Propagation of solar cosmic rays: Diffusion versus focused diffusion

This paper is designed to bring to the attention the fact that the effect of focusing of solar energetic particles is always essential as compared with scattering, no matter how small the value of the mean free path may be. That is why, an ordinary (focusing-free) diffusion approach can not be applied to the solar cosmic ray transport. In the case of high-energy solar particles, the focused diffusion is demonstrated to lead to a power law decay of energetic particle intensity much like an ordinary diffusion. However, the power law index of the decay is renormalized by the focusing.     

Shock acceleration of solar cosmic rays

We investigated the acceleration of solar cosmic rays (SCRs) by the shock waves produced by coronal mass ejections. We performed detailed numerical calculations of the SCR spectra produced during the shock propagation in the solar corona in terms of a model based on the diffusive transport equation using a realistic set of physical parameters for the corona. The resulting SCR energy spectrum N(ε) ∝ ε^?γ exp [? (ε/ε_max)^α] is shown to include a power-law portion with an index γ?2 that ends with an exponential tail with α ? 2.5 ? β, where β is the spectral index of the background Alfvén turbulence. The maximum SCR energy lies within the range ε_max = 1–300 MeV, depending on the shock velocity. Because of the steep spectrum of the SCRs, their backreaction on the shock structure is negligible. The decrease in the Alfvén Mach number of the shock due to the increase in the Alfvén velocity with heliocentric distance r causes the efficient SCR acceleration to terminate when the shock reaches a distance of r = 2–3R_⊙. Since the diffusive SCR propagation in this case is faster than the shock expansion, SCR particles intensively escape from the shock vicinity. A comparison of the calculated SCR fluxes expected near the Earth’s orbit with available experimental data indicates that the theory satisfactorily explains all of the main observed features.     

Shock acceleration of solar cosmic rays

The solar cosmic ray (SCR) acceleration by the shocks driven by coronal mass ejections is studied by taking into account the generation of Alfvén waves by accelerated particles. Detailed numerical calculations of the SCR spectra produced during the shock propagation through the solar corona have been performed within a quasi-linear approach with a realistic set of coronal parameters. The resultant SCR energy spectrum is shown to include a power-law part N ∝ ?^-γ with an index γ = 1.7–3.5 that ends with an exponential tail. The maximum SCR energy lies within the range ? _max = 0.01–10 GeV, depending on the shock velocity V _S = 750–2500 km s^?1. The decrease of the shock Alfvénic Mach number due to the increase Alfvén velocity with heliocentric distance r leads to the end of the efficient SCR acceleration when the shock size reaches R _S ≈ 4R _⊙. In this case, the diffusive SCR propagation begins to exceed the shock velocity; as a result, SCRs escape intensively from the shock vicinity. The self-consistent generation of Alfvén waves by accelerated particles is accompanied by a steepening of the particle spectrum and an increase of their maximum energy. Comparison of the calculated SCR fluxes expected near the Earth’s orbit with the available experimental data shows that the theory explains the main observed features.     

Sectorial anisotropy of solar cosmic rays

A hitherto unobserved sectorial pattern of anisotropy that was limited to a narrow and stable region was displayed by the ground level event (GLE) of January 24, 1971. For the entire 11/2 h interval following onset before isotropy set in, the anisotropy was limited to a 10° cone centered about 60° from the spiral magnetic field line. It is also the first solar particle event for which it is possible, by analytical procedures based upon a theoretical propagation model, to distinguish between two rival candidates for the parent flare.This research was sponsored by the National Science Foundation and Air Force Cambridge Research Laboratories, Office of Aerospace Research, under Contract No. F19628-70-C-0190, but the report does not necessarily reflect endorsement by the sponsor.     

Non-linear momentum diffusion of heliospheric cosmic rays

Besides parallel and perpendicular spatial diffusion, momentum diffusion can be seen as the third important process of cosmic ray transport. In this paper, the recently derived weakly non-linear theory is applied for a simple quasi-magnetostatic composite model to determine the momentum diffusion coefficient. It is demonstrated that non-linear effects are essential and cannot be neglected. Therefore, the weakly non-linear transport theory has to be preferred over the traditional quasi-linear approach. Within this improved theory, we find for the rigidity dependence of the momentum diffusion coefficient   A ∼ R ^1.4  for relativistic and   A ∼ R ^0.4  for non-relativistic cosmic rays.     

Transport of cosmic rays in the solar corona

A model is proposed to explain the transport of energetic protons in the solar corona. The particles are assumed to undergo an enhanced gradient-B drift along thin current sheets separating discontinuous field structures in the corona. These discontinuities may represent the extension into the corona of photospheric granular and supergranular cell boundaries. We have made a quantitative analysis of this process by assuming that the particle propagation can be described by a diffusion equation. Comparison of predictions of the model with cosmic ray observations at 1 AU provide some support for the model.     

