Time-dependent Turbulent Electron Acceleration and Transport in Solar Flares

Abstract

Solar flares are explosive releases of magnetic energy stored in the solar corona, driven by magnetic reconnection. These events accelerate electrons, generating hard X-ray emissions and often display Quasi Periodic Pulsations (QPPs) across the energy spectra. However, the energy transfer process remains poorly constrained, with competing theories proposing different acceleration mechanisms. We investigate electron acceleration and transport in a flaring coronal loop by solving a time-dependent Fokker-Planck equation. Our model incorporates transient turbulent acceleration, simulating the effects of impulsive energy input to emulate the dynamics of time-dependent reconnection processes. We compute the density-weighted electron flux, a diagnostic directly comparable to observed X-ray emissions, across the energy and spatial domains from the corona to the chromosphere. We investigate different time-dependent functional forms of the turbulent acceleration, finding that the functional form of the acceleration source maintains its signature across energy bands (1 to 100 keV) with a response time that is energy dependent (with higher energy bands displaying a longer response time). In addition, we find that (a) for a square pulse the switch on and off response time is different; (b) for a sinusoidal input the periodicity is preserved; and (c) for a damped sinusoidal the decay rate increases with density and higher energy bands lose energy faster. This work presents a novel methodology for analyzing electron acceleration and transport in flares driven by time-dependent sources.

Figures (15)

• Figure 1: The evolution of the turbulent acceleration time amplitude, $g(t)$, described by Eqs. (\ref{['eq:g2']})-(\ref{['eq:g4']}).
• Figure 2: The two atmospheric models used in the simulations, showing the plasma number density (blue) and temperature distribution (orange). In the coronal region, the continuous blue line represents atmospheric model 1 and dashed blue atmospheric model 2. Outside the corona models 1 and 2 are identical.
• Figure 3: $nVF$ spectra as function of (a) the electron energy, the shaded areas represent the estimated uncertainty multiplied by 100 for better visualization, (b) loop position, and (c) pitch angle cosine. In panel (b) the black-dashed line represents the chromosphere boundary and the cyan-dashed line is the acceleration region, $\sigma=3^{\prime\prime}$.
• Figure 4: $nVF$ energy spectra for different regions of the solar flare for source $g_2$ (described in Eq. (\ref{['eq:g2']})). Left column panels (a), (c) and (e), contain results for $n=3\times 10^{10}$ cm$^{-3}$, while right column panels (b), (d) and (f) for $n=5\times 10^{9}$ cm$^{-3}$. The shaded regions represent the estimated uncertainty of each bin, multiplied by 100 for better visualization. $\tau_{\rm{acc}}$ color code in panel (a) stays the same across every panel. $\delta_{nVF}$ represents the spectral indices obtained applying a fitting on $nVF$ between 15-25 keV and 40-90 keV.
• Figure 5: $nVF$ at different energy bands for a square pulse (source $g_2$, Eq. (\ref{['eq:g2']})), where turbulent acceleration turns on at $t=$ 10 s and turns off at $t=$ 30 s. Continuous lines show results for $\tau_{\rm{acc}}=$ 5 s, dashed lines show results for $\tau_{\rm{acc}}=$ 25 s. Left column, panels (a), (c) and (e), contain results for $n=3\times 10^{10}$ cm$^{-3}$, while right column, panels (b), (d) and (f) contain results for $n=5\times 10^{9}$ cm$^{-3}$.