M. S. Potgieter

A transport model for the diffusive shock acceleration and modulation of anomalous cosmic rays in the heliosphere

Standard transport theory is used to investigate the acceleration and modulation of anomalous cosmic rays during solar minimum periods. With the developed anomalous cosmic ray transport model, pickup ions are transformed into anomalous cosmic rays by diffusive shock acceleration at the heliospheric termination shock. The emphasis is on explaining the 1987 cosmic ray spectra at large radial distances in the equatorial plane. Energy density calculations of cosmic ray spectra at the termination shock suggest that the termination shock might be strongly mediated by anomalous cosmic rays. Therefore, in the spirit of nonlinear shock acceleration theory, the termination shock is modeled as a hyperbolic tangent. Calculations show that realistic anomalous cosmic ray intensity radial gradients can only be achieved, within the framework of our assumptions, for a termination shock structure with a compression ratio of 3.2 <s <4.0 and a scale length of ∼1.2 AU. In addition, simulations of anomalous and galactic cosmic ray spectra revealed as follows: (1) For radial distances of less than at least 24 AU from the Sun it should be generally difficult, although not impossible, to identify anomalous protons, for example, in the observed 1987 proton spectra. (2) Anomalous protons should be more easily detectable, however, in the proton spectra seen at 42 AU by Pioneer 10 during 1987 and should be clearly visible at larger radial distances. (3) The best way to identify anomalous protons in observed proton spectra is to look for a clear flattening in the spectral slopes in the energy interval of ∼20–100 MeV. Recent 1994 observations by Pioneer 10 at ∼60 AU and by Voyager 1 and Voyager 2 support the model prediction of very flat proton spectra in the above mentioned energy interval at large radial distances. Within the framework of our assumptions this might be an indication that the average termination shock position during 1994 was in the vicinity of 80 AU as assumed in the model.

MODELING THE MODULATION OF GALACTIC AND JOVIAN ELECTRONS BY STOCHASTIC PROCESSES

We present a newly developed numerical modulation model to study the transport of galactic and Jovian electrons in the heliosphere. The model employs stochastic differential equations (SDEs) to solve the corresponding transport equation in five dimensions (time, energy, and three spatial dimensions) which is difficult to accomplish with the numerical schemes used in finite difference models. Modeled energy spectra for galactic electrons are compared for the two drift cycles to observations at Earth. Energy spectra and radial intensity profiles of galactic and Jovian electrons are compared successfully to results from previous studies. In line with general drift considerations, it is found that most 100 MeV electrons observed at Earth enter the heliosphere near the equatorial regions in the A > 0 cycle, while they enter mainly over the polar regions in the A < 0 cycle. Our results indicate that 100 MeV electrons observed at Earth originate at the heliopause with ~600 MeV undergoing adiabatic cooling during their transport to Earth. The mean propagation time of these particles varies between ~180 and 300 days, depending on the drift cycle. For 10 MeV Jovian electrons observed at Earth, a mean propagation time of ~40 days is obtained. During this time, the azimuthal position of the Jovian magnetosphere varies by ~1°. At a 50 AU observational point, the mean propagation time of these electrons increases to ~370 days with an azimuthal position change of Jupiter of ~20°. The SDE approach is very effective in calculating these propagation times.

The simulated features of heliospheric cosmic-ray modulation with a time-dependent drift model. I: General effects of the changing neutral sheet over the period 1985-1990

A time-dependent drift model is used to simulate the heliospheric modulation of galactic cosmic rays, with emphasis on the effects of the wavy heliospheric neutral sheet (HNS) as a function of time during the period 1985-1990. The model predicted a clearly defined time difference between the time when minimum modulation occurred at earth and at radial distances farther away from the sun, indicating that the HNS plays an important part in establishing this observed time delay. The movements of the Voyager 1 and 2 and Pioneer 10 spacecraft were simulated in order to calculate instantaneous radial and latitudinal gradients. The time dependencies of these gradients were found to follow the observations well. The model study indicates that the HNS cannot be ignored, especially during periods of low to moderate solar activity, when the modulation of cosmic rays is described.

