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Solar Energetic Particles (SEP) events are interesting from a scientific perspective as they are the product of a broad set of physical processes from the corona out through the extent of the heliosphere, and provide insight into processes of particle acceleration and transport that are widely applicable in astrophysics. From the operations perspective, SEP events pose a radiation hazard for aviation, electronics in space, and human space exploration, in particular for missions outside of the Earth’s protective magnetosphere including to the Moon and Mars. Thus, it is critical to imific understanding of SEP events and use this understanding to develop and improve SEP forecasting capabilities to support operations. Many SEP models exist or are in development using a wide variety of approaches and with differing goals. These include computationally intensive physics-based models, fast and light empirical models, machine learning-based models, and mixed-model approaches. The aim of this paper is to summarize all of the SEP models currently developed in the scientific community, including a description of model approach, inputs and outputs, free parameters, and any published validations or comparisons with data.

Understanding the physics of Solar Energetic Particle (SEP) events is of importance to the general question of particle energization throughout the cosmos as well as playing a role in the technologically critical impact of space weather on society. The largest, and often most damaging, events are the so-called gradual SEP events, generally associated with shock waves driven by coronal mass ejections (CMEs). We review the current state of knowledge about particle acceleration at evolving interplanetary shocks with application to SEP events that occur in the inner heliosphere. Starting with a brief outline of recent theoretical progress in the field, we focus on current observational evidence that challenges conventional models of SEP events, including complex particle energy spectra, the blurring of the distinction between gradual and impulsive events, and the difference inherent in particle acceleration at quasi-parallel and quasi-perpendicular shocks. We also review the important problem of the seed particle population and its injection into particle acceleration at a shock. We begin by discussing the properties and characteristics of non-relativistic interplanetary shocks, from their formation close to the Sun to subsequent evolution through the inner heliosphere. The association of gradual SEP events with shocks is discussed. Several approaches to the energization of particles have been proposed, including shock drift acceleration, diffusive shock acceleration (DSA), acceleration by large-scale compression regions, acceleration by random velocity fluctuations (sometimes known as the “pump mechanism”), and others. We review these various mechanisms briefly and focus on the DSA mechanism. Much of our emphasis will be on our current understanding of the parallel and perpendicular diffusion coefficients for energetic particles and models of plasma turbulence in the vicinity of the shock. Because of its importance both to the DSA mechanism itself and to the particle composition of SEP events, we address in some detail the injection problem. Although steady-state models can improve our understanding of the diffusive shock acceleration mechanism, SEP events are inherently time-dependent. We therefore review the time-dependent theory of DSA in some detail, including estimating possible maximum particle energies and particle escape from the shock complex. We also discuss generalizations of the diffusive transport approach to modeling particle acceleration by considering a more general description based on the focused transport equation. The escape of accelerated particles from the shock requires that their subsequent transport in the interplanetary medium be modeled and the consequence of interplanetary transport can lead to the complex spectra and compositional profiles that are observed frequently. The different approaches to particle transport in the inner heliosphere are reviewed. The various numerical models that have been developed to solve the gradual SEP problem are reviewed. Explicit comparisons of modeling results with observations of large SEP events are discussed. A summary of current progress and the outlook on the SEP problem and remaining open questions conclude the review.

The perpendicular diffusion coefficient is calculated by combining a recently developed Unified Nonlinear Transport (UNLT) theory with the Newton–Lorentz equation. The total perpendicular mean free path can be described as a combination of a guiding center contribution and a microscopic contribution. It is shown that the total mean free path depends strongly on the energy range of the turbulence power spectrum and on particle energy. Further, a slab/2D composite model is used to investigate the influence of each contribution to the total mean free path for a quasi-3D turbulence model. For pure 2D turbulence the UNLT reduces to the NLGC-theory. For pure slab turbulence the guiding center contribution is subdiffusive in accordance with simulations and the theorem on reduced dimensionality. Conversely, the microscopic contribution is non-zero, which has to be interpreted as normal diffusion.

