Compton backscattered and primary X-rays from solar flares: angle dependent Green's function correction for photospheric albedo

The observed hard X-ray (HXR) flux spectrum $I(\epsilon)$ from solar flares is a combination of primary bremsstrahlung photons $I_P(\epsilon)$ with a spectrally modified component from photospheric Compton backscatter of downward primary emission. The latter can be significant, distorting or hiding the true features of the primary spectrum which are key diagnostics for acceleration and propagation of high energy electrons and of their energy budget. For the first time in solar physics, we use a Green's function approach to the backscatter spectral deconvolution problem, constructing a Green's matrix including photoelectric absorption. This approach allows spectrum-independent extraction of the primary spectrum for several HXR flares observed by the {\it Ramaty High Energy Solar Spectroscopic Imager} (RHESSI). We show that the observed and primary spectra differ very substantially for flares with hard spectra close to the disk centre. We show in particular that the energy dependent photon spectral index $\gamma (\epsilon)=-d \log I/d \log \epsilon$ is very different for $I_P(\epsilon)$ and for $I(\epsilon)$ and that inferred mean source electron spectra ${\bar F}(E)$ differ greatly. Even for a forward fitting of a parametric ${\bar F}(E)$ to the data, a clear low-energy cutoff required to fit $I(\epsilon)$ essentially disappears when the fit is to $I_P(\epsilon)$ - i.e. when albedo correction is included. The self-consistent correction for backscattered photons is thus shown to be crucial in determining the energy spectra of flare accelerated electrons, and hence their total number and energy.