Energy and pitch angle distributions for auroral ions using the current sheet acceleration model

Using a dipole plus tail magnetic field model, H⁺, He⁺⁺ and O ₁₆ ⁺⁶ ions are followed numerically, backward in time, from an output plane perpendicular to the axis of the geomagnetic tail, to their point of entrace to the magnetosphere as solar wind particles in the magnetosheath. An adiabatic or guiding center approximation is used in regions where the particles do not interact directly with the current sheet. A Maxwellian distribution with bulk flow is assumed for solar wind particles in the magnetosheath. Bulk velocity, density, and temperature along the magnetopause are taken from the fluid calculations of Spreiter. Using Liouville's theorem, and varying initial conditions at the output plane, the distribution function is found as a function of energy and pitch angle at the output plane. These results are then mapped to the auroral ionosphere using guiding center theory. Results show that the total precipitation rate is sufficient only for particles which enter the magnetosphere near the edges of the current sheet. Small pitch angles are favored at the output plane, but mappings to the auroral ionosphere indicate isotropic pitch angle distributions are favored with some peaking of the fluxes parallel or at other angles to the field lines. Perpendicular auroral pitch angle anisotropies are at times produced by the current sheet acceleration mechanism. Therefore, caution must be used in interpreting all such observations as ‘loss cone-trapping’ distributions. Energy spectra appear to be quite narrow for small cross-tail electric fields, and a little broader as the electric field increases. Comparisons of these results with experimental observations are presented.     

Rocket observations of electron spectral and angular characteristics in an “inverted V” event

Energy spectra and pitch angle distributions of auroral electrons in the energy range 2.5–11 keV observed on a rocket flight launched from Andøya on 13 November 1970 are presented. Strong rapidly fluctuating fluxes during the first part of the flight were succeeded by fluxes below or close to the level of detectability. Before the rocket passed through the northern precipitation boundary two spectral events of “inverted V” character occurred. Both events were associated with field aligned pitch angle distributions. While anisotropies with the flux peaked near 0° were in general associated with the spectral peak energy, isotropy over the upper hemisphere was the dominant distribution for other energies. The observations made during these events provide strong support for the theory of a parallel potential drop close to the ionosphere as an important accelerating mechanism for auroral electrons in connection with “inverted V” events.     

Particle Interactions with Single or Multiple 3D Solar Reconnecting Current Sheets

The acceleration of charged particles (electrons and protons) in flaring solar active regions is analyzed using numerical experiments. The acceleration is modeled as a stochastic process taking place by the interaction of the particles with local magnetic reconnection sites via multiple steps. Two types of local reconnecting topologies are studied: the Harris-type and the X-point. A formula for the maximum kinetic energy gain in a Harris-type current sheet, found in a previous work of ours, fits well the numerical data for a single step of the process. A generalization is then given approximating the kinetic energy gain through an X-point. In the case of the multiple step process, in both topologies the kinetic energy distribution of the particles is found to acquire a practically invariant form after a small number of steps. This tendency is interpreted theoretically. Other characteristics of the acceleration process are given, such as the mean acceleration time and the pitch angle distributions of the particles.     

A note on adiabatic solutions of the one-dimensional current sheet problem

It has recently been shown that adiabatic solutions of the one-dimensional current sheet problem exist provided that magnetically trapped particles are included in the model together with the current-carrying untrapped “beam” particles. We show here that a formulation of the problem in terms of particle velocity and pitch angle is advantageous, and we derive some general properties of the solutions. In particular it is shown that there is, in general, no discontinuity in the value of the particle distribution function ? across the boundary in velocity space between “beam” and trapped particles, but that there will be a discontinuity in the gradients of ?. An example is given in which the beam population is of bi-Maxwellian form at the outer boundary of the current sheet.     

Reflection of pre-accelerated pick-up ions at the solar wind termination shock: The seed for anomalous cosmic rays

It has been hypothesized for quite some time that interplanetary pick-up ions due to energization taking place in the region close to the solar wind termination shock, at some fraction and as an outcome of a complicated chain of processes, eventually are converted into species of the anomalous cosmic-ray particles. For the actual conversion efficiency it is of great importance to know the energy distribution of these pick-up ions upon their arrival at the shock. It turns out that pre-acceleration of these ions during their passage through the heliosphere shall substantially increase their chances to become reflected at the shock into the upstream direction which is a prerequisite for a further climb-up in energy by virtue of Fermi-1 acceleration processes. In this paper we start out from stochastically pre-accelerated pick-up ions and investigate their behaviour at the shock. With the use of adiabatic approaches in the de Hoffman-Teller frame of the shock, we calculate the energy distribution function of the reflected part of pick-up ions. From the calculated distribution functions it turns out that the reflected ions in the average suffer an energy increase by about a factor of 10, still not enough to let them move off the shock by spatial diffusion in the upstream direction. Thus, since converted back into the shock, they can undergo repeated reflections and energy gains till the diffusion-convection limit is reached. As we show in addition, the reflection probability for pick-up ions is about a factor of 10 higher than expected from the present literature and strongly varies with the off-axis angle, pointing to the fact that the termination shock represents a surface with a three-dimensionally varying source strength for the production of anomalous cosmic rays. The ACR source pattern is also expected to vary during the solar cycle and the relevant injection energies are expected to be larger by factors of 10 to 100 than the canonically adopted 1 keV nucl^–1.Institute for Problems of Mechanics of the Russian Academy of Sciences, Prospect Vernadskogo 101, 117526, Moscow, Russia.     