In gradual solar energetic particle (SEP) events, protons and heavy ions are often accelerated to >100 MeV/nucleon at a CME-driven shock. In this work, we study particle acceleration at an oblique shock by extending our earlier particle acceleration and transport in heliosphere (PATH) code to include shocks with arbitrary θ BN , where θ BN is the angle between the upstream magnetic field and the shock normal. Instantaneous particle spectra at the shock front are obtained by solving the transport equation using the total diffusion coefficient κ , which is a function of the parallel diffusion coefficient κ ∥ and the perpendicular diffusion coefficient κ ⊥ . In computing κ ∥ and κ ⊥ , we use analytic expressions derived previously. The particle maximum energy at the shock front as a function of time, the time intensity profiles and particle spectra at 1 AU for five θ BN ’s are calculated for an example shock.

A nonlinear time dependent fluid simulation model is developed that describes the evolution of magnetohydrodynamic waves in the presence of collisional and charge exchange interactions of a partially ionized plasma. The partially ionized plasma consists of electrons, ions and a significant number of neutral atoms. In our model, the electrons and ions are described by a single fluid compressible magnetohydrodynamic (MHD) model and are coupled self-consistently to the neutral gas, described by the compressible hydrodynamic equations. Both the plasma and neutral fluids are treated with different energy equations that describe thermal energy exchange processes between them. Based on our self-consistent model, we find that propagating Alfv\'enic and fast/slow modes grow and damp alternately through a nonlinear modulation process. The modulation appears to be robust and survives strong damping by the neutral component., Comment: The paper has been accepted in Physics Letters A

We discuss the asymmetry of the heliospheric discontinuities obtained from the analysis of 3D modeling of the solar wind (SW) interaction with local interstellar medium (LISM). The flow of charged particles is governed by the ideal MHD equations and the flow of neutral particles is described by the Boltzmann equation. The emphasis is made on the asymmetries of the termination shock (TS) and the heliopause under the combined action of the interstellar and interplanetary magnetic fields (ISMF and IMF) in the presence of neutral hydrogen atoms whose transport through the heliosphere is modeled kinetically, using a Monte Carlo approach. We show that the deflection of neutral hydrogen flow from its original direction in the unperturbed LISM is highly anisotropic and evaluate a possible angle between the hydrogen deflection plane measured in the SOHO SWAN experiment and the plane containing the ISMF and LISM velocity vectors for different ISMF strengths. It is shown that the ISMF of a strength greater than 4 μG can account for the 10 AU difference in the TS heliocentric difference observed during its crossing by the Voyager 1 and Voyager 2 spacecraft, which however results in a larger discrepancy between the calculated and observed velocity distributions. The effect of a strong ISMF on the distribution of plasma quantities in the inner heliosheath and on 2–3 kHz radio emission is discussed.

Low-energy termination shock particle populations observed by the Voyagers upstream of the shock exhibited strong field-aligned beaming with anisotropies of the order of unity. The Parker transport equation is valid only for nearly isotropic phase space distributions and is inapplicable to these highly beamed populations. The usual approach is to revert to the more general focused transport equation retaining pitch-angle information. We developed a complimentary technique employing a three-moment expansion of the Skilling equation using Legendre polynomials. We investigate the effects of adiabatic focusing and reflection on the diffusive acceleration process at oblique shock waves. It is shown that low-energy particle intensities are discontinuous and sharply peaked at the shock, consistent with the observations. Particle spectra are not only harder than the power laws predicted from diffusive transport theory, but also exhibit spectral gaps near the low-energy acceleration threshold due to more efficient acceleration by scattering and mirroring. Our model also predicts upstream anisotropies as high as 100% for highly oblique shocks whereas downstream distributions are nearly isotropic.

An overview is presented of magnetic-field-related effects in the solar wind (SW) interaction with the local interstellar medium (LISM) and the different theoretical approaches used in their investigation. We discuss the possibility that the interstellar magnetic field (ISMF) introduces north–south and east–west asymmetries of the heliosphere, which might explain observational data obtained by the Voyager 1 and Voyager 2 spacecraft. The SW–LISM interaction parameters that are responsible for the deflection of the interstellar neutral hydrogen flow from the direction of propagation of neutral helium in the inner heliosheath are outlined. The possibility of a strong ISMF, which increases the heliospheric asymmetry and the H–He flow deflection, is discussed. The effect of the combination of a slow-fast solar wind during solar minimum over the Sun’s 11-year activity cycle is illustrated. The consequences of a tilt between the Sun’s magnetic and rotational axes are analyzed. Band-like areas of an increased magnetic field distribution in the outer heliosheath are sought in order to discover regions of possible 2–3 kHz radio emission.

