H.-Q. He

PROPAGATION OF SOLAR ENERGETIC PARTICLES IN THREE-DIMENSIONAL INTERPLANETARY MAGNETIC FIELDS: IN VIEW OF CHARACTERISTICS OF SOURCES

In this paper, a model of solar energetic particle (SEP) propagation in the three-dimensional Parker interplanetary magnetic field is calculated numerically. We study the effects of the different aspects of particle sources on the solar surface, which include the source location, coverage of latitude and longitude, and spatial distribution of source particle intensity, on propagation of SEPs with both parallel and perpendicular diffusion. We compute the particle flux and anisotropy profiles at different observation locations in the heliosphere. From our calculations, we find that the observation location relative to the latitudinal and longitudinal coverage of particle source has the strongest effects on particle flux and anisotropy profiles observed by a spacecraft. When a spacecraft is directly connected to the solar sources by the interplanetary magnetic field lines, the observed particle fluxes are larger than when the spacecraft is not directly connected. This paper focuses on the situations when a spacecraft is not connected to the particle sources on the solar surface. We find that when the magnetic footpoint of the spacecraft is farther away from the source, the observed particle flux is smaller and its onset and maximum intensity occur later. When the particle source covers a larger range of latitude and longitude, the observed particle flux is larger and appears earlier. There is east-west azimuthal asymmetry in SEP profiles even when the source distribution is east-west symmetric. However, the detail of particle spatial distribution inside the source does not affect the profile of the SEP flux very much. When the magnetic footpoint of the spacecraft is significantly far away from the particle source, the anisotropy of particles in the early stage of an SEP event points toward the Sun, which indicates that the first arriving particles come from outside of the observer through perpendicular diffusion at large radial distances.

AN EFFECT OF PERPENDICULAR DIFFUSION ON THE ANISOTROPY OF SOLAR ENERGETIC PARTICLES FROM UNCONNECTED SOURCES

Recently, Tan and coworkers studied the 2001 September 24 solar energetic particle (SEP) event observed by the Wind spacecraft at 1 AU and found that there is a counter-streaming particle beam with a deep depression of flux at 90 degrees pitch angle during the beginning of the event. They suggested that it is a result of a reflecting boundary at some distance outside of 1 AU. While this scenario could be true under some specific configuration of an interplanetary magnetic field, in this paper we offer another possible explanation. We simulated the SEP event by solving the five-dimensional focused transport equation numerically for 40 keV electrons with perpendicular diffusion. We find that a counter-streaming particle beam with deep depression at 90 degrees pitch angle can form on Parker magnetic field lines that do not directly connect to the main particle source on the Sun in the beginning of an SEP event. It can happen when a significant number of observed particles come from adjacent field lines through parallel transport to large radial distance first, hopping across field lines through perpendicular diffusion, and then getting scattered back to 1 AU, where they combine with the particles directly coming from the Sun to form a counter-streaming beam.

A DIRECT METHOD TO DETERMINE THE PARALLEL MEAN FREE PATH OF SOLAR ENERGETIC PARTICLES WITH ADIABATIC FOCUSING

The parallel mean free path of solar energetic particles (SEPs), which is determined by physical properties of SEPs as well as those of solar wind, is a very important parameter in space physics to study the transport of charged energetic particles in the heliosphere, especially for space weather forecasting. In space weather practice, it is necessary to find a quick approach to obtain the parallel mean free path of SEPs for a solar event. In addition, the adiabatic focusing effect caused by a spatially varying mean magnetic field in the solar system is important to the transport processes of SEPs. Recently, Shalchi presented an analytical description of the parallel diffusion coefficient with adiabatic focusing. Based on Shalchis results, in this paper we provide a direct analytical formula as a function of parameters concerning the physical properties of SEPs and solar wind to directly and quickly determine the parallel mean free path of SEPs with adiabatic focusing. Since all of the quantities in the analytical formula can be directly observed by spacecraft, this direct method would be a very useful tool in space weather research. As applications of the direct method, we investigate the inherent relations between the parallel mean free path and various parameters concerning physical properties of SEPs and solar wind. Comparisons of parallel mean free paths with and without adiabatic focusing are also presented.

Modification of the Parallel Scattering Mean Free Path of Cosmic Rays in the Presence of Adiabatic Focusing

The cosmic ray mean free path in a large-scale nonuniform guide magnetic field with superposed magnetostatic turbulence is calculated to clarify some conflicting results in the literature. A new, exact integro-differential equation for the cosmic-ray anisotropy is derived from the Fokker-Planck transport equation. A perturbation analysis of this integro-differential equation leads to an analytical expression for the cosmic ray anisotropy and the focused transport equation for the isotropic part of the cosmic ray distribution function. The derived parallel spatial diffusion coefficient and the associated cosmic ray mean free path include the effect of adiabatic focusing and reduce to the standard forms in the limit of a uniform guide magnetic field. For the illustrative case of isotropic pitch angle scattering, the derived mean free path agrees with the earlier expressions of Beeck & Wibberenz, Bieber & Burger, Kota, and Litvinenko, but disagrees with the result of Shalchi. The disagreement with the expression of Shalchi is particularly strong in the limit of strong adiabatic focusing.

