Pressure: It’s complicated

Lies, Damned Lies and Statistical Mechanics – 5

Reading standard physics texts one could be forgiven for believing pressure is a simple physical property. You heat something in a container and the pressure rises. Cool it and the pressure reduces. Similarly for changing the volume, such as by pushing in or pulling out a piston. The pressure described there is thermal pressure due to the kinetic interactions between the molecules in the substance.  There is another form of pressure we experience on Earth: atmospheric pressure is actually, in effect, the weight of the atmosphere. Oceanic pressure is similar, being the weight of the atmosphere plus the weight of the water. Another unhelpful feature is that “pressure” can mean the force per unit area impinging on a body immersed in a fluid or some applied force that results in the former or the effective momentum of a moving body of fluid.

In space there are no containers. Thermal pressure tends to dissipate by expanding the gas until its pressure matches that of its surroundings. Where pressure exists in astrophysical objects it is caused mainly by gravity, in the same way as atmospheric and oceanic pressure here on Earth. Gravity, of course, tends to be somewhat higher (than on Earth) around stars and major planets. From this one can appreciate that stability occurs when the pressures balance. In laboratory experiments thermal pressure is balanced by inward pressure provided by the container, but this tends to go unnoticed. To make sense of this, I refer to inward pressures, like gravity and our background container, as impressure and pressures that push outwards, like thermal pressure, I call expressures. (Think of implode and explode.) Mathematically, expressure is positive while impressure is negative.

We now have a rule: in a stable fluid impressure equals expressure. Or, mathematically, impressure+ expressure=0.

There are other pressures.

Electron degeneracy pressure is a force between atoms borne of the Pauli Exclusion Principle and depends on the number density of electrons and the electron energies. Again, it is not something we directly experience in daily life, but it is omnipresent and, for example, contributes to the sensation of a solid object. In a star this expressure can be very high due to the high density, under gravitational pressure.

Radiation pressure, almost undetectable here in daily life, occurs when electromagnetic radiation impacts atoms and raises the energies of the latter. It tends to push the atoms away from the radiation source. It can therefore act as expressure in a star’s corona or as impressure in the case of stellar radiation affecting a nearby planet’s atmosphere. This pressure is dependent on the radiation flux, the atom density (the number of atoms per cubic metre) and other factors such as the probability of a photon colliding with an electron (which is where quantum mechanics come in). Near an intense source of radiation, such as a star, this pressure can be enormous. In the solar corona, it is largely radiation pressure that stops the corona collapsing under its own weight.

There are also electrostatic pressures affecting charged gases, and let’s not forget inter-atomic gravity, small though it is.

In a star, the pressure balancing act entails the combination of electron degeneracy, radiation and heat pitted against gravitational pressure. Close to the solar surface the density is high so radiation and electron degeneracy pressures are high and almost match the weight of the full height of the corona above. If the expressures start to win the balancing act, the gas/plasma expands, becoming less dense, so the expressures decrease accordingly and the balance is restored. The solar wind is an extension of the corona and the pressure driving it also reacts backwards adding to the gravitational pressure.

Pretty much the same principle applies to Earth’s atmosphere. Atmospheric pressure is largely controlled by gravitational pressure. This results also in control of the mean atmospheric temperature at sea level. When the air is heated, either by a hot surface or by solar radiation, it expands upwards (not to be confused with convection, which also occurs), reducing its density and thermal pressure until the latter balances gravitational pressure. Radiation (outwards) and electron degeneracy pressures play little if any part in Earth’s atmospheric environment. Solar radiation reaching Earth causes radiation pressure towards the surface of the planet so acts to assist gravity. Since air density is greatest close to the surface, this is where solar radiation pressure is highest. So air pressure is a function not just of heating but also of gravity.

I wonder if meteorologists take gravitational pressure into account in their (rather complicated) calculations: I suspect not.

Observation: the combined effect of these pressures is that kinetic temperature in a gaseous region, not constrained by a container, is inversely proportional to the density, as is evident in the solar corona and inter-stellar gas clouds.  The gas law does allow for this, but, in containers pressure rises with temperature so this effect is not normally seen. The behaviours of planetary atmospheres are far too complex to show this property.

The upshot of all the above is that any analysis of pressure in a fluid must be undertaken with due regard to the combination of different types of pressure present in the environment under examination and not to make premature assumptions based on relatively simplistic laboratory measurements. Otherwise we haven’t moved on from the nineteenth century.