The electron injection function and energy-dependent delays in thick-target hard X-rays

We analytically and numerically study the relationship between an energy-de pendent electron injection spectrum, F-0(E-0, t), and the resulting bremsst rahlung photon spectrum, J(epsilon, t), With the goal of exploring whether injection functions could explain energy-dependent time delays observed in solar hare hard X-rays (HXRs) without any time-of-flight effects. We calcul ate the inversion of the bremsstrahlung photon spectrum (for the Kramers cr oss section) and find that the timing of the electron injection function de pends on the time derivative of the second spectral derivative of the photo n spectrum. To match the observed delays, a systematic softening of the ele ctron injection spectrum is required over the duration (approximate to 1 s) of individual HXR pulses. This requirement is exactly the same as that whi ch occurs in the time-of-flight model, except there the softening is due to spatial dispersion of injected electrons of different energy E-0. We show that such a softening injection rate is not consistent with acceleration mo dels where the electron acceleration times are comparable with the HXR puls e lengths, but it can be consistent with models where the acceleration time s are very short since the injection spectrum variations are then governed by spectral variations in the acceleration rate. We conclude that accelerat ion mechanisms cannot be ruled out on the basis of HXR light curves alone a s an alternative to time-of-flight effects. Observations of HXR images and of the relationship of HXRs to soft X-ray loops strongly suggest, however, that time-of-flight effects must be important and must be included in attem pts to infer primary accelerator properties from HXR light curves. We also conclude that the agreement of the time-of-flight model with observed energ y-dependent HXR delays, and the properties of any acceleration model contri buting to this trend, puts strong constraints on the timescales involved in the accelerator.