Introduction

Solar flares - RHESSI's main business - turn out to be powerful sources of high-energy radiation, specifically X-rays and γ-rays. The hard X-ray (>20 keV) and direct γ-ray (MeV) radiations directly reveal the acceleration and transport processes of the energetic particles - electrons and ions, respectively. Hard X-ray emissions are produced by electron-ion bremsstrahlung of supra-thermal electrons and direct γ-ray lines are produced by the nuclear de-excitation of the "target" ions after their collisions with accelerated ions, at typical energies many MeV per nucleon.

Analyses of particular hard X-ray and γ-ray bursts have shown that the time of the peak intensity depends on the photon energy. Such observations, initially explained by a "two-step" acceleration process, can be reproduced more simply by taking into account transport effects during the propagation of the particles trapped inside closed magnetic structures.

In this nugget, we show how to produce the delay observed by RHESSI between the X-ray and γ-ray fluxes during its first γ-ray flare (the event of 23 July 2002) by a "trap-plus-precipitation" model.

Model

The principle of the model is illustrated by the following cartoon.

Particles injected into the corona propagate at a velocity v in a closed magnetic loop of length L and produce hard X-ray and γ-ray radiation calculated with a "thin target" (instantaneous emission) approximation. The magnetic mirror force, acting in the convergent chromospheric magnetic field, defines the size α of the loss cone and hence determines the trapping. Particles whose pitch angles scatter into the loss cone precipitate into the denser chromosphere. There they produce hard X-ray and γ-ray radiations as they stop collisionally and come to rest. The free parameters of the model are the precipitation rate α²v/L of the particles into the chromosphere, the density of the ambient medium, the spectral index and the time evolution of the injected particles.

The 23 July 2002 event

Figure 1 shows the hard X-ray time profile at 150 keV and the γ-ray time profile in the nuclear lines during the 23 July event 2002, following the analysis of Share et al (2003). This event shows a nice delay between the hard X-rays (which precede) and the γ-rays, as shown by the cross-correlation analysis in Figure 2. The delay is about 12 s, much longer than the time for these nearly relativistic particles to transit the flare dimension. We want to know if we can reproduce this delay with the trap-plus-precipitation model described above.

Figure 1: Time evolution of the hard X-ray bremsstrahlung at 150 keV and of the nuclear de-excitation lines (mainly 1-10 MeV) during the 2002 July 23 flare, observed by RHESSI at 20 s time resolution.

Figure 2: Cross correlation calculated with a time resolution of 20s between the 150-keV hard X-ray continuum and the γ-ray lines. These cross correlation factors are interpolated with a quadratic function (dashed line).

To answer to this question, we start with the assumption that the time profile of the injected electrons and ions follows the time profile of the hard X-ray flux at 150 keV. We reproduce the first part of the hard X-ray flux (between 0s and 400s) with a parabolic function of 400s duration. The other parameter that we can fix is the spectral index of the accelerated protons and alpha particles, determined to be about 3.75 from the published analysis of the RHESSI data.

We use these two parameters to compute the X-ray/γ-ray time delay (δt) as a function of the density and the precipitation rate. The strong de-excitation lines at 6.129 MeV from ¹⁶O, 4.438 MeV from the first excited state of ¹²C populated directly and by the spallation reactions on ¹⁶O, 1.779 MeV from ²⁸Si, 1.634MeV from ²⁰Ne, 1.369 MeV from ²⁴Mg are computed. The results, presented in Figure 3, shows that the observed time delay of 12s can be reproduced for a density between
2 x 10¹¹ cm^-3 and 10¹⁰ cm^-3 and a ratio α²/L between 10^-7 km^-1 and
5 x 10^-5 km^-1. This result agrees with the density 6 x 10¹⁰ cm^-3 deduced from soft X-ray observations.

Figure 3: Computed peak time delay δt between hard X-ray emission at 150 keV and γ-ray line emission expected from trap-plus-precipitation model. δt is function of the density and of the ratio α²/L. The black line represents the average observed delay of 12s determined from the cross correlation factor presented in the figure 2. The time delay of 12s can be reproduced for a range of density values and precipitation rates values.

Conclusions

Our simple model assumes that both kinds of particles (electrons and ions) get accelerated simultaneously, but that the ions remain for a certain time in coronal magnetic loops due to magnetic mirroring. With reasonable parameters for this process we can reproduce the observed delay; this lends credence to the idea that the energetic ions responsible for the γ-ray emission might appear in the actual flare loops as observed in soft X-rays. We cannot yet infer this directly from γ-ray images very well (this should be a future nugget) because of limited angular resolution.