Absorption and Addition of OpacitiesInitial questions: Why does the spectrum of the Sun show dark lines? Are therecircumstances in which the spectrum of something should be a featureless continuum? Whatcan we learn from spectra?If the only way photons could interact was via simple scattering, there would be noblackbodies. We’ll go into that in much more detail in the next lecture, but the basic reasonis that in simple scattering there is no creation or destruction of photons. For that we haveto treat emission and absorption, respectively. We’ll start with absorption, and talk aboutthe important principles of how opacities are combined.To be ridiculously basic, absorption happens when a photon gets eaten by something. Itcould be by an atom or molecule, in which case the atom or molecule changes atomic state oris even ionized or dissociated. It could be by a free electron moving near an ion, in a processinverse to bremsstrahlung. Whatever the detailed process, though, it is rarely frequency-independent (unlike scattering, which is often nearly independent of photon frequency). Inaddition, multiple different sources of opacity are present simultaneously, meaning that toget a qualitative handle on how spectra will look, how radiation will move through a medium,etc., we need to know how to add opacities.For the following, it is useful to start by imagining a hypothetical situation in which allenergy transport is done with photons of a fixed energy and polarization. Of course, Dopplershifts mean that the photon energy “seen” by an interacting particle (an electron, an atom,a molecule, ...) could vary depending on the motion of the particle, but we’ll ignore that.Thus we’ll suppose that after scatterings, and even after absorption and re-emission, thephoton energy and polarization are the same as they were before. This would be an exampleof a fixed channel of energy transfer. For a fixed channel, the opacities add linearly (highestopacity dominates). Even though we have imagined photons forced to retain their energyand polarization, for our analysis a given energy and polarization counts as a fixed channel;that is, if we consider photons that currently have a given energy and polarization, that’sthe channel.Now, in reality there will always be processes that can change the energy of a photon oreven its polarization. Doppler shifts will do it, for example, and absorption and re-emissionwill certainly do it. Thus the photon energies and polarizations with low total opacity willbe more effective at transporting energy than will the photon energies and polarization withhigh total opacity. For the discussion below, different energies and polarizations count asindependent channels of energy transfer. Between independent channels, therefore, it is thelowest opacity that matters most.We can even go beyond photons. Consider the interior of a star, in which the temperaturevaries as a function of position and thus energy is transported. At a given location the energywill be shared by photons, electrons, and ions (and occasionally more exotic particles). If youconsider, say, the photons and electrons, they can both transport energy (by radiation andby conduction, respectively). If the effective opacity to photons is a lot less than the effectiveopacity to electron conduction, then most of the energy is transported via radiation. Thereverse can happen if the effective opacity to conduction is much less than that to radiation.Because these concepts have posed great difficulties throughout many classes I havetaught, I will now present them again in a slightly different way:• The cross section is the “effective area” of a single particle/atom/molecule/whatever.Equivalently, it’s the number of interactions of that particle per second divided by theflux of incoming photons (number per area per second).• The opacity is the “effective area” per gram of material. This is not equivalent to crosssection. For example, the ionization cross section can be much larger than the Thomsoncross section for some photon energies. However, the ionization opacity also dependson the number of atoms with electrons; the ionization opacity of fully ionized materialis zero, even though the ionization cross section is still the same as it was.• For a fixed channel of energy transfer, highest opacity dominates. For example, supposewe are interested in the opacity to photons of a given frequency. The photons willinteract first via the highest opacity process. If, e.g., there is a line at that frequency,the photons will interact quickly. This means that for a fixed channel, opacities addlinearly. Example: suppose we construct a half-sphere that is mirrored on the inside,and put a light source in the middle. We poke some holes in the sphere, and surroundthe sphere with transparent glass. Ask class: In a given direction, what determineswhether light shows through? If the sphere does not have a hole (the opacity is high),then it is dark in that direction. The presence of a low opacity along that line of sight(the transparent glass) doesn’t make a difference. For a fixed channel, opacities addlike resistors in series.• For independent channels of energy transfer, lowest opacity dominates. For example,consider the “average” opacity over a full blackbody spectrum. Photon energies withlow opacity are easy for photons to travel in. So, photons tend to diffuse into thoseenergies away from energies where the opacity is high. That’s why in optically thickmaterial, lines are dark. Example: in the half-sphere example above, Ask class:where does the light come out? It comes out where there are holes (i.e., low opacity).In directions where there are no holes, the light reflects back and eventually escapesvia a hole. Another example is conduction versus radiative energy transport. Whenenergy is easily transferred via conduction, that’s what determines the overall energytransfer rate. When energy is easily transferred via radiation, that’s what dominates.Example: if you’re standing in cold air, you lose heat at a certain rate. If you put yourhand on a piece of metal at the same temperature as the air, you lose heat a lot morerapidly! It’s an independent channel of energy loss. Ask class: mathematically, howwould one weight the smaller opacities more? For independent channels, opacities addharmonically, that is, 1/κ =P(1/κi) for different channels i, weighted appropriately.For independent