Bidirectional distribution of energetic protons in the 18 February 1979 interplanetary shock event

Pitch angle distributions of 35–1000 keV protons in the 18 January 1979 interplanetary shock event are presented and discussed in this paper. The observed, apparently unusual distributions can be well explained by assuming a magnetic bottle configuration of the interplanetary magnetic field immediately prior to the shock passage. The implication of such a magnetic configuration on the acceleration of charged particles by shock waves is also discussed.     

Element Abundances in Solar Energetic Particles and the Solar Corona

This is a study of abundances of the elements He, C, N, O, Ne, Mg, Si, S, Ar, Ca, and Fe in solar energetic particles (SEPs) in the 2?–?15 MeV?amu^?1 region measured on the Wind spacecraft during 54 large SEP events occurring between November 1994 and June 2012. The origin of most of the temporal and spatial variations in abundances of the heavier elements lies in rigidity-dependent scattering during transport of the particles away from the site of acceleration at shock waves driven out from the Sun by coronal mass ejections (CMEs). Variation in the abundance of Fe is correlated with the Fe spectral index, as expected from scattering theory but not previously noted. Clustering of Fe abundances during the “reservoir” period, late in SEP events, is also newly reported. Transport-induced enhancements in one region are balanced by depletions in another, thus, averaging over these variations produces SEP abundances that are energy independent, confirms previous SEP abundances in this energy region, and provides a credible measure of element abundances in the solar corona. These SEP-determined coronal abundances differ from those in the solar photosphere by a well-known function that depends upon the first ionization potential (FIP) or ionization time of the element.     

Particle acceleration by ultrarelativistic shocks: theory and simulations

We consider the acceleration of charged particles near ultrarelativistic shocks, with Lorentz factor     . We present simulations of the acceleration process and compare these with results from semi-analytical calculations. We show that the spectrum that results from acceleration near ultrarelativistic shocks is a power law,     , with a nearly universal value     for the slope of this power law.
We confirm that the ultrarelativistic equivalent of the Fermi acceleration at a shock differs from its non-relativistic counterpart by the occurrence of large anisotropies in the distribution of the accelerated particles near the shock. In the rest frame of the upstream fluid, particles can only outrun the shock when their direction of motion lies within a small loss cone of opening angle     around the shock normal.
We also show that all physically plausible deflection or scattering mechanisms can change the upstream flight direction of relativistic particles originating from downstream by only a small amount:     . This limits the energy change per shock crossing cycle to     , except for the first cycle where particles originate upstream. In that case the upstream energy is boosted by a factor     for those particles that are scattered back across the shock into the upstream region.     

A nonlinear theory of stellar wind with allowance for the influence of relativistic particles

We solve the nonlinear problem of the dynamics of a steady-state, spherically symmetric stellar wind by taking into account particle acceleration to relativistic energies near the shock front. The particles are assumed to be accelerated through the Fermi mechanism, interacting with stellar-wind turbulence and crossing many times the shock front that separates the supersonic and subsonic stellar-wind regions. We take into account the influence of the accelerated particles on hydrodynamic plasma-flow parameters. Our method allows all hydrodynamic parameters of the shock front and plasma in the supersonic region to be determined in a self-consistent way and the accelerated-particle energy spectrum to be calculated. Our numerical and analytic calculations show that the plasma compression ratio at the shock front increases compared to the case where there are no relativistic particles and that the velocity profile in the supersonic region acquires a characteristic kink. The shape of the energy spectrum for the accelerated particles and their pressure near the front are essentially determined by the presumed dependence of the diffusion coefficient on particle energy, which, in turn, depends on the scale distribution of turbulent pulsations and other stellar-wind inhomogeneities.     

