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.