Astronomical X-ray observatories with grazing incidence optics face the problem of pseudo-focusing of low energy protons from the mirrors towards the focal plane. Those protons constitute a variable, unpredictable component of the non X-ray background that strongly affects astronomical observations and a correct estimation of their flux at the focal plane is then essential. For this reason, we investigate how they are scattered from the mirror surfaces when impacting with grazing angles. We compare the non-elastic model of reflectivity of particles at grazing incidence proposed by Remizovich et al. (1980) with the few available experimental measurements of proton scattering from X-ray mirrors. We develop a semi-empirical analytical model based on the fit of those experimental data with the Remizovich solution. We conclude that the scattering probability weakly depends on the energy of the impinging protons and that the relative energy losses are necessary to correctly model the data. The model we propose assumes no dependence on the incident energy and can be implemented in particle transport simulation codes to generate, for instance, proton response matrices for specific X-ray missions. Further laboratory measurements at lower energies and on other mirror samples, such as ATHENA Silicon Pore Optics, will improve the resolution of the model and will allow us to build the proper proton response matrices for a wider sample of X-ray observatories.
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