Consequently the motion of electrons in

Consequently, the motion of electrons in the first stage is the free scattering of accelerated electrons in the coronal part of the magnetic field in the low-density plasma. The accelerated electrons may be subsequently captured into a magnetic trap and ‘leave’ to the loss cone according to the pitch-angle distribution and the conditions in the loop plasma, namely, to the distribution of plasma concentration and configuration of the magnetic field. Notice that in zvad fmk with earlier works on beam kinetics [5–9], the model we propose involves a combined initial condition for time that also depends on electron energy, which is associated with electron transit from the acceleration to the injection region. As it is rather difficult to find the spatial localization of the acceleration and injection regions of accelerated electrons from the observation data, we examined models with spatially separated ( ≠ 0) and combined ( = 0) acceleration and injection regions.
The results of numerically modeling the time-delay spectra The kinetics of accelerated electrons is described by a non-stationary relativistic Fokker–Planck equation [5]. In the model with separated acceleration and injection regions the time term of the electron distribution function takes the form S4(t(E)), the argument t has a shift depending on energy as Е–1/2. We examined three types of injection at the top of a closed loop:
Systematized curves of the calculated time-delay spectra are shown in Fig. 7, while the initial parameters of the models with the curve numbers are listed in the Table. Fig. 7, а demonstrates the delay spectrum integral over the loop that was calculated using the isotropic injection model. It follows from the plots that time delays are increasing with an increase in quantum energy both for the model with a remote electron accelerator ( = 1.5۰107 cm, curve 2), and the model with the accelerator combined with the injector ( = 0, curve 1); the delays monotonously increase in value up to 520 and 620 ms, respectively. For a model with a shorter injection duration (t0 = 0.5 s, curve 3) the delay spectrum increases to smaller values (50 ms). However, it is impossible to obtain the decreasing and the U-shaped spectra that we found from the observations earlier using these models. The behavior of the delays in the looptop follows the integral curves of the delay spectra (see Fig. 7b), while the delays are somewhat longer: the maximum values are about 1 s. Fig. 7с shows the delay curves from the footpoints of the magnetic loop. It is important to trace the changes in behavior of the spectrum for short injection pulses (t0 = 0.5 s); evidently, the increasing spectrum was replaced by a decreasing one for the model with the particle accelerator that is close to the looptop. This, however, did not affect the trend of the integral delay spectrum (see Fig. 7а, curve 3), which confirms that the preeminent contributions to the overall HXR flow is from the top and the intermediate parts of the magnetic loop. The calculations for anisotropic two-beam injection in case of a remote accelerator showed a U-shaped species of integral time-delay spectra (Fig. 7d). The difference between these spectra is, primarily, in the duration of the injection process. In case of a lengthier injection (t0 ≥ 1 s), the delay spectra become U-shaped, with the energy corresponding to zero-time delays in the 60–80 keV range. When the injection time is shorter, this energy shifts to be greater or equal to 140 keV. The delay spectrum is increasing for the model of combined acceleration and injection regions. Fig. 7е shows HXR delay spectra obtained for the looptop. The value of the delays is slightly higher: it varies from tens to 500 ms; the HXR delay spectra with longer injection times are, as for the isotropic case, increasing with an increase in energy and decreasing for a shorter injection (t0 = 0.5 s). The spectra in the footpoints are drastically different, i.e. they decrease with an increase in energy (Fig. 7f). The delay values vary from ones to 50 ms and are similar to the HXR delay values for the whole loop.