The heliopause, a surface separating the tenuous hot heliosheath flow and the dense, strongly magnetized interstellar flow, is subject to instabilities of the Rayleigh–Taylor and Kelvin–Helmholtz types. The dynamic evolution of this discontinuity is of considerable importance for understanding the neutral atom and cosmic-ray filtration at the interface. Here, we investigate the stability of the upwind heliopause in the presence of charge exchange collisions using both an analytic (dispersion relation) approach and a numerical model that includes the interstellar magnetic field. The linear analysis yields a cubic dispersion relation that admits imaginary solutions for the full range of wavenumbers, implying that the stagnation point on the heliopause is unconditionally Rayleigh–Taylor unstable to small perturbations propagating parallel to the discontinuity surface. We confirm this result by following the nonlinear development of the instability with a time-dependent simulation using a four fluid MHD-neutral numerical code. For the typical solar wind and LISM conditions, we obtain cyclical evolution of the upwind heliopause with a period of the order of 100 years. We also identify two areas of space physics where the instability may have important implications.

We present numerical results showing the effect of neutral hydrogen atoms on the solar wind (SW) interaction with the local interstellar medium (LISM), where the interstellar magnetic field (ISMF) is coupled to the interplanetary magnetic field (IMF) at the surface of the heliopause. The IMF on the inner boundary surrounding the Sun is specified in the form of a Parker spiral and self-consistently develops in accordance with the SW motion inside the heliopause. The model of the SW–LISM interaction involves both plasma and neutral components which are treated as fluids. The configuration is chosen where the ISMF is orthogonal to the LISM velocity and tilted 60° to the ecliptic plane. This orientation of the magnetic field is a possible explanation of the 2–3 kHz emission data which is believed to originate ahead of the heliopause. It is shown that the topology of the heliospheric current sheet is substantially affected by the ISMF. The interaction pattern dependence on the neutral hydrogen density is analyzed.

Charge exchange processes in the outer heliosphere produce a population of hot hydrogen gas within the heliosphere, creating a “hydrogen wall” between the heliopause and bow shock. The heliospheric hydrogen wall scatters Lyα photons passing through it, producing a detectable absorption signature in observations of H I Lyα emission from nearby stars. This heliospheric absorption has been observed using observations from the Hubble Space Telescope (HST), and these observations have also yielded detections of analogous “astrospheric” absorption from material surrounding the observed stars. The astrospheric detections dramatize the importance of understanding the heliospheric interaction, since similar interactions exist around other stars and can now be studied with HST. We review comparisons that have been made between the observed heliospheric absorption and the predictions of various models. The astrospheric absorption provides a way to empirically estimate the mass loss rates of solar-like stars, leading to the first empirical estimates of how solar-like winds vary with stellar age and activity. Thus, we also review these astrospheric results and discuss their ramifications for solar, stellar, and planetary science.

Historically, solar energetic particle (SEP) events are classified in two classes as “impulsive” and “gradual”. Whether there is a clear distinction between the two classes is still a matter of debate, but it is now commonly accepted that in large “gradual” SEP events, Fermi acceleration, also known as diffusive shock acceleration, is the underlying acceleration mechanism. At shock waves driven by coronal mass ejections (CMEs), particles are accelerated diffusively at the shock and often reach > MeV energies (and perhaps up to GeV energies). As a CME-driven shock propagates, expands and weakens, the accelerated particles can escape ahead of the shock into the interplanetary medium. These escaping energized particles then propagate along the interplanetary magnetic field, experiencing only weak scattering from fluctuations in the interplanetary magnetic field (IMF). In this paper, we use a Monte-Carlo approach to study the transport of energetic particles escaping from a CME-driven shock. We present particle spectra observed at 1 AU. We also discuss the particle “crossing number” at 1AU and its implication to particle anisotropy. Based on previous models of particle acceleration at CME-driven shocks, our simulation allows us to investigate various characteristics of energetic particles arriving at various distances from the sun. This provides us an excellent basis for understanding the observations of high-energy particles made at 1 AU by ACE and WIND.

