An Empirical Method to Determine Electron Energy Modification Rates from Spatially Resolved Hard X‐Ray Data

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Abstract
We discuss a technique for determining the energy loss (or gain) rates affecting high-energy electrons from spatially resolved observations of the hard X-ray bremsstrahlung signature that they produce. The procedure involves two main steps—determining the local electron flux spectrum from inversion of the hard X-ray spectrum using a matrix technique, and evaluating the changes (due to energy losses) in the electron flux spectra at different positions in the source via the continuity equation for total electron flux. In order to test the viability of this numerical technique, we generate a set of simulated hard X-ray photon count spectra, corresponding to different models of electron energy loss, characterized parametrically through an exponent α in the energy loss rate equation, including the case α = 1, which corresponds to the electrons losing energy solely through Coulomb collisions in an ionized target. We then add Poisson noise in the hard X-ray count rate spectra, based on a nominal detector area and observation integration interval, and use the above procedure on this simulated noisy data set to determine the energy-loss rate as a function of energy in each model. For count rates associated with large flares, the procedure reproduces well the collisional energy loss profile for electron energies up to about 40 keV, even when no statistical smoothing (regularization) methodology is applied. Above this energy, the method breaks down due to the data noise present, but the method could be extended to higher energies by use of a suitable regularized inversion technique. When other (noncollisional) models of energy loss are used to generate the simulated hard X-ray data, the procedure produces energy loss forms that are demonstrably and quantifiably different from the purely collisional case. This shows that even using a simple, unregularized inversion procedure, spatially resolved hard X-ray spectra can indeed be used to compare models of energy transport in solar flares. We discuss our results with reference to the forthcoming High Energy Solar Spectroscopic Imager mission, which will provide data of the necessary quality for the application of our technique.