Precipitation in Trap Models for Solar Hard X-ray Bursts

Precipitation of electrons due to collisions necessarily occurs in trap models for hard X-ray bursts and the thick-target emission from the precipitating electrons produces an X-ray spectrum similar in intensity and in spectral shape to that from the trapped electrons. Such a trap-plus-precipitation model combines attractive features and removes some of the difficulties of thick-target and simple trap models. The evolution of the trap-plus-precipitation model is amenable to an analytic description, which is presented, including inversion, to find the injection electron spectrum from the X-ray spectrum. Streaming instabilities are unlikely to be important and resonant scattering is also probably not important for the electrons |(E<̃\,100\,keV)| which emit most of the X-rays but may well be important for the higher energy electrons which generate microwave bursts.