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 A_ji(Z,I) (aka the Einstein coefficient), the number of particles of the species that exists in the upper level of the transition N_j(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, e_ji(N_e,T_e,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 A_Z (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(T_e,Z,I)=N_Z,I/N_Z is strongly variable, as is the so-called emission measure, EM = N_e²dV, which btw is also a function of T_e. 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 (N_e~10^8-12 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 e_ji 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.

Here is a specific example that came up due to a comment by a referee on a paper with David G.-A. We had said that the O abundance is dominated by uncertainties in the DEM at low temperatures because that is where most of the emission from O is formed. The referee disputed this, saying yeah, but O is also present at higher temperatures, and since the DEM is much higher there, that should be the predominant contribution to the estimate. In effect, the referee said, “show me!” The problem is, how? The measured fluxes are:

fO7_obs = 2 +- 0.75

fO8_obs = 4 +- 0.88

The predicted fluxes are:

fO7_pred = 1.8 +- 0.72

fO8_pred = 3.6 +- 0.96

where the error bars here come from the stddev of the fluxes predicted by each DEM realization that comes out of the MCMC analysis. On the face of it, it looks like a pretty good match to the observations, though a slightly different picture emerges if one were to look at the distribution of the predicted fluxes:

mode(fO7_pred)=0.76 (95% HPD interval = 0.025:2.44)

mode(fO8_pred)=2.15 (95% HPD interval = 0.95:4.59)

What if one computed the flux at each temperature and did the same calculation separately? That is shown in the following plot, where the product of the DEM and the line emissivity computed at each temperature bin is shown for both O VII (red) and O VIII (blue). The histograms are for the best-fit DEM solution, and the vertical bars are stddevs on the product, which differs from the flux only by a constant. The dashed lines show the 95% highest posterior density intervals.

Figure 1: Fluxes from O VII and O VIII computed at each temperature from DEM solution of RST 137B. The solid histograms are the fluxes for the best-fit DEM, and the vertical bars are the stddev for each temperature bin. The dotted lines denote the 95% highest-posterior density intervals for each temperature.

But even this tells an incomplete tale. The full measure of the uncertainty goes unseen until all the individual curves are seen, as in the animated gif below which shows the flux calculated for each MCMC draw of the DEM:

Figure 2: Predicted flux in O VII and O VIII lines as a product of line emissivity and MCMC samples of the DEM for various temperatures. The dashed histogram is from the best-fit DEM, the solid histograms are for the various samples (the running number at top right indicates the sample sequence; only the last 100 of the 2000 MCMC draws are shown).

So this brings me to my question. How does one represent this kind of uncertainty in a static plot? We know what the uncertainty is, we just don’t know how to publish them.