Multi-parametric effect of solar activity on cosmic rays

The long-term modulation of cosmic ray intensity (CRI) by different solar activity (SA) parameters and an inverse correlation between individual SA parameter and CRI is well known. Earlier, it has been suggested that the concept of multi-parametric modulation of CRI may play an important role in the study of long-term modulation of CRI. In the present study, we have tried to investigate the combined effect of a set of two SA parameters in the long-term modulation of CRI. For this purpose, we have used a new statistical technique called “Running multiple correlation method”, based on the “Running cross correlation method”. The running multiple correlation functions among different sets of two SA parameters (e.g., sunspot numbers and solar flux, sunspot numbers and coronal index, sunspot numbers and grouped solar flares, etc.) and CRI have been correlated separately. It is found that the strength of multiple correlation (among two SA parameters and CRI) and cross correlation (between individual SA parameter and CRI) is almost similar throughout the period of investigation (1955–2005). It is also found that the multiple correlations among various SA parameters and CRI is stronger during ascending and descending phases of the solar cycles and it becomes weaker during maxima and minima of the solar cycles, which is in accordance with the linear relationship between SA parameters and CRI. The values of multiple correlation functions among different sets of SA parameters and CRI fall well within the 95% confidence interval. In the view of odd-even hypothesis of solar cycles, the strange behaviour of present cycle 23 (odd cycle), as this is characterized by many peculiarities with double peaks and many quiet periods (Gnevyshev gaps) interrupted the solar activity (for example April 2001, October–November 2003 and January 2005), leads us to speculate that the solar cycle 24 (even cycle) might be of exceptional nature.     

The effect of a bounded interplanetary diffusion medium on the propagation of solar flare cosmic rays

An analysis of solar flare data indicates that the graph of log(nt ^3/(2–)) deviates late in the solar event from the straight line predicted for the infinite, unbounded interplanetary medium. It is shown by mathematical analysis, utilizing a model based on the radial diffusion coefficient D = Mr , with 1, that the deviation can be ascribed to the loss of flare particles through an external boundary at about 5–6 AU from the Sun. An inner region terminating at 5–6 AU, followed by an extensive region of increasingly less resistance to the diffusion of flare particles is also feasible and it is shown that measurements taken at the Earth cannot predict the extent of this outer region. The results are applicable to either the isotropic or highly anisotropic models. The constant diffusion model is shown to be inadequate since it requires a boundary 1.5 AU from the Sun. In view of the present and previous studies of solar flare data, it is asserted that the fundamental principle governing the diffusion of solar flare particles through interplanetary space is the radial diffusion coefficient mode of propagation.     

Energy losses of solar cosmic rays in interplanetary space

A simple model of solar cosmic ray propagation which includes diffusion, convection, and energy loss by adiabatic deceleration is studied. A Monte Carlo technique is employed to investigate the variation of mean particle energy in the interplanetary medium after the impulsive release of mono-energetic particles at the Sun. At 1 AU typical energy losses are 43% at 20 h and 64% at 60 h after particle release for a diffusion coefficient (r)= ₀r with =+1/2 and ₀=1.33 × 10²¹ cm² s^–1. When ₀ in this model is reduced by a factor of 4, the energy loss is greater by a factor of 2 at 60 h after particle release. When is increased, the energy losses are greater. Using the model parameters above, an increase in solar wind speed from 300 to 600 km s^–-1 gives rise to energy losses that are greater again by factor of 2 at a time of 60 h. Results are compared with an observation by Murray et al. (1971) of a knee in the energy spectrum of solar protons. It is not considered likely that the change in the energy of the knee with time requires, in addition to adiabatic deceleration, another energy change process which acts to increase the energy of particles.Part of this work was performed while the author was at CSIRO, Division of Radiophysics, Epping, NSW, Australia; also supported in part by the U.S. Atomic Energy Commission.     

Influence of solar heliospheric parameters on cosmic rays anisotropy

In the present work the cosmic ray data of three different neutron monitoring stations, Deep River, Inuvik, and Tokyo, located at different geomagnetic cutoff rigidities and altitudes have been harmonically analyzed for the period 1980–95 for a comparative study of diurnal semi-diurnal and tri-diurnal anisotropies in cosmic ray intensity in connection with the change in interplanetary magnetic field Bz component and solar wind velocity on 60 quietest days. It is observed that the amplitudes of all the three harmonics increase during the period 1982–84 at all the stations during the high speed solar wind stream epoch and remain low during the declining phase of the stream. The amplitudes of the three harmonics have no obvious characteristics associated with the time variation of magnitude of the Bz component. The phases of all the three harmonics have no time variation characteristics associated with solar wind velocity and Bz. Published in Astrofizika, Vol. 49, No. 4, pp. 651–664 (August 2006).     

