Power-law spectra of energetic electrons in solar flares from the maximum entropy and dimensional considerations

The maximum entropy formalism and dimensional analysis are used to derive a power-law spectrum of accelerated electrons in impulsive solar flares, where the particles can contain a significant fraction of the total flare energy. Entropy considerations are used to derive a power-law spectrum for a particle distribution characterised by its order of magnitude of energy. The derivation extends an earlier one-dimensional argument to the case of an isotropic three-dimensional particle distribution. Dimensional arguments employ the idea that the spectrum should reflect a balance between the processes of energy input into the corona and energy dissipation in solar flares. The governing parameters are suggested on theoretical grounds and shown to be consistent with solar flare observations. The flare electron flux, differential in the non-relativistic electron kinetic energy E, is predicted to scale as $E^{-3}$. This scaling is in agreement with RHESSI measurements of the hard X-ray flux that is generated by deka-keV electrons, accelerated in intense solar flares.