Spectral lines are a ubiquitous feature of astronomical data. This week, we explore the special case of optically thin emission from low-density and high-temperature plasma, and consider the component factors that determine the line intensity.
The flux [ergs s-1 cm-2 sr-1] from an optically thin emission line that arises due to a transition between energy levels j and i in an ionic species Z+I is simply written. It is the product of the probability of the transition Aji(Z,I) (aka the Einstein coefficient), the number of particles of the species that exists in the upper level of the transition Nj(Z,I), the volume of the emission dV, and the energy of the emitted photon hc/lambda, scaled down by the distance to the source (4 pi d2; note that the factor 4 pi is due to the emission being radially symmetric).
But this apparently purely atomic calculation can be reformed and rewritten, after some algebra, in terms of quantities that are astrophysically more meaningful. The equations below walk you through the tranformation from atomic physics to quantities that can be separated out into different hierarchies of astrophysical source properties, from things that change not at all from one source to another, to things that are likely not the same even along the line-of-sight.
All of the quantities that depend only on the atomic physics can be pulled together into the emissivity of the transition, eji(Ne,Te,Z,I). This is (mostly) independent of the physical conditions at the source, and is generally treated as invariant except for changes due to the electron number density. These can therefore be calculated beforehand, and indeed, codes such as CHIANTI, SPEX, and APEC do just that. The abundance AZ (note, not the Einstein coefficient: apologies for the overlapping notation, can’t be helped for historical reasons) changes from source to source, and sometimes even within a source, but is the stablest of the factors after the emissivity. The ion balance i(Te,Z,I)=NZ,I/NZ is strongly variable, as is the so-called emission measure, EM = Ne2dV, which btw is also a function of Te. The atomic emissivity and the ion balance are sometimes combined together and the product is also confusingly referred to as the emissivity. Strictly speaking, the level population is dependent on the ion fractions and therefore the emissivity cannot be exactly separated from the ion balance. However, this dependence is weak in the density limits we are usually interested in (Ne~108-12 cm-3, as in the solar corona), and the two can be separated.
It is important to note that each of the terms listed above have associated model or measurement uncertainties. Often, the Einstein coefficients and the energy of the emission are not experimentally verified, and the level populations are approximate calculations due to the complexity of the level structure of the species in question. Typical ion balance calculations assume that the plasma is in thermodynamic equilibrium, which is often not a good assumption. Abundances are known to vary radically (by factors greater than 2x) across the source. And finally, except at high temperatures and low density (such as stellar coronae), the assumption of zero opacity (i.e., that any emitted photon escapes to infinity without any scatterings) is not applicable, and radiative transfer effects must be included.
A brief word about the units. Astronomers tend to use cgs, not SI. So the flux usually has units [ergs/s/cm2/sr], the emissivity eji is in [ph cm3/s] (unless the factor hc/lambda is included in the emissivity, in which case the units are [ergs cm3/s]), and the emission measure is in [cm3].
The emission measure is a story by itself, one best left alone for another time.