An Analytic Description of Coronal Proton Trapping

Abstract

Analytic steady state solutions of the focused diffusion equation are used to deduce proton trapping time, τ_trap, an average residence time for all particles injected into the magnetic loop. We take into account the effect of magnetohydrodynamic (MHD) turbulence and the divergence of magnetic field lines in both the corona and chromosphere of the Sun. Numerical simulations of pitch-angle scattering have then been used to check the analytic solutions and to ascertain boundary conditions for the focused diffusion equation. Results are obtained for several different functional forms of B(ζ), the magnetic field as a function of distance along a particular coronal field line. Five cases have been studied, from B constant along the coronal portion of the loop to B(ζ) corresponding to a force-free magnetic structure. The results indicate that a divergence of the magnetic field in the coronal portion of the loop can significantly increase the trapping time, no matter how small the mean free path may be. Derived analytic expressions for τ_trap can be used to calculate the intensity of secondary emissions from the loop-top sources. Analytic time-dependent solutions of the focused diffusion equation are considered in the case of constant coronal B to find the basic decay time of the trapped proton number, τ_decay, an asymptotic value of the exponential decay time when the time tends to infinity. In the case of variable B in the coronal portion of the loop, Monte Carlo simulations of pitch-angle scattering have been employed to calculate τ_decay in a wide range of parameters. However, we have also obtained analytic expressions for how the characteristic time scales with parameters of the magnetic loop. Deduced analytic expressions for τ_decay can be used to calculate ion acceleration in the escape-time approximation and to interpret the decay phase of solar gamma-ray flares. Magnetic focusing in the coronal portion of the loop makes acceleration more efficient than would be expected in the approximation without coronal focusing.