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The scaling laws that relate the peak temperature TM and volumetric heating rate EH to the pressure P and length L for static coronal loops were established over 40 yr ago; they have proved to be of immense value in a wide range of studies. Here we extend these scaling laws to dynamic loops, where enthalpy flux becomes important to the energy balance, and study impulsive heating/filling characterized by upward enthalpy flows. We show that for collision-dominated thermal conduction, the functional dependencies of the scaling laws are the same as for the static case, when the radiative losses scale as T−1/2, but with a different constant of proportionality that depends on the Mach number M of the flow. The dependence on the Mach number is such that the scaling laws for low to moderate Mach number flows are almost indistinguishable from the static case. When thermal conduction is limited by turbulent processes, however, the much weaker dependence of the scattering mean free path (and hence thermal conduction coefficient) on temperature leads to a limiting Mach number for return enthalpy fluxes driven by thermal conduction between the corona and chromosphere. [ABSTRACT FROM AUTHOR]

We extend a recently published analytic model for the intensity–time profile of solar energetic particle (SEP) events, in which the dominant physical mechanism is turbulent pitch-angle scattering of a collimated distribution of particles accelerated at the Sun. The present model includes the effect of magnetic focusing in the expanding magnetic field geometry of the inner heliosphere. For a power-law variation of the magnetic field strength with distance (B ∼ s−α) that lacks a characteristic focusing length scale, the fundamental shape of the intensity–time profile (i.e., a Lévy distribution at times up to and just past the time of peak intensity, followed by an exponential decay) is preserved. The effect of magnetic focusing is essentially to produce a rescaling of the stochastic process describing the angular diffusion of the particles, making the typical time that characterizes the SEP time profile quantitatively lower by a factor of (α + 1), 3 for the radial field geometry B ∼ s−2. [ABSTRACT FROM AUTHOR]

Recent observations from RHESSI have revealed that the number of non-thermal electrons in the coronal part of a flaring loop can exceed the number of electrons required to explain the hard X-ray-emitting footpoints of the same flaring loop. Such sources cannot, therefore, be interpreted on the basis of the standard collisional transport model, in which electrons stream along the loop while losing their energy through collisions with the ambient plasma; additional physical processes, to either trap or scatter the energetic electrons, are required. Motivated by this and other observations that suggest that high-energy electrons are confined to the coronal region of the source, we consider turbulent pitch-angle scattering of fast electrons off low-frequency magnetic fluctuations as a confinement mechanism, modeled as a spatial diffusion parallel to the mean magnetic field. In general, turbulent scattering leads to a reduction of the collisional stopping distance of non-thermal electrons along the loop, and hence to an enhancement of the coronal hard X-ray source relative to the footpoints. The variation of source size L with electron energy E becomes weaker than the quadratic behavior pertinent to collisional transport, with the slope of L(E) depending directly on the mean free path λ associated with the non-collisional scattering mechanism. Comparing the predictions of the model with observations, we find that λ ∼ (108-109) cm for ∼30 keV, less than the length of a typical flaring loop and smaller than, or comparable to, the size of the electron acceleration region. [ABSTRACT FROM AUTHOR]

Driven by recent RHESSI observations of confined loop-top hard X-ray sources in solar flares, we consider stochastic acceleration of electrons in the presence of Coulomb collisions. If electron escape from the acceleration region can be neglected, the electron distribution function is determined by a balance between diffusive acceleration and collisions. Such a scenario admits a stationary solution for the electron distribution function that takes the form of a kappa distribution. We show that the evolution toward this kappa distribution involves a “wave front” propagating forward in velocity space, so that electrons of higher energy are accelerated later; the acceleration timescales with energy according to τacc ∼ E3/2. At sufficiently high energies escape from the finite-length acceleration region will eventually dominate. For such energies, the electron velocity distribution function is obtained by solving a time-dependent Fokker-Planck equation in the “leaky-box” approximation. Solutions are obtained in the limit of a small escape rate from an acceleration region that can effectively be considered a thick target. [ABSTRACT FROM AUTHOR]