Spatial Distribution of Solar Energetic Particles in the Inner Heliosphere

We study the spatial distribution of solar energetic particles (SEPs) throughout the inner heliosphere during six large SEP events from the period 1977 through 1979, as deduced from observations on the Helios 1 and 2, IMP 7 and 8, ISEE 3, and Voyager 1 and 2 spacecraft. Evidence of intensity maxima associated with the expanding shock wave is commonly seen along its central and western flanks, although the region of peak acceleration or “nose” of the shock is sometimes highly localized in longitude. In one event (1 January 1978) a sharp peak in 20?–?30 MeV proton intensities is seen more strongly by Voyager at ～?2 AU than it is by spacecraft at nearby longitudes at ～?1 AU. Large spatial regions, or “reservoirs,” often exist behind the shocks with spatially uniform SEP intensities and invariant spectra that decrease adiabatically with time as their containment volume expands. Reservoirs are seen to sweep past 0.3 AU and can extend out many AU. Boundaries of the reservoirs can vary with time and with particle velocity, rather than rigidity. In one case, a second shock wave from the Sun reaccelerates protons that retain the same hard spectrum as protons in the reservoir from the preceding SEP event. Thus reservoirs can provide not only seed particles but also a “seed spectrum” with a spectral shape that is unchanged by a weaker second shock.     

Acceleration of relativistic charged particles by supersonic hydromagnetic turbulence

The acceleration of relativistic particles is considered during their intersection with hydromagnetic shock fronts in the presence of randomly distributed large-scale magnetic fields. In a series of astronomical objects, the Larmor radius of the relativistic particles exceeds the width of the shock front. In this case there is a change in the adiabatic invariant which results in an increase in the energy of the particle when it crosses the front in any direction. We have proved that the adiabatic part of the energy change will be partially or completely compensated by its reverse change in the weaker regions of the magnetic field. The acceleration mechanism considered is found to be more effective than the Fermi mechanism.If the mean free path of the particles is much less than the distance between the shock fronts, magnetic small-scale fluctuations cause further scattering of the particles. In this case the particles following and crossing the front will return to it. After reversed crossing, a fraction of the particles-defined by the ratio of the front speed to the particle velocity or of the distance between the fronts to the free path — will not return to the front. It is proved that for both large and small free paths the rates at which the particle gains energy are nearly the same.     

Pitch angle and energy distributions of auroral electrons measured by the ESRO 4 satellite

Energy spectra and angular distributions of auroral electrons in the energy range 0.2–16 keV measured by the low-altitude polar orbiting satellite ESRO 4 are presented. The observations were made in the altitude range 800–1000 km near magnetic midnight. Energy-time spectrograms show inverted-V structures with peaked energy spectra. The inverted-V events are associated with anisotropic electron pitch angle distributions peaked at 0 deg. Frequently these distributions have a maximum also at 90 deg. Measurements of >43 keV electrons indicate that the acceleration probably occurs on closed field lines. It is found that many properties of the observed particle distributions can be explained by acceleration in an electric field parallel to the magnetic field lines, if trapping of particles under an increasing potential drop is included in the model.     

A Numerical Study on the Ion Acceleration in Parallel Shock Wave

Accompanied by flares and coronal mass ejections, the symbiotic coronal and interplanetary fast shock waves have become a hot topic in particle acceleration. Under the condition of quasi-parallel propagation, we construct a numerical method of solving one-dimensional transport equation, and thereby exploring the physical relationships between the distribution of the accelerated ions and the parameters of the shock waves and background plasmas. Considering the diffusive coefficient as a constant and a function of energy, respectively, the results of calculations of finite free escape boundaries show that (1) the energetic particles approximately exhibit a double power-law distribution and the spectral index decreases gradually from 10.2 to 2.4 in the low energy region of 3-10MeV with the increase of the acceleration time; (2) with the increase of the compression ratio of the shock from 2 to 4, the spectral index decreases from 3.2 to 2.2 at a given time and in the same energy range; (3) when the finite upstream and downstream escape boundaries decrease from 5 to 2, the spectral index of the particle energy spectrum increases from 2.4 to 3.3, and the acceleration efficiency of particles decreases; (4) the spectral index decreases from 2.4 to 0.9 with the increase of the initial inject energy of the particles; (5) when the diffusive coefficient is directly proportional to the energy of the particles, the spectral index increases from 2.2 to 4.3 as compared with the case of constant diffusive coefficient.     

Observation of a flux transfer event on the earthward side of the magnetopause

The medium energy particle spectrometer (electrons of energy > 20 keV, protons > 25 keV) on board ISEE-2 has measured very similar pitch angle distributions and intensities during “flux transfer” events in the magnetosheath and events previously designated as “inclusion” events in the magnetosphere on a single pass through the magnetopause. This is interpreted as strong evidence that magnetic field lines in the magnetosphere can connect to field lines in the magnetosheath, at least locally and for brief times, allowing the same population ofparticles to be observed on both sides of the boundary. In addition, a simple mathematical model is provided incorporating a time constant for the process re-supplying particles to the open flux tube. The observed data are satisfactorily reproduced using a time constant of 46 s, which is comparable to the half-bounce time of protons at this position.     

