Lecture #4b-- Attenuation and optical depth



1) If we track the removal of ionizing photons as a function of radius, we have that if there are N photons impinging radially onto a shell of width dr, then the photons dN lost in that shell is given by dN/N = -dr/l where l is the mean free path, so dr/l is the fraction of the mean free path each shell represents, since dr/l is the fraction of radially propagating photons that are absorbed as they pass through the (thin) shell dr.

2) Since r and l can be scaled to different problems, it is useful to instead define dtau = dr/l, so that dN/N = -dtau, giving the obvious exponentially dwindling solution N(tau), where tau is called "optical depth". Note this works in any symmetry also, the only complication from the geometry comes in if photons impinge on the absorbing layer obliquely-- but then the absorption probability in that layer is simply augmented by the appropriate trigonometric factor 1/cosine(theta).

3) "Effective" thickness: Note that tau normally includes both true absorption (photon destruction) and scattering, but if we are simply tracking the total flux of photons out of some volume, then we only want to account for true absorption. In such a case, scattering does not by itself alter the flux out of the volume, but it does change the path taken by the photons, turning the process into a random walk that spends much more time inside the volume and thereby increasing the chance of a true absorption. This scattering introduces the possibility that a volume could be "effectively thin" to absorbtion even though optically thick to scattering.

Even in radiative equilibrium, where radiative energy entering and leaving a volume are the same, you can have photon "destruction". In that case, "destruction" usually involves taking the photon energies down by a significant amount. For example, in the Stromgren analysis of an HII region, we simply do not count the recombinations and ionizations that directly involve the ground state, as those leave the photons in the UV range and are counted as a scattering process. The "destruction" process there is when we have a UV photon causing an ionization, but when it recombines a lower energy (typically visible) photon is produced. This is also why HII regions "glow" in the visible (typically in the red, due to Balmer alpha).

In the case of the radiative cooling function, we count any lines that allow the photons to random walk their way out of the volume before being destroyed by a collisional transition, i.e., we consider the collisional excitations as the creation of a photon, and the collisional de-excitations as the destruction of one, and only count the former in the radiative cooling function if photon escape generally occurs prior to the latter. In this context, note that the "random walk" in a spectral line typically takes a number of scatterings of order 1/tau, where tau is the optical depth at line center, not 1/tau^2, as occurs for a random walk with continuum opacity. This different behavior, which allows escape from large tau to occur much more easily in lines, is due to the way the thermal motions of the ions in the lab frame redistribute the photon frequencies over the line, allowing the photons to reach the wings of the line where the optical depth is much less. The escape effectively occurs in frequency space before it does in real space. Lines are said to be "effectively thin" when the photon is not reabsorbed during those ~1/tau scatterings it needs to escape, and the line cooling function applies to all effectively thin lines.