A dynamical time-dependent model of particle acceleration at coronal mass ejection (CME) driven shock waves is presented. We consider first a one-dimensional model applicable to shock waves of arbitrary strength. The model includes the determination of the particle's injection energy, the maximum energy of particles accelerated at the shock, energetic particle spectra at all spatial and temporal locations, and the dynamical distribution of particles that are trapped in the post shock flow and those that escape upstream and downstream of the evolving shock complex. We then consider the extension of this model to two dimensions. The 2-dimensional model computes the variation of the shock strength both temporally and spatially and illustrates how this effects the evolution of the energetic particle spectra and intensities.

Historically, solar energetic particle (SEP) events are classified in two classes as “impulsive“ and “gradual”. Whether there is a clear distinction between the two classes is still a matter of debate, but it is now commonly accepted that in large “gradual” SEP events, Fermi acceleration, also known as diffusive shock acceleration, is the underlying acceleration mechanism. At shock waves driven by coronal mass ejections (CMEs), particles are accelerated diffusively at the shock and often reach > MeV energies (and perhaps up to GeV energies). As a CMEdriven shock propagates, expands and weakens, the accelerated particles can escape ahead of the shock into the interplanetary medium. These escaping energized particles then propagate along the interplanetary magnetic field, experiencing only weak scattering from fluctuations in the interplanetary magnetic field (IMF). In this paper, we use a Monte-Carlo approach to study the transport of energetic particles escaping from a CMEdriven shock. We present particle spectra observed at 1 AU. We also discuss the particle “crossing number” at 1AU and its implication to particle anisotropy. Based on previous models of particle acceleration at CMEdriven shocks, our simulation allows us to investigate various characteristics of energetic particles arriving at various distances from the sun. This provides us an excellent basis for understanding the observations of high-energy particles made at 1 AU by ACE and WIND.

In this paper we use the best currently available estimates for interstellar parameters to calculate several models of the interaction between the solar wind and the magnetized interstellar plasma in the presence of LISM neutral hydrogen. We show how the heliosphere may be modified by both the galactic magnetic field and neutral particles.

The flux of energetic particles at the termination shock, preaccelerated at interplanetary shocks, is calculated using Voyager 2 measurements in the outer heliosphere and by assuming a subsequent r −2 radial dependence. The spectrum of particles accelerated at the termination shock is then determined using the above spectrum as a source in the steady state diffusion convection equation. The accelerated spectrum is then modulated back to 57 AU and compared with measured anomalous cosmic ray (ACR) fluxes. For injection energies, at the termination shock, of about 50 keV the computed ACR flux at 57 AU compares well with the measured flux. Calculations, however, suggest that the injection energy should be considerably higher than 50 keV. If this is the case then the calculated ACR flux is considerably lower than expected. Voyager 2 observations beyond 40 AU and subsequent modeling suggest that interplanetary shocks may become too weak to continue accelerating energetic particles. Their flux should then fall off faster than r −2 and hence the flux at the termination shock may then be insufficient to explain the observed ACR fluxes. Using the injection energies computed for various species and by assuming that injection and acceleration takes place directly at the termination shock, we find acceleration efficiencies and fluxes consistent with those calculated from ACR measurements in the outer heliosphere.

The companion paper of Fisk (1999) in this volume convincingly argues that the theory of cosmic ray transport in the heliosphere has probably been complete for almost three decades. The outstanding problem is to find the magnitude, spatial, and energy dependence of the convection, diffusion, drift, adiabatic cooling, and acceleration mechanisms. This problem can be addressed in two complementary ways: the relevant parameters can be calculated from theory, or the observed particle spectra can be used to deduce them. This paper emphasises the second approach, i.e., we review to what extent one is able to fit cosmic ray observations self-consistently with a single set of modulation parameters. For this purpose we mainly concentrate on model fits to the intensity spectra of cosmic ray protons, helium, and oxygen as observed during the solar minimum periods of 1987 and 1997. The main results are that: (1) the scattering parameters deduced from the particle spectra broadly fall in the same range as the newest values derived from theory, but (2) the separate sets of scattering parameters needed to simulate the 1987 and 1997 solar minimum spectra differ so much from one another that it is unlikely that the final solution has been found.

A brief overview of the physics thought to determine the properties of the heliospheric termination shock is given and the central role of hot interstellar pick-up ions and cosmic rays is emphasized.