Numerical Study of the Longitudinally Asymmetric Distribution of Solar Energetic Particles in the Heliosphere

Solar energetic particles (SEPs) affect the solar-terrestrial space environment and become a very important aspect in space weather research. In this work, we numerically investigate the transport processes of SEPs in three-dimensional interplanetary magnetic field, with an emphasis on the longitudinal distribution of SEPs in the heliosphere. We confirm our previous finding that there exists an east-west longitudinal asymmetry in the SEP intensities, i.e., with the same longitude separations between the solar source centers and the magnetic footpoint of the observer, the fluxes of SEP events originating from solar sources located on the eastern side of the nominal magnetic footpoint of the observer are systematically larger than those of the SEP events originating from sources located on the western side. We discuss the formation mechanism of this phenomenon, and conclude that the longitudinally asymmetric distribution of SEPs results from the east-west azimuthal asymmetry in the topology of the heliospheric magnetic field as well as the effects of perpendicular diffusion on the transport of SEPs in the heliosphere. Our results will be valuable to understanding Sun-Earth relations and useful for space weather forecasting.

A Direct Approach for Determining the Perpendicular Mean Free Path of Solar Energetic Particles in a Turbulent and Spatially Varying Magnetic Field

A direct approach for explicitly determining the perpendicular mean free path of solar energetic particles (SEPs) influenced by parallel diffusion and composite dynamical turbulence in a spatially varying magnetic field is presented. As theoretical applications of the direct approach, we investigate the inherent relations between the perpendicular mean free path and various parameters concerning physical properties of SEPs as well as those of interplanetary conditions such as the solar wind and the turbulent magnetic field. Comparisons of the perpendicular mean free paths with and without adiabatic focusing are also presented. The direct method shows encouraging agreement with spacecraft observations, suggesting it is a reliable and useful tool for use in theoretical investigations and space weather forecasting.

PERPENDICULAR DIFFUSION IN THE TRANSPORT OF SOLAR ENERGETIC PARTICLES FROM UNCONNECTED SOURCES: THE COUNTER-STREAMING PARTICLE BEAMS REVISITED

In some solar energetic particle (SEP) events, a counter-streaming particle beam with a deep depression of flux near 90 degrees pitch angle during the beginning phase is observed. Two different interpretations exist in the community to explain this interesting phenomenon. One explanation invokes the hypothesis of an outer reflecting boundary or a magnetic mirror beyond the observer. The other one considers the effect of the perpendicular diffusion on the transport process of SEPs in the interplanetary space. In this work, we revisit the problem of the counter-streaming particle beams observed in SEP events and discuss the possible mechanisms responsible for the formation of this phenomenon. We clarify some results in previous works.

The dependence of the parallel and perpendicular mean free paths on the rigidity of the solar energetic particles: Theoretical model versus observations

Context. The dependence of the parallel and perpendicular mean free paths on the particle rigidity is an important topic in the studies of the diffusion and propagation of charged energetic particles in a large-scale turbulent magnetic field. Aims. In this work, we investigate the dependence of the parallel and perpendicular mean free paths on the rigidity of solar energetic particles (SEPs) by means of both the theoretical model and spacecraft observations with regard to several typical SEP events. Methods. A direct method developed by previous studies and derived from the focused transport equation and Taylor-Green-Kubo (TGK) formulation is applied to explicitly determine the parallel and perpendicular mean free paths of SEPs in a turbulent and spatially varying magnetic field. Results. We find that the parallel and perpendicular mean free paths, λ‖ and λ⊥, of energetic protons monotonically decrease with increasing particle rigidity, and the ratio λ⊥/λ‖ monotonically increases with particle rigidity, when the magnetic turbulence is weak. Taking a series of typical SEP events together, it can also be seen that the ratio λ⊥/λ‖ of the perpendicular to the parallel mean free paths remains in the range of 0.001−0.2.

EFFECTS OF LATITUDINALLY DEPENDENT SOLAR WIND SPEED ON DIFFUSION COEFFICIENTS OF COSMIC RAYS IN THE PRESENCE OF ADIABATIC FOCUSING

The solar wind is observed to display high speeds in high heliolatitude coronal holes and low speeds near the ecliptic plane. The heliospheric magnetic field associated with the solar wind plays a very important role in the transport and modulation of charged energetic particles, including galactic cosmic rays (GCRs) and solar energetic particles (SEPs), in the three-dimensional heliosphere. In previous studies, a constant solar wind speed, which is independent of heliolatitude, was assumed and commonly used in modulation modeling of cosmic rays. In this work, we investigate the realistic latitudinally dependent solar wind speed and utilize the theoretical models in hyperbolic and piecewise polynomial forms to explore the important effects on the heliospheric magnetic field and the diffusion coefficients (parallel, perpendicular, and drift scale) of cosmic rays in the presence of adiabatic focusing. Comparisons of the diffusion coefficients derived from standard Parker field and modified magnetic fields are presented. Since the structures and properties of different solar wind sources (coronal streamer belt, polar coronal hole, and transition region between them) differ from each other in essence, we suggest that the latitudinally dependent solar wind speed and the corresponding heliospheric magnetic field and diffusion coefficients with adiabatic focusing should be employed in the global modeling studies of GCRs and SEPs in the heliosphere.