Moments of energetic particle distributions in the solar atmosphere

This paper describes a theory that provides a direct link between observabl e properties of energetic particle distributions and the unobservable prope rties of the initial distributions. A general approach is first outlined an d then applied to the specific situation of collisional electron transport in the solar atmosphere. By building on previous results, this treatment do es not require explicit solution of the Fokker-Planck equation. In particul ar, we derive results for two important consequences of an energetic electr on distribution: 1) The emission of hard X-ray bremsstrahlung radiation and 2) energy deposition in the background plasma by Coulomb collisions. For 1 ) we describe the energy spectrum of the emission completely and provide a description of its spatial and pitch angle properties in terms of the first and second order moments of the injected electron distribution. This means that for any assumed initial electron distribution, expressions are obtain ed that predict the HXR source's spatial position and extent for any photon energy. The results have most immediate application to phenomena associate d with the early impulsive stage of solar flares involving energetic electr ons. Other potential uses include: calculation of Ha impact polarisation; a pplication to polarisational and directional properties of hard X-rays; and diagnostics of unobservably high energy features in the electron spectrum.