The Electron Injection Function and Energy‐dependent Delays in Thick‐Target Hard X‐Rays

Abstract

We analytically and numerically study the relationship between an energy-dependent electron injection spectrum, F0(E0, t), and the resulting bremsstrahlung photon spectrum, J(e, t), with the goal of exploring whether injection functions could explain energy-dependent time delays observed in solar flare hard X-rays (HXRs) without any time-of-flight effects. We calculate the inversion of the bremsstrahlung photon spectrum (for the Kramers cross section) and find that the timing of the electron injection function depends on the time derivative of the second spectral derivative of the photon spectrum. To match the observed delays, a systematic softening of the electron injection spectrum is required over the duration (≈ 1 s) of individual HXR pulses. This requirement is exactly the same as that which occurs in the time-of-flight model, except there the softening is due to spatial dispersion of injected electrons of different energy E0. We show that such a softening injection rate is not consistent with acceleration models where the electron acceleration times are comparable with the HXR pulse lengths, but it can be consistent with models where the acceleration times are very short since the injection spectrum variations are then governed by spectral variations in the acceleration rate. We conclude that acceleration mechanisms cannot be ruled out on the basis of HXR light curves alone as 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 attempts to infer primary accelerator properties from HXR light curves. We also conclude that the agreement of the time-of-flight model with observed energy-dependent HXR delays, and the properties of any acceleration model contributing to this trend, puts strong constraints on the timescales involved in the accelerator.