel, Nick. (May, 2001). Nick Strobel's Astronomy Notes.

Fluid mechanics | Astrophysics | Hydrostatics

Hydrostatique