The Stellar Formation and Structure Delusions

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.

Crunch time!

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.

The Photosphere Delusion

Lies, Damned Lies and Statistical Mechanics

(Aimed at a broad readership)

We have a problem with the Sun. The temperatures don’t seem to add up. Whereas the middle and periphery are immensely hot the intervening, visible, surface, known as the photosphere, is relatively cool. This set-up does not accord with the laws of thermodynamics. So what’s wrong?

First, some history: it has been known, for some time, that the apparent temperature of the solar surface is around 6kK (6 kilo Kelvin or 6000 Kelvin. If you’re not used to Kelvins, a temperature change of 1 Kelvin is the same as a change of 1 degree Celsius and 0 Kelvin, or “absolute zero”, is equal to -272.8 degrees Celsius). It was Arthur Stanley Eddington who deduced that the source of the Sun’s energy must be hydrogen fusion, because no other form of energy generation could account for the amount of energy output and the length of time for which it has been “burning”. Experiences on Earth suggest that hydrogen fusion requires a temperature of more than 20 MK (20 MegaKelvin or 20,000,000 Kelvin) for its initiation. This level of concentrated energy is necessary to overcome the electrostatic repulsion between two hydrogen nuclei, known as the Coulomb barrier and calculated by Gamow’s theorem. The heat produced by such fusion is immense, as witnessed in hydrogen bomb explosions, with temperatures around 100MK.  Naturally, it was assumed that the Sun was entirely comprised of a ball of superheated plasma that cooled towards its periphery. So then, how does all this energy NOT heat up the photosphere to more than 6kK? The presumption was that the dense layers of hydrogen beneath the photosphere must be optically opaque so radiation must be giving way to convection as the heat transport mechanism. (Radiation warms up atoms by a process known as Compton Scattering.) Convection is a slow process and, therefore, would not be passing energy through fast enough to heat up the atoms. Then, when the temperatures and densities of the solar corona were measured for the first time, in the 1970s, it was a complete surprise that the coronal periphery was at around 5MK. (Temperatures and other coronal characteristics have been confirmed by extensive further analysis since then, using increasingly sophisticated satellite-based instruments.) How could this be?

Something "unknown" is going on here. It was known that considerable electromagnetic activity goes on in the corona, so extensive efforts have been made, without universally accepted result, to explain how this activity could explain the energy transfer from the centre to the corona and drive the solar wind. (The relevant science is known as magnetohydrodynamics or MHD for short.)

At this point, consider that the definition of temperature arises from the study of thermodynamics and statistical mechanics, based largely on heat engines, experiences with smelting metals and Earth-bound laboratory experiments. But the environment on the Sun is different from Earth’s in two significant ways: much higher ranges of both gravity and radiation. (For which reason I call the laws of thermodynamics “the steam engine laws”. Anyone who can cook an omelette can cook a feast?)

Whichever way you look at it, the solar surface should be hotter than is apparent, being heated either by the central fusion or even by the hot outer corona. Logically, something is preventing the atoms in the photosphere getting and/or appearing “hotter” and the effective “temperature” must be different from that defined by statistical mechanics.

In statistical mechanics, temperature is representative of the kinetic energy of the molecules (or, in this case, atoms), observable as the atoms/molecules “jiggling about”: the hotter they get, the more they jiggle.

The temperature of the photosphere is obtained by means of spectroscopy from which we can measure the kinetic energy of the atoms, but also we get an overall radiation profile from which we can use Wien’s Displacement Law to derive a temperature. Both agree on the temperature of about 6kK. But could this agreement just be a coincidence? Remember that we see the solar surface through the corona, which itself intervenes, actively, between ourselves and the photosphere. The plot thickens.

Now for some clues.

Quantum mechanics 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 external forces 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 as they become more compressed. 

This degeneracy pressure prevents densely packed atoms absorbing more energy, because, otherwise, the size of the 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, regardless of how much radiation energy is present.

It is also known that the hydrogen atom is 99.99999% empty space, plenty of room for radiation photons to pass through, so the opacity idea (mentioned in the second paragraph at the beginning of the blog) seems flawed.

We can now put forward a proposition.

The photosphere and the regions immediately above and beneath it, constrained by gravitational pressure (the same thing as atmospheric pressure on Earth but augmented by the reaction from ejecting the solar wind), are being maintained at sufficient density to limit the absorption of radiation emitted by the solar core, whilst being transparent to radiation due to the “emptiness” of these atoms.
Consequently the kinetic energy of the atoms is not representative of the total energy present and the apparent temperature is inaccurate. The rate of absorption of radiant energy increases as the density reduces, with growing height above the photosphere, hence resulting in the increase in temperature with height.

Since the radiation is largely absorbed before leaving the corona, or in driving the solar wind, it is not observable in the solar spectrum. The correct temperature of the photosphere is, therefore, not directly deducible from the solar spectrum, but must be calculated from the total energy flux (including the energy of the solar wind) exiting the corona. An alternative would be to calculate the considerable radiation pressure necessary to balance the difference between the gravitational and thermal pressures, at the photosphere, and calculate the equivalent temperature of that. The correct temperature is probably in the region of 75 MK. (Precise calculations are complicated, involving factors not discussed here, are ongoing and are still subject to extensive cross-checking).

This is not the whole story, there are other delusions: the internal structure of the Sun is another. This is just one component of my proposed solution to the solar heat transfer problem.