Neon produced by solar cosmic rays in ordinary chondrites

Solar‐cosmic‐ray‐produced Ne (SCR‐Ne), in the form of low cosmogenic ²¹Ne/²²Ne ratios (²¹Ne/²²Ne_cos <0.8), is more likely to be found in rare meteorite classes, like Martian meteorites, than in ordinary chondrites. This may be the result of a sampling bias: SCR‐Ne is better preserved in meteorites with small preatmospheric radii and these specimens are often only studied if they belong to unusual or rare classes. We measured He and Ne isotopic concentrations and nuclear tracks in 25 small unpaired ordinary chondrites from Oman. Most chondrites have been intensively heated during atmospheric entry as evidenced by the disturbed track records, the low ³He/²¹Ne ratios, the low ⁴He concentrations, and the high peak release temperatures. Concentration depth profiles indicate significant degassing; however, the Ne isotopes are mainly undisturbed. Remarkably, six chondrites have low ²¹Ne/²²Ne_cos in the range 0.711–0.805. Using a new physical model for the calculation of SCR production rates, we show that four of the chondrites contain up to ~20% of SCR‐Ne; they are analyzed in terms of preatmospheric sizes, cosmic ray exposure ages, mass ablation losses, and orbits. We conclude that SCR‐Ne is preserved, regardless of the meteorite class, in specimens with small preatmospheric radii. Sampling bias explains the predominance of SCR‐Ne in rare meteorites, although we cannot exclude that SCR‐Ne is more common in Martian meteorites than it is in small ordinary chondrites.     

Heavy solar cosmic rays in the January 25, 1971 solar flare

A detailed study of the charge composition of heavy solar cosmic rays measured in the January 25, 1971 solar flare including differential fluxes for the even charged nuclei from carbon through argon is presented. The measurements are obtained for varying energy intervals for each nuclear species in the energy range from 10 to 35 MeV nucleon^?1. In addition, abundances relative to oxygen are computed for all the above nuclei in the single energy interval from 15 to 25 MeV nucleon^?1. This interval contains measurements for all of the species and as a result requires no spectral extrapolations. An upper limit for the abundance of calcium nuclei is also presented. These measurements, when combined with other experimental results, enable the energy dependence of abundance measurements as a function of nuclear charge to be discussed. It is seen that at energies above about 10 MeV nucleon^?1, the variations of abundance ratios are limited to about a factor of 3 from flare to flare, in spite of large variations in other characteristics of these solar events.     

Diurnal variations of cosmic rays during a solar magnetic cycle

We have used data from five neutron monitor stations with primary rigidity (Rm) ranging from 16 GeV to 33 GeV to study the diurnal variations of cosmic rays over the period: 1965–1986 covering one 22-year solar magnetic cycle. The heliosphere interplanetary magnetic field (IMF) and plasma hourly measurements taken near Earth orbit, by a variety of spacecraft, are also used to compare with the results of solar diurnal variation. The local time of maximum of solar diurnal diurnal variations displays a 22-year cycle due to the solar polar magnetic field polarities. In general, the annual mean of solar diurnal amplitudes, magnitude of IMF and plasma parameters are found to show separte solar cycle variations. Moreover, during the declining period of the twenty and twenty-ne solar cycles, large solar diurnal amplitudes are observed which associated with high values of solar wind speed, plasma temperature and interplanetary magnetic field magnitude B3.     

Motion of solar cosmic rays in the coronal magnetic field

Trajectories of solar cosmic rays have been calculated in a static ninth-order coronal magnetic field. It is found that as a result of field curvature and gradients, protons drift across the field lines at a rate of up to 200 ² deg hr^–1. These drift rates are of the same order as, but somewhat smaller than, empirically derived rates. Localized enhancements of magnetic field have been inserted into the ninth-order field in order to model (in a highly idealized manner) the effects of the small-scale magnetic features which give rise to X-ray bright points. The motions of the particles in the presence of these scattering centers can be parameterized approximately by a cross-field diffusion coefficient. Our estimates of this coefficient, although crude, overlap with empirical values which have been deduced over a wide range of energies.We propose that coronal propagation of solar cosmic rays has two components. One is independent of particle velocity, and is associated with dynamic field phenomena (such as an expanding magnetic bottle): this is the only component which is important in flares which occur close to the foot-point of the Sun-Earth field line. The second component is velocity dependent, but is independent of mass, and is associated with scattering off (relatively static) magnetic inhomogeneities with scale sizes of at least 500 km: the second component contributes to coronal propagation if the flare occurs more than about 50–60 deg away from the Sun-Earth field line.     

