An empirical method to determine electron energy modification rates from spatially resolved hard X-ray data

We discuss a technique for determining the energy loss (or gain) rates affe cting high-energy electrons from spatially resolved observations of the har d 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 cha nges (due to energy losses) in the electron flux spectra at different posit ions 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 mo dels of electron energy loss, characterized parametrically through an expon ent alpha in the energy loss rate equation, including the case alpha = 1, w hich corresponds to the electrons losing energy solely through Coulomb coll isions in an ionized target. We then add Poisson noise in the hard X-ray co unt rate spectra, based on a nominal detector area and observation integrat ion 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. Fo r count rates associated with large flares, the procedure reproduces well t he collisional. energy loss profile for electron energies up to about 40 ke V, even when no statistical smoothing (regularization) methodology is appli ed. 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 ener gy loss are used to generate the simulated hard X-ray data, the procedure p roduces energy loss forms that are demonstrably and quantifiably different from the purely collisional case. This shows that even using a simple, unre gularized inversion procedure, spatially resolved hard X-ray spectra can in deed be used to compare models of energy transport in solar flares. We disc uss our results with reference to the forthcoming High Energy Solar Spectro scopic Imager mission, which will provide data of the necessary quality for the application of our technique.