Fiat Lux

Why can't we see past the sun's photosphere?

We learned in class that optical depth depends on the opacity of a gas, and that opacity is determined by energy, essentially, whether it be from bound-bound transitions of elements in atoms, bound-free or free-free absorption, or from electron scattering. Furthermore, we learned about thermodynamic equilibrium, specifically, local thermodynamic equilibrium (LTE) and the "random walk" that photons take on their journey to freedom. If the path of the average length of one of these little strolls is much shorter than the distance over which the temperature changes significantly, the photons experience LTE, meaning they will be quite comfortable and needn't dress in massless layers or bother bringing a massless coat. In fact, they needn't wear anything at all, for they won't even be seen. They will be traveling under the cloak of opacity. This occurs in the interior of stars where the temperature is so high that photons can get into their birthday suits without worrying about peeping tom's engaging in voyeurism due to the fact that all of the atoms will be ionized, and there will be no wavelengths which permit transparency from a bound-bound transition, for example. (As of now, scientists have not been able to disprove the fact that photons living inside stars have formed nudist colonies, and that they simply don't want to be seen.)

But at the photosphere, these jaybirds exchange their safe LTE, which only depends on single parameter temperature, for a more risqué set of conditions, which includes many more degrees of freedom and greatly increases their chances of being seen. According to Harvard's "Breakdown of Local Thermodynamic Equilibrium" by George W. Collins, II, Maxwell-Boltzmann statistics and the Saha-Boltzmann ionization-excitation are sufficient to describe the distribution of energy levels, and that these levels will be constant under LTE. However, Collins states that in the upper atmosphere of stars, the density is not enough to support LTE, and here we must use different equations to lure our little photon friends from their primitive way of life and get them to realize their full potential in the form of useful light. Okay, so I'm paraphrasing. However, Collins does discuss some of the "phenomena which produce departure from Local Thermodynamic Equilibrium": the principle of detailed balancing, interlocking, and collisional ionization and photoionization.

In detailed balancing, every process, such as absorption and emission, must be balanced, and the jumps in energy levels must also balanced. In the upper atmospheres of stars, the rate of energy emitted per wavelength through atomic collisions is not able to be represented by the Planck function, and since this  is not possible under thermal equilibrium, the fact that these collisions are able to occur are evidence of a breakdown in LTE. The example Collins gives is when transition from the first to the third energy level is more favorable than a transition to the second energy level in a hypothetical atom which only has three energy levels, and the fact that the transitions back down to the second and first energy levels will not be balanced.

Interlocking is when two different absorption lines have the same upper energy level. To be honest, I barely understood detailed balancing, and interlocking is even more confusing to me. Basically, the same breakdown of LTE occurs when there is an imbalance of the cyclical processes as there was in detailed balancing, only in addition, strong lines which are formed under non-LTE conditions appear weaker because they share the highest energy level with weaker lines from inside the star. The example given was that red, singly ionized Calcium lines appear abnormally weak due to the photons given off by their interlocked lines. Also, to summarize both detailed balancing and interlocking, when there are more reactions among photons than there are of particles, there will be a breakdown of LTE. It makes sense that this would happen when there are comparatively less ions, since that means there will be less particles zipping about.

In the uppermost atmospheres of stars, there is still ionization occurring, through collisions and photoionization, although at a greatly reduced rate than that inside the star. And this is very interesting, but not all particles depart from LTE at the same temperature! But this makes sense. Remember that local thermodynamic equilibrium depends on the mean free path, and in a gas, different particles have different cross sections and number densities, therefore different mean free paths. Since electrons are the last particles to remain in thermodynamic equilibrium, when comparing collisions and photoionization, only at high temperatures and densities can there be more reactions among particles than among photons, so it only makes sense that at the cooler, less dense outer atmospheres of stars can photoionization dominate over collisional ionization. In conclusion, this is additional evidence of departure from LTE in the upper atmosphere of most stars.

Although scientists haven't been able to prove that photons aren't little nudists, I'm going to be bold and conjecture that they are not. I pretty much just added that bit to see if anyone was actually reading this.