High-energy particles carry a significant fraction of energy released in solar flares (e.g. ). Most of the energetic electrons and ions accelerated in solar flares precipitate in the corona, producing bright microwave, HXR and gamma-ray emissions, heating the corona and chromosphere and, in some cases, causing sunquakes. However, some of these particles propagate upward and escape into the heliosphere, becoming observable first via metric and deca-metric radio emission in the upper corona, then directly in the interplanetary space and near the Earth. These particles contribute to space weather and can be used as a diagnostic tool for the processes in flaring corona. Hence, we need to understand how energetic particles accelerated in solar flares escape into the heliosphere.
Comparison of the characteristics of nonthermal electrons observed in situ near the Earth in impulsive solar energetic particle events (when particles are accelerated in solar flares) with the characteristics of electrons obtained from HXR observations shows that the energy spectra of these two populations are different . It is not clear what caused this difference: acceleration in different places or different mechanisms. In addition, the spectra may change during particle transport, for instance, due to scattering, strongly non-uniform magnetic field, or the return current effect. To establish the effect(s) primarily responsible for the difference between particles in the corona and the heliosphere, we use computational models of individual solar flares combining magnetohydrodynamic and test-particle methods.
Solar flare models
Gordovskyy et al.  constructed an observationally-based model of a real solar flare, combining the magnetohydrodynamic (MHD) and test particle (TP) methods to simultaneously describe the evolution of the magnetic field, thermal plasma, and energetic particles. The synthetic observables (mainly HXR radiation) obtained from the model were in good agreement with the observations of the flare, demonstrating that MHD-TP is a viable approach for modeling individual flares. In the present study ( see Gordovsky et al.  for details), we used the MHD-TP approach to simulate two separate solar flares and investigate the behaviour of energetic particles in them to understand how particles accelerated in flares are transported into the heliosphere.
The MHD-TP simulation of an individual flare begins with a nonlinear force-free (NLFF) reconstruction of the magnetic field in the active region where the flare occurred. This is done using the approach developed by Inoue et al.  with HMI/SDO vector magnetograms observed approximately 1 hour before the event. The resulting NLFF magnetic field configuration is embedded in a plane-parallel gravity-stratified corona, thus, forming the initial conditions for MHD simulation.
Evolution of magnetic and parallel electric fields in the model of an X-flare that occurred on September 6, 2011 over the AR11283 active region is shown in Fig.1. The coronal magnetic field in this active region is determined by two close sunspots with magnetic field strengths of about 2.0 kG of opposite polarities, and mostly positive diffuse photospheric network flux. The total negative flux through the photosphere in this region is less than the total positive flux, and part of the photospheric positive flux opens towards the heliosphere. The energy release and particle acceleration region (i.e. the volume containing strong parallel electric field) is located approximately 10 Mm above the negative sunspot in closed magnetic field. As the reconnection progresses, the magnetic connectivity in the region changes. Importantly, some closed magnetic flux (solid lines in Fig. 1) becomes open (dashed lines in Fig. 1).
The resulting MHD models are used to trace a large number (~10^6) of test electrons and protons, which are initially statistically-representative of the thermal plasma in the MHD model. We are interested in particles accelerated to energies above 8 keV.
A comparison of populations of particles precipitating in the corona (i.e. lost through the lower boundary of the domain) and particles ejected towards the heliosphere (i.e. lost through the upper boundary of the domain) reveals two interesting features. First, only a small fraction of energetic electrons escape to the heliosphere (no more than 20% at any given time). This is due to the fact that practically all particles in the considered models are accelerated in closed magnetic fields. Particles ejected towards the heliosphere are accelerated in a closed field, but manage to drift across the magnetic field from closed to open magnetic flux.
The fraction of escaping electrons changes with time depending on the evolution of the magnetic field in the flaring corona. For instance, in the event shown in Fig.1 closed magnetic flux becomes open as the reconnection proceeds, and the fraction of escaping particles increases with time (Fig. 2).
Secondly, the energy spectra of escaping and precipitating particles are different: the energy spectra of escaping protons and electrons are softer in both events under consideration (Fig. 3). Thus, for the energy spectra approximated by the power functions E^-delta, the power-law indices delta are about 2 for precipitating electrons and about 2.5 for electrons escaping through the upper boundary. This is because particles with lower parallel (with respect to the magnetic field) velocities spend more time around the acceleration region, where they can drift from the closed magnetic field to the open one, and therefore are more likely to enter the open magnetic field and, hence, go to the upper boundary of the domain. This, in principle, can explain the difference in the energy spectra of precipitating electrons. If the HXR in this event was produced in a “thin target”, then its power-law index gamma would be ~3, i.e. ~0.5 units lower than the power-law index of the energy spectrum of escaping electrons, which is in a good agreement with observations .
Our models demonstrate how energetic particles accelerated predominantly in closed magnetic field in solar flares are transported into open magnetic field and escape into the heliosphere. It is shown that energetic particles precipitating in the corona and escaping into the heliosphere have different energy spectra, similar to those observed. This difference is mainly due to the structure of the magnetic field in the energy release region and immediately around it. In other words, in the considered events under consideration, the structure of the magnetic field in the flaring corona, on its own, is sufficient to explain the difference between the properties of precipitating and escaping populations of energetic particles.
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