The power law fits for the box plots are intriguing, but fitting the box centers feels artificial, and assigning a specific temperature to a line is difficult, as line emissivity functions are fairly wide. I recapitulate a plot of the ratios between line cooling power and total cooling below, to illustrate the difficulty of disentangling flux data and the temperature distribution hidden behind it.
Because radiative cooling gets less efficient at higher temperatures, the ratios can be substantial at temperatures much higher than their peaks. A log-log plot can maybe be misleading, but we can see this problem even on a linear plot! In fact, it looks worse.
Thus, the FWHM-based box plots are unfortunately very necessary.
However, it’s then difficult to pin down specific points for curve fitting. One method for doing such a fit, or checking results, is forward modeling. David has talked about this in the past–he considers it “more honest”, as we make fewer assumptions. The basic method is to generate spectra from an assumed distribution, rather than generating a distribution from spectra.
After a few emails with the coauthors, I think we’ve found a good method of doing this. We generate our data by stepping through shock data in time. Since we don’t really care about time (ergodicity in the wind), the scaling on the time becomes arbitrary.
- Set up a temperature grid spanning the temperature range we’re interested in. I go from K to K.
- Set up an array full of zeroes, with each array element referring to a different line. This represents our detector.
- Start at the highest temperature in the grid of step 1, and iterate down. For each temperature, we accumulate flux in each line.
The timestep is calculated from the total cooling function. Here, C is a proportionality constant we scale out.
- When you’re done, scale the fluxes by a constant factor to match the observed data. We’re mostly interested in the relative differences between lines right now.
With relatively few parameters, the final generated fluxes are fairly good. Below is a plot with an assumed exponent of 0.8.
This artificially generated “spectrum” of line fluxes matches observation pretty well, with just a power law exponent derived from our earlier box plot fits. David has suggested a different method of comparing the fits–not connecting the lines, but instead using vertical bars between model and observation. The second plot is a “residual plot”, but of the error as multiple of observed.
The green bars indicate the model is overestimating, while the red bars indicate an underestimation. The result seems quite reasonable, except for the discrepancy at the lowest wavelength lines. The low wavelength issue is probably from a high temperature cutoff.