Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Particle acceleration at astrophysical shocks: A theory of cosmic ray origin

Pulsar surveys offer a rare opportunity to monitor the radio sky for impulsive burst-like events with millisecond durations. We analyzed archival survey data and found a 30-jansky dispersed burst, less than 5 milliseconds in duration, located 3 degrees from the Small Magellanic Cloud. The burst properties argue against a physical association with our Galaxy or the Small Magellanic Cloud. Current models for the free electron content in the universe imply that the burst is less than 1 gigaparsec distant. No further bursts were seen in 90 hours of additional observations, which implies that it was a singular event such as a supernova or coalescence of relativistic objects. Hundreds of similar events could occur every day and, if detected, could serve as cosmological probes.

https://science.sciencemag.org/content/sci/318/5851/777.full.pdf

A Bright Millisecond Radio Burst of Extragalactic Origin

The central idea of diffusive shock acceleration is presented from microscopic and macroscopic viewpoints; applied to reactionless test particles in a steady plane shock the mechanism is shown to produce a power law spectrum in momentum with a slope which, to lowest order in the ratio of plasma to particle speed, depends only on the compression in the shock. The associated time scale is found (also by a macroscopic and a microscopic method) and the problems of spherical shocks, as exemplified by a point explosion and a stellar-wind terminator, are treated by singular perturbation theory. The effect of including the particle reaction is then studied. It is shown that if the scattering is due to resonant waves these can rapidly grow with unknown consequences. The possible steady modified shock structures are classified and generalised Rankine-Hugoniot conditions found. Modifications of the spectrum are discussed on the basis of an exact, if rather artificial, solution, a high-energy asymptotic expansion and a perturbation expansion due to Blandford. It is pointed out that no steady solution can exist for very strong shocks; the possible time dependence is briefly discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/46/8/002/pdf

An introduction to the theory of diffusive shock acceleration of energetic particles in tenuous plasmas

We survey the theory and experimental tests for the propagation of cosmic rays in the Galaxy up to energies of 10 15 eV. A guide to the previous reviews and essential literature is given, followed by an exposition of basic principles. The basic ideas of cosmic-ray propagation are described, and the physical origin of its processes is explained. The various techniques for computing the observational consequences of the theory are described and contrasted. These include analytical and numerical techniques. We present the comparison of models with data, including direct and indirect—especially γ-ray—observations, and indicate what we can learn about cosmic-ray propagation. Some important topics, including electron and antiparticle propagation, are chosen for discussion.

/pdf/cosmic-ray-propagation-and-interactions-in-the-galaxy-52wrtrud7h.pdf

Cosmic-Ray Propagation and Interactions in the Galaxy

Cosmic-ray propagation. i. charged particles in a random magnetic field.

We present a new analysis of the transport of cosmic rays in a turbulent magnetic field that varies in all three spatial dimensions. The analysis utilizes a numerical simulation that integrates the trajectories of an ensemble of test particles from which we obtain diffusion coefficients based on the particle motions. We find that the diffusion coefficient parallel to the mean magnetic field is consistent with values deduced from quasi-linear theory, in agreement with earlier work. The more interesting and less understood transport perpendicular to the average magnetic field is found to be enhanced (above the classical scattering result) by the random walk, or braiding, of the magnetic field. The value of κ⊥ obtained is generally larger than the classical scattering value but smaller than the quasi-linear value. The computed values of κ⊥/κ∥, for a representation of the interplanetary magnetic field, are 0.02-0.04; these values are of the same general magnitude as those assumed in recent numerical simulations of cosmic-ray modulation and transport in the heliosphere, and give reasonable agreement with spacecraft observations of cosmic rays. Some consequences of these results for the interpretation of heliospheric observations are discussed.

https://iopscience.iop.org/article/10.1086/307452/pdf

The Transport of Cosmic Rays across a Turbulent Magnetic Field

The problem of diffusive shock acceleration of fast charged particles is reexamined with emphasis on the rate of energy gain, and the maximum energy which can be attained in a given circumstance. The direction of the average magnetic field at the shock is shown to have a large effect. If the perpendicular diffusion coefficient is much smaller than the parallel coefficient, particles can gain much more energy if the shock is quasi-perpendicular than if it is quasi-parallel. The maximum energy attainable can be substantially higher (by a factor of 100 or more) than previous discussions would predict, in cases where the shock is quasi-perpendicular. The energy gain increases as kappa-perpendicular decreases. The principal limitation comes from the requirement that diffusion be a valid approximation to the particle motion, and that the particle be able to diffuse fast enough to encounter the shock many times.

Rate of energy gain and maximum energy in diffusive shock acceleration

This paper presents a coherent exposition of the modern statistical theory of the transport of fast charged particles (cosmic rays) in the solar wind. Observations are discussed only as they illustrate the phenomena under discussion. A brief introductory section surveys the historical development of the theory. The dominant effect on the motion of cosmic rays in the solar wind is the interplanetary magnetic field, which is irregular and which is therefore best treated statistically, using random functions. The magnetic irregularities scatter the cosmic rays in pitch angle, so that to a good approximation the cosmic rays diffuse through the irregular magnetic field. Using a statistical analysis of the equations of motion, one may relate the diffusion tensor to the power spectrum of the magnetic field, which is in principle measurable. The resulting general transport theory relates the motion of cosmic rays, statistically, to the solar-wind velocity and magnetic field. Application of the theory both to the modulation of galactic cosmic rays by the solar wind and to the propagation of solar cosmic rays is discussed in detail. It is concluded that the present theory explains the principal phenomena quite well. Future theoretical work will probably be devoted to obtaining better solutions of the equations, to obtaining better values of the parameters, and to studying higher-order or more subtle effects.

Propagation of cosmic rays in the solar wind

Although gradient and curvature drifts are explicitly contained in the general equations of cosmic-ray transport, they have been almost universally neglected in applications of these equations. The drifts are evaluated explicitly for the Parker (1965) spiral magnetic field, and it is shown that for particles with rigidities greater than about 0.3 GV in the solar wind, they are larger than the solar-wind velocity over much of the heliosphere. Hence most current models of solar modulation and solar-flare particle events neglect terms which in many cases are as important as those retained. Calculations are presented which demonstrate the importance of the effects for simple modulation models. It is concluded that comparisons of presently available model calculations with observations do not provide a fair test of transport theory since they neglect drifts, which may substantially reduce the net modulation.

J. R. Jokipii

Papers

Cosmic-ray propagation. i. charged particles in a random magnetic field.

The Transport of Cosmic Rays across a Turbulent Magnetic Field

Rate of energy gain and maximum energy in diffusive shock acceleration

Propagation of cosmic rays in the solar wind

Effects of particle drift on cosmic-ray transport. I. General properties, application to solar modulation