Electrons accelerated in solar flares are injected into the surrounding plasma, where they are subjected to the influence of collisional (Coulomb) energy losses. Their evolution is modeled by a partial differential equation describing continuity of electron number. In a recent paper, Dobranskis & Zharkova claim to have found an “updated exact analytical solution” to this continuity equation. Their solution contains an additional term that drives an exponential decrease in electron density with depth, leading them to assert that the well-known solution derived by Brown, Syrovatskii & Shmeleva, and many others is invalid. We show that the solution of Dobranskis & Zharkova results from a fundamental error in the application of the method of characteristics and is hence incorrect. Further, their comparison of the “new” analytical solution with numerical solutions of the Fokker-Planck equation fails to lend support to their result. We conclude that Dobranskis & Zharkova's solution of the universally accepted and well-established continuity equation is incorrect, and that their criticism of the correct solution is unfounded. We also demonstrate the formal equivalence of the approaches of Syrovatskii & Shmeleva and Brown, with particular reference to the evolution of the electron flux and number density (both differential in energy) in a collisional thick target. We strongly urge use of these long-established, correct solutions in future works. [ABSTRACT FROM AUTHOR]

Observations with RHESSI have enabled the detailed study of the structure of dense hard X-ray coronal sources in solar flares. The variation of source extent with electron energy has been discussed in the context of streaming of non-thermal particles in a one-dimensional cold target model and the results used to constrain both the physical extent of, and density within, the electron acceleration region. Here, we extend this investigation to a more physically realistic model of electron transport that takes into account the finite temperature of the ambient plasma, the initial pitch angle distribution of the accelerated electrons, and the effects of collisional pitch angle scattering. The finite temperature results in the thermal diffusion of electrons, which leads to the observationally inferred value of the acceleration region volume being an overestimate of its true value. The different directions of the electron trajectories, a consequence of both the non-zero injection pitch angle and scattering within the target, cause the projected propagation distance parallel to the guiding magnetic field to be reduced, so that a one-dimensional interpretation can overestimate the actual density by a factor of up to ∼6. The implications of these results for the determination of acceleration region properties (specific acceleration rate, filling factor, etc.) are discussed. [ABSTRACT FROM AUTHOR]

Many astrophysical observations are characterized by a single, non-repeatable measurement of a source brightness or intensity, from which we are to construct estimates for the true intensity and its uncertainty. For example, the hard X-ray count spectrum from transient events such as solar flares can be observed only once, and from this single spectrum one must determine the best estimate of the underlying source spectrum I(ε), and hence the form of the responsible electron spectrum F(E). Including statistical uncertainties on the measured count spectrum yields a “confidence strip” that delineates the boundaries of electron spectra that are consistent with the observed photon spectrum. In this short article, we point out that the expectation values of the source brightness and its variance in a given photon energy bin are in general not (as has been assumed in prior works) equal to n, the number of counts observed in that energy bin. Rather, they depend both on n and on prior knowledge of the overall photon spectrum. Using Bayesian statistics, we provide an explicit procedure and formulas for determining the “confidence strip” (Bayesian credible region) for F(E), thus providing rigorous bounds on the intensity and shape of the accelerated electron spectrum. [ABSTRACT FROM AUTHOR]

We establish a classification scheme for stochastic acceleration models involving low-frequency plasma turbulence in a strongly magnetized plasma. This classification takes into account both the properties of the accelerating electromagnetic field, and the nature of the transport of charged particles in the acceleration region. We group the acceleration processes as either resonant, non-resonant, or resonant-broadened, depending on whether the particle motion is free-streaming along the magnetic field, diffusive, or a combination of the two. Stochastic acceleration by moving magnetic mirrors and adiabatic compressions are addressed as illustrative examples. We obtain expressions for the momentum-dependent diffusion coefficient D(p), both for general forms of the accelerating force and for the situation when the electromagnetic force is wave-like, with a specified dispersion relation ω = ω(k). Finally, for models considered, we calculate the energy-dependent acceleration time, a quantity that can be directly compared with observations of the time profile of the radiation field produced by the accelerated particles, such as those occuring during solar flares. [ABSTRACT FROM AUTHOR]

We present electron images of an extended solar flare source, deduced from RHESSI hard X-ray imaging spectroscopy data. We apply the electron continuity equation to these maps in order to determine empirically the form of the energy loss rate for the bremsstrahlung-emitting electrons. We show that this form is consistent with an energy transport model involving Coulomb collisions in a target with a temperature of about 2 × 107 K, with a continuous injection of fresh deka-keV electrons at a rate of approximately 10–2 electrons s–1 per ambient electron. [ABSTRACT FROM AUTHOR]