Lies, Damned Lies and Statistical Mechanics - 2
(Aimed at a broad readership)
The key to the solar structure lies in the way quantum mechanics controls the stellar formation process.
First, some history: In 1902 James Jeans published his theory of instability of a gas cloud that was sufficiently dense and cold. If it is cold enough and dense enough, gravity can overcome thermal pressure and cause the cloud to collapse and become a star. Such a collapsing cloud, it was thought, would become increasingly hot at the centre, due to increasing pressure, until hot enough to initiate fusion. Consequently a hot gaseous sphere would result, with plasma at the centre and cooling to gas at its extremities. However this theoretical structure does not concur with the evidence of solar measurements. When the (verified) data disagrees with the theory, the theory must be wrong.
Physics has moved on, including the discovery of quantum mechanics, and will always move on, so we need to re-evaluate in that light.
Quantum mechanics (recapitulation) tells us that the electron in a hydrogen atom inhabits a cloudy region surrounding its nucleus, referred to as a “probability distribution”, whose radius depends on the electron’s quantum energy level – the higher the energy: the larger the radius. The Uncertainty Principle, in the guise of Pauli’s Exclusion Principle, says that two electrons of the same quantum energy configuration cannot occupy the same “system”, or in this case probability distribution. When you try to compress atoms into very high density, this principle causes a force between the atoms known as electron degeneracy pressure. It has the effect of restricting the size of the electron cloud surrounding each atom’s nucleus, rather like squeezing soft rubber balls together, to the point where the atoms behave increasingly like solid spheres.
This degeneracy pressure prevents densely packed atoms absorbing more energy, because, otherwise, the size of the individual electron clouds would have to increase and breech the Pauli Exclusion Principle by overlapping each other. Hence, at high density, constrained by externally imposed pressure, atoms cannot have their temperatures increased. Indeed they can even become colder.
During stellar formation, gravitational pressure (which is like Earth’s atmospheric and oceanic pressures) puts more of a squeeze on the centre as the protostar continues to grow, in mass and size, due to the incoming gas/dust from the remainder of the protostellar cloud. Gravitational potential energy possessed by the cloud constituents prior to the collapse gets converted to kinetic energy as it falls in and when the incoming matter reaches the protostar some of this energy is converted to heat on the outside of the protostar. This heat cannot permeate to the inner regions due to the Pauli Exclusion Principle being invoked by the density and pressure as stated above.
With much earlier beginnings: In 1611, Kepler had conjectured that identical solid spheres could not be packed together more densely than to 74% of the total enclosing volume. This idea was supported by Gauss but only proven (computationally) in 1998 and accepted as a theorem in 2014. (You thought Peter Higgs had a long wait?!) Now, if this principle is extended to densely packed atoms, it can be seen that at 74% packing density, the atoms are unable to move in relation to each other meaning that the matter is solid. The packing density has to be reduced to about 50% to permit full fluidity and around 25% to allow gas formation (my informed estimates). (Matter can still be solid at lower densities and similarly for liquids; it’s the maxima that are important in this case.)
The radii of hydrogen atoms at different energy levels can be readily calculated and applying the principles above we can determine the relation between atom energy, density and phase of matter. The maximum possible density of hydrogen is found to be 1029 atoms per cubic metre and only if it is cold. Anything hotter, at that density, or denser than that isn’t hydrogen.
So the pre-stellar sphere, prior to fusion initiation, has a cold solid centre but as it gets less dense away from the centre it can get hotter and, as it does so, goes through the phases from solid to liquid to gas. This structure is, of course, completely opposite to the conventional model but it complies with the measured data. But that is before it has fired up.
Pressure is the same thing as energy density. This can be derived, from the ideal gas equation of state, and, also, the physical dimensions of these measures are identical. In order for two hydrogen atoms to fuse, some force or pressure is needed to overcome the electrostatic repulsion between the two positively charged nuclei. This is called the Coulomb barrier and the energy needed to overcome it is calculated by Gamow’s theorem. If therefore, at the centre of the star, the pressure divided by the atomic density (giving the energy per atom) exceeds the Gamow energy (and, of course, electron degeneracy pressure) we can have fusion. Clearly there is a critical mass that must be present for fusion to initiate and new versions of the Jeans’ equations are being sought on this new basis. This critical mass for fusion start-up is irrespective of the eventual mass of the complete star.
Initially, fusion may be started before the star is sufficiently massive to maintain the central gravitational pressure against the outward thermal pressure caused by the fusion. So there can be some false starts before continuous fusion gets under way. Once fusion has been established it starts eating away the surrounding solid hydrogen from the inside. Heat is borne from the centre to the exterior by radiation, heating the outer layers progressively, as permitted by the Pauli Exclusion Principle and as stated in my previous blog (The Photosphere Delusion). In the biggest stars, the fusion-capable region is correspondingly bigger so the rate of fusion is higher than for smaller stars, making them hotter and brighter, which is confirmed by observations. Brown dwarfs may be objects that had insufficient mass for fusion to initiate and their observed radiation comes from the hot matter at their outer regions (gravitational potential energy converted to heat as described earlier) as they cool.
Consequently, the resulting star has fusion at the centre surrounded by a solid shell followed by a liquid layer, gas, and then plasma. The densities of the solar structure from the photosphere outward have been measured repeatedly and the inner areas are too dense to be gaseous, let alone plasma. The granulation evident on the solar “surface” is vigorously boiling liquid hydrogen.
In a mature star, fusion products, i.e. denser elements fall to the innermost centre of the star. During hydrogen fusion helium atoms are produced. Helium atoms are slightly smaller that hydrogen and are equivalent to four hydrogen atoms in content and mass. So the mass density of helium is much higher than that of hydrogen. Consequently, hydrogen fusion creates material that can reach higher densities (both mass and number density), making room, as it were, for further fusion to occur. The hydrogen shell surrounding the fusion zone is kept in place by the balance between internal thermal and radiation pressures and external gravitational pressure, which also accounts for its spherical shape. The star will eventually die when this balance is lost, either by implosion or explosion, depending on which pressure wins.
The stellar structure described here is inevitable from the measured data.
There are still more delusions to deal with including the solar spectrum itself (next blog) and sunspots.
This is one component of my solution to the solar heat transfer problem.