A non-uniqueness problem in solar hard X-ray spectroscopy

Description

We consider the hard x-ray emission process by interaction between the electrons and the ions in the solar atmosphere. We provide the integral equations describing this process as an inverse problem in the case of uniform ionization of the plasma and of a simple but rather realistic approximation of non-uniform conditions. The singular system of the integral operators is computed analytically in the continuous case for the uniform ionization model and numerically in the case of discrete data for both uniform and non-uniform ionization conditions. By analytical arguments and analysis of the singular spectrum we point out that non-uniform ionization results in an ambiguous interpretation of the solution of the integral equation, this solution not being unique. Finally, we briefly recall that this analysis facilitates methods for recovering unique and regularized solutions from high-resolution hard x-ray spectral data soon to be forthcoming from the HESSI space mission.

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