Resonant interaction of hydromagnetic waves with charged particles

The effect of the resonant interaction of a distribution of hydromagnetic waves on a distribution of particles is described by a diffusion equation in momentum space. The diffusion coefficients and other coefficients describing systematic acceleration and diffusion in energy space are derived in general and for a number of particular cases. It is shown that the resonant acceleration of slow ions by hydromagnetic waves is ineffective. The time evolution of the energy spectrum for ultrarelativistic particles due to interaction with hydromagnetic waves is found and applied to the case of the Crab Nebula in an accompanying article.     

Field line topology in the dayside cusp region inferred from low altitude particle observations

Dayside low altitude satellite observations of the pitch angle and energy distribution of electrons and protons in the energy range 1 eV to 100 eV during quite geomagnetic conditions reveal that at times there is a clear latitudinal separation between the precipitating low energy (keV) electrons and protons, with the protons precipitating poleward of the electrons. The high energy (100 keV) proton precipitation overlaps both the low energy (keV) electron and proton precipitation. These observations are consistent with a model where magnetosheath particles stream in along the cusp field lines and are at the same time convected poleward by an electric field.The electrons with energies of a few keV move fast and give the “ionospheric footprint” of the distant cusp. The protons are partly convected poleward of the cusp and into the polar cap. Here the mirroring protons populate the plasma mantle. Equatorward of the cusp the pitch angle distribution of both electrons and protons with energies above a few keV is pancake shaped indicating closed geomagnetic field lines. The 1 keV electrons, penetrate, however, into this region of closed field line structure maintaining an isotropic pitch angle distribution. The intensity is, however, reduced with respect to what it was in the cusp region. It is suggested that these electrons, the lowest energies measured on the satellite, are associated with the entry layer.     

Energization of positive ions in the cometary foreshock region

The energization of positive ions in front of a cometary bow shock is investigated. Ions produced by ionization of the cometary neutrals interact with the solar wind protons to produce, among other waves, large amplitude oscillations of the ambient magnetic field. Such oscillations are convected towards the comet at the unperturbed solar wind speed far from the shock and at a lower speed closer to the shock (due to the solar wind mass loading) ; hence, they can energize the suprathermal ions by Fermi acceleration. The spatial extension of the acceleration region is of the order of 10⁶ km and the resulting ion energy spectrum is harder than in the Earth's bow shock case. The energization of cometary ions produces an additional deceleration of the solar wind, such that the cometary bow shock of Halley-type comet may be regarded as a “cosmic ray shock”.     

Test Particle Simulations of Interaction Between Monochromatic Chorus Waves and Radiation Belt Relativistic Electrons

Chorus waves have been suggested to be effective in acceleration of radiation belt electrons. Here we perform gyro-averaged test-particle simulations to calculate the bounce-averaged pitch angle and energy diffusion coefficients for parallel-propagating monochromatic chorus waves, and perform a comparison of test-particle (TP) model with quasi-linear (QL) theory to evaluate the influence of nonlinear processes. For small amplitude chorus waves, the diffusion coefficients of TP and QL models are in good agreement. As the wave amplitude reaches a threshold value, two nonlinear processes (phase trapping and phase bunching) start to occur, especially at large equatorial pitch angles. Phase trapping yields rapid increases in pitch angle and kinetic energy. In contrast, phase bunching causes overall decreases in pitch angle and kinetic energy. For the waves with amplitudes slightly above the threshold value, the average behavior is dominated by the phase trapping, and TP diffusion coefficients are larger than QL ones. As wave amplitude increases, TP diffusion coefficients become smaller than QL ones, indicating that phase trapping gradually reduces the dominance over phase bunching.     

Using X‐ray observations to identify the particle acceleration mechanisms in VHE SNRs and “dark” VHE sources

Very high energy (VHE) γ‐ray observations have proven to be very successful in localizing Galactic acceleration sites of VHE particles. Observations of shell‐type supernova remnants have confirmed that particles are accelerated to VHE energies in supernova blast waves; the interpretation of the γ‐ray data in terms of hadronic or leptonic particle components in these objects relies nevertheless strongly on input from X‐ray observations. The largest identified Galactic VHE source class consists of pulsar wind nebulae, as detected in X‐rays. Many of the remaining VHE sources remain however unidentified until now. With X‐ray observations of these enigmatic “dark” objects one hopes to solve the following questions: What is the astrophysical nature of these sources? Are they predominantly electron or hadron accelerators? And what is their contribution to the overall cosmic ray energy budget? The paper aims to provide an overview over the identification status of the Galactic VHE source population. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)