Inversion of thick-target bremsstrahlung spectra from nonuniformly ionised plasmas

The effects of non-uniform plasma target ionisation on the spectrum of thick-target HXR bremsstrahlung from a non-thermal electron beam are analysed. In particular the effect of the target ionisation structure on beam collisional energy losses, and hence on inversion of an observed photon spectrum to yield the electron injection spectrum, is considered and results compared with those obtained under the usual assumption of a fully ionised target. The problem is formulated and solved in principle for a general target ionisation structure, then discussed in detail for the case of a step function distribution of ionisation with column depth as an approximation to the sharp coronal-chromospheric step structure in solar flare plasmas. It is found that such ionisation structure has very dramatic effects on derivation of the thick-target electron injection spectrum F₀(E₀) as compared with the result F*₀ (E₀) obtained under the usual assumption of a fully ionised target: (a) Inferred F*₀ contain more electrons than F₀ and in some cases include electrons at energies where none are actually present. Although the total (energy-integrated) beam fluxes in the two cases do not differ by a factor of more than Λ_ee/Λ_eH, the spectral shapes can differ greatly over finite energy intervals resulting in the danger of misleading results for total fluxes obtained by extrapolation, (b) The unconstrained mathematical solution for F₀ for any photon spectrum is never unique, while that for F*₀ is unique. When the physical constraint F₀ ≥ 0 is added, for some photon spectra solutions for F₀ may not exist or may not be unique. (This is not an effect of noise but of real analytic ambiguity.) (c) For data corresponding to F*₀ with a low-energy cut-off, or a cut-off or rapid enough exponential decline at high energies, a unique solution F₀ does exist and we obtain a recursive summation for its evaluation. Consequently, in future work on the inversion of HXR bremsstrahlung spectra it will be vital for algorithms to include the effects of target ionisation if spurious results on thick-target electron spectra are not to be inferred. Finally it is pointed out that the depth of the transition zone, and its evaporative evolution during flares may be derivable from its effect on the HXR spectrum.