Ground level enhancements of cosmic rays in solar cycle 24

Using data from ground-based observations of cosmic rays (CRs) on the worldwide network of stations and spacecraft, we have investigated the proton spectra and the CR anisotropy during the ground level enhancements of CRs on May 17, 2012 (GLE71) and January 6, 2014 (GLE72) occurred in solar cycle 24 by the spectrographic global survey method. We provide the CR rigidity spectra and the relative changes in the intensity of CRs with a rigidity of 2 GV in the solar–ecliptic geocentric coordinate system in specific periods of these events. We show that the proton acceleration during GLE71 and GLE72 occurred up to rigidities R ~ 2.3?2.5 GV, while the differential rigidity spectra of solar CRs are described neither by a power nor by an exponential function of particle rigidity. At the times of the events considered the Earth was in a loop-like structure of the interplanetary magnetic field.     

The continuous emission of low energy cosmic rays during solar flares

It has been recently suggested by several investigators that the accelerated charged particles provide the energy of the optical flare by the ionization loss process. We have examined this mechanism assuming different forms of the spectrum of the accelerated protons at lower chromosphere. The flux and the energy spectrum of protons of energy 0.1–100 MeV have been calculated at successive heights, from 10³ to 40 × 10³ km from the solar surface taking into account the ionization loss, pitch angle distribution and density distribution of the neutral and ionized hydrogen in the chromosphere and lower corona. Hence the energy spectrum of the protons escaping from the Sun and the amount of energy dissipated in the solar chromosphere are computed. Comparing the calculated results with the observational data on the solar event of September 28, 1961 it is found that the ionization loss of the accelerated protons and heavier nuclei in the solar atmosphere may supply a significant part of the energy of the optical flare assuming that the fraction, f, of magnetic tubes of force extending out of the solar atmosphere is about 1 %. The accelerated proton spectrum in the form of power law in kinetic energy seems to be the most appropriate form. In the event of September 28, 1961 best estimates are made on this basis of the total number and the energy spectrum of protons at injection, the flux and energy spectrum of escaping protons and the energy dissipated in the solar atmosphere by the accelerated ions. It is found that the possible range of variation of the height of injection level hardly affects the total energy dissipated. The high variability of the intensity of protons released by the Sun is interpreted in terms of the variations of the parameter, f, determined by the configurations of the magnetic field lines.Preliminary results were presented at the International Symposium on Solar-Terrestrial Physics, Leningrad, May, 1970.Presently at NASA/Goddard Space Flight Center, Greenbelt, Maryland, U.S.A., on leave from T.I.F.R., Bombay.     

Statistical acceleration and spatial diffusion of cosmic rays in turbulent medium

Statistical acceleration of cosmic rays in a turbulent medium is considered. Charged particles are assumed to acquire energy in a bounded region of space and leave the acceleration region due to spatial diffusion caused by the scattering of cosmic rays in turbulent magnetic fields. Analytical solutions of the cosmic ray transport equation are obtained and equilibrium space-energy distributions of high-energy particles are studied in the acceleration region and beyond.     

Propagation effects in the hydrogen-to-helium ratio in the solar cosmic rays

Taking as basis my dimensional solution of the propagation equation of solar cosmic rays in interplanetary space and using the equivalent diffusion coefficient found from observed data on solar protons, I discuss the effects of propagation on the hydrogen-to-hellum ratio Including its variation with the speed of the solar wind and with location in space. After eliminating the variations with solar distance and energy from the HEOS and PIONEER data collected by Perron et al., this ratio will be seen to increase with the magnetic longitude of the parent flare. The Initial value of the hydrogen-to-helium ratio at the Sun's surface, obtained after applying the corrections for propagation, is close to the value in the solar wind.     

The peculiarities of diffusion of the galactic cosmic rays at low energies

I give an interpretation of a result of Simpsonet al. (1988) on the variation with kinetic energyT _i of the mean pathlengthX _m (T _i ) of the galactic cosmic rays (CGRs) in the range 0.1T _i 10.0 GeV nucl^–1. I argue that the data onX _m (T _i ) may be interpreted in terms of a model of GCR diffusion on the one-dimensional Alfvén-wave turbulence, having a cutoff in the spectrum at frequencies _h , where _h is the proton gyrofrequency. The cutoff results in changing of the character of variation of the GCR diffusion coefficientD(T _i )T ^a in the rangeT _i 1 GeV nucl^–1 towards some more complicated variation at 0.1T _i 1.0 GeV nucl^–1 due to the peculiarities of the pitch-angle scattering at 90⁰.