## 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.

# Lies, Damned Lies and Statistical Mechanics - 4

#### The Story So Far:

The three previous blogs explained how the Sun came to be a hydrogen shell with a fusion zone supporting it by thermal pressure from below while the weight of the coronal constituents bears down on it from above. Also how the photosphere and lower corona, squeezed by gravitational pressure, have their temperatures (i.e. kinetic energies) kept low by the Pauli Exclusion Principle. The solar spectrum is then manufactured by the progress of Compton collisions through the photosphere and corona. This left us with puzzles about the nature of sunspots and the presence of elements heavier than hydrogen in the corona, bar those in the protostellar cloud that arrived late on the scene.

The delusion is that sunspots are said to be dark because “they are colder material than the super bright solar surface”. However, if that was the case, there would be a point above the surface where the sunspot outflows reached the same temperature as the photosphere and this would be visible. It seems more likely that sunspots are hotter than the photosphere. Also, why is there all that electromagnetic activity? And why do pairs of sunspots often form?

Magma Plumes on Earth

In some senses, the interior of the Earth is a structural analogue of the Sun’s interior. Think of the Hydrogen shell mimicking Earth’s crust and the internal plasma matching the Earth’s magma (molten rock, or lava beneath the crust). Then there is helioseismology. While Earth’s seismic shocks are transmitted through magma, they originate in the crust. So where do the Sun’s shocks originate?

You may be familiar with the volcanic hotspots on Earth, which arise where plumes of magma burst through the Earth’s crust, often with dramatic effect, building new islands in the Pacific or creating enormous volcanic structures like the Yellowstone caldera.

Plasma Plumes

Deep in the solar interior, much violent fusion is going on, which probably rises and falls in its intensity, occasionally building so much pressure that it bursts upwards, in the form of a plume, and breaks through the photosphere. It is also possible that stray asteroids puncture the Sun’s surface, allowing some of the plasma to escape. (Is it just a co-incidence that the solar cycle corresponds with Jupiter’s orbital period? Jupiter’s gravitational field acts as a magnet to re-focus the more eccentric small orbiting bodies)

Ions, travelling, are, in effect, electric currents. The positive ions work like a reverse direction current. So since antiparallel currents repel each other, whilst parallel currents attract each other, the two streams of positive and negative ions tend to separate streams that can become twin plumes, then each plume follows its own path along a radius out to the photosphere. Hence twin sunspots are observed. Once at the surface, the plasma from below merges with that from above, but at the point of outflow, the opposing magnetic fields, generated by the plumes create substantial magnetic fields that use the other plasma as conduits.

One might wonder why the hydrogen shell below the photosphere doesn’t disintegrate, but volcanoes on Earth don’t have this effect on Earth’s crust.

Heavy Elements in the Corona

These plasma plumes may be bringing helium and other elements from the solar centre to the corona, accounting for some of the non-hydrogen lines in the solar spectrum.

This is part 4 of my solution to the Solar temperature and heat transfer problem and completes the set.

# Lies, Damned Lies and Statistical Mechanics - 3

## Recapitulation

The earlier blogs in this series explained how quantum mechanics determines the matter phases of the hydrogen in the Sun and the inability of densely packed atoms to absorb heat due to the Pauli Exclusion Principle. Re-examining the star formation process in this light reveals how fusion is possible in cold hydrogen if it is sufficiently densely packed and under sufficiently high gravitational pressure. These conditions would be extremely difficult, if not impossible, to replicate in Earth-bound laboratories. (Having said that, any facility that stores very cold, or liquid, hydrogen under pressure needs to beware.) It was concluded that the physical structure of a star consists of the following main shells/zones in sequence from the centre (heavy elements e.g. fusion products at the centre), fusion, solid, liquid, gas and plasma. The granular appearance of the photosphere is owed to the boiling surface of the liquid. Effects similar to the majority of solar prominences can be seen above vigorously boiling water.

We now turn to the matter of how these revelations affect the solar spectrum.

It is widely stated that orbital electrons can only jump quantum levels if photons of exactly the correct energy collide with them. This is not correct. If sufficiently high energy photons are present, Compton scattering can occur in which an exchange of energy occurs. If the electron is boosted by an inexact energy match, the excess energy is re-emitted.

There is no black body here. Radiation emitted by the hydrogen fusion zone occupies four distinct wavelengths in the gamma region. In the hydrogen shell, where the atoms are too tightly packed, no absorption is possible. As the pressure eases slightly, Compton collisions occur semi-productively. Electrons may initially absorb energy and jump to a higher energy level, but the pressure forces them back again, emitting hydrogen Balmer line radiation as they do so. The overall effect on the photon compliment is shown in the diagram:

In the above event diagram, time runs left to right. Vertical bars mark the occurrence of events. The black path shows the progress of the status of an electron with time while the red paths represent those of photons. The first event represents the collision between photon and electron, at which the electron gains energy at the expense of the incoming photon. The second event represents the electron rebounding to its original energy level and in doing so emitting a new photon. Effectively the collision has resulted in splitting the original photon into two, and the original spectrum has been altered so that one high energy line has been dimmed, and two new emission lines have appeared.

The directions of travel of the photons are also changed so that photons are now much more likely to undergo collisions amongst themselves, resulting in a spread of wavelengths either side of their original values, due to energy/momentum swapping amongst the photons. This gives rise to a pronounced bump around most of the Balmer region, in the spectrograph. Looking at Jack Martin’s Book “A Spectroscopic Atlas of Bright Stars” the Hydrogen Balmer bulge is plain to see in many of the spectrographs.

As the atom density decreases further, longer term absorption can occur, and it is at this stage that the earlier hydrogen emissions can be absorbed, giving the characteristic absorption spike in the spectrograph. Further rebounds can still occur at the lower levels. It is not necessary for the incoming photon to exactly match the energy required for an electron quantum jump, as long as it has sufficient energy for a productive collision to occur.

This process repeats progressively throughout the height of the corona, and beyond, until the matter and photons are too sparse to collide. The original photon energies are therefore progressively reduced through several energy levels. Similar effects are experienced by the atoms of any other elements that may be present in the above-surface medium.

Thus, the solar spectrum is constructed. Observed absorption lines at higher energies are, in some cases, due to absence, or lower rates, of emissions, leading potentially to false identification of elements in the corona. Since the Sun will have formed from a cloud of mixed gas and dust there are probably several trace elements, other than hydrogen, in the corona however one should always look at forbidden lines with suspicion.

Given that each successive shell, working outwards, is hotter than its inner neighbour, convection does not occur anywhere in the Sun.

Another source of real elements, other than hydrogen, in the corona may be sunspots, which are discussed in the next blog.

This is one component of my solution to the solar heat transfer problem.