The modelling of the latitude dependence of cosmic ray protons and electrons in the inner heliosphere

Abstract Observations during the fast latitude scan of Ulysses show that the latitude dependence of cosmic ray protons is significantly less than predicted by drift models which were based on the Parker geometry for the heliospheric magnetic field (HMF) and diffusion coefficients ∞ 1/Bp. Recent modeling with a modified HMF and enhanced turbulence in the polar zones of the heliosphere can reproduce the observed latitude dependence for protons reasonably well at high rigidities. It is shown here that incorporating enhanced perpendicular diffusion in the polar direction in a drift model also gives a latitude dependence for protons, over a wide range of rigidities, and a latitude dependence for electrons and therefore a charge-dependence that are compatible with observations.

A Cosmic-Ray Positron Anisotropy due to Two Middle-Aged, Nearby Pulsars?

Geminga and B0656+14 are the closest pulsars with characteristic ages in the range of 100 kyr to 1 Myr. They both have spin-down powers of the order 3 × 1034 ergs s−1 at present. The winds of these pulsars had most probably powered pulsar wind nebulae (PWNe) that broke up less than about 100 kyr after the birth of the pulsars. Assuming that leptonic particles accelerated by the pulsars were confined in the PWNe and were released into the interstellar medium (ISM) on breakup of the PWNe, we show that, depending on the pulsar parameters, both pulsars make a nonnegligible contribution to the local cosmic ray (CR) positron spectrum, and they may be the main contributors above several GeV. The relatively small angular distance between Geminga and B0656+14 thus implies an anisotropy in the local CR positron flux at these energies. We calculate the contribution of these pulsars to the locally observed CR electron and positron spectra depending on the pulsar birth period and the magnitude of the local CR diffusion coefficient. We further give an estimate of the expected anisotropy in the local CR positron flux. Our calculations show that within the framework of our model, the local CR positron spectrum imposes constraints on pulsar parameters for Geminga and B0656+14, notably the pulsar period at birth, and also the local interstellar diffusion coefficient for CR leptons.

A stochastic differential equation code for multidimensional Fokker-Planck type problems

Abstract We present a newly developed numerical code that integrates Fokker–Planck type transport equations in four to six spatial dimensions (configuration plus momentum space) and time by means of stochastic differential equations. In contrast to other, similar approaches our code is not restricted to any special configuration or application, but is designed very generally with a modular structure and, moreover, allows for Cartesian, cylindrical or spherical coordinates. Depending on the physical application the code can integrate the equations forward or backward in time. We exemplify the mathematical ideas the method is based upon and describe the numerical realisation and implementation in detail. The code is validated for both cases against an established finite-differences explicit numerical code for a scenario that includes particle sources as well as a linear loss term. Finally we discuss the new possibilities opened up with respect to general applications and newly developed hardware.

The role of merged interaction regions and drafts in the heliospheric modulation of cosmic rays beyond 20 AU - A computer simulation

The time-dependent, axially symmetric transport equation of cosmic rays in the heliosphere was solved numerically using the Voyager 2 magnetic field measurements to simulate merged interaction regions (MIRs) and rarefaction regions (RRs) for the period 1985-1989. Concurrently, the wavy neutral sheet was used as a time-dependent, drift parameter. The two approaches were first modeled separately and then in combination in this exploratory study. The drift approach proved to be the more successful for 1985 to early 1988 because it could reproduce the intensity levels, the factor modulation, and latitudinal gradients for 1 GeV protons at 23 AU without difficulties

A comparison of predictions of a wavy neutral sheet drift model with cosmic-ray data over a whole modulation cycle - 1976-1987

A comprehensive set of predictions using a newly developed, axially symmetric, steady state, drift model with a wavy neutral sheet is compared with data obtained during an entire 11 yr solar modulation cycle from 1976 to 1987. The model accurately predicts the 1976 modulated proton and helium spectra starting with realistic interstellar spectra for these species, but only when drift effects are somewhat reduced. For 1981-1987, the proton intensities with energy of 60 MeV and higher and E = 120-227 MeV observed by IMP, and those predicted by this model as a function of neutral sheet tilt agree fairly well. These results suggest that the correlation between intensity and the observed tilt reported by other groups has a theoretical explanation in terms of a drift and a wavy neutral sheet picture. The model can explain many of the observed features of the 11 yr cycle, showing that drift together with the wavy neutral sheet probably play a significant role. 21 refs.

On the local interstellar spectrum for cosmic ray electrons

Abstract Galactic propagation models for cosmic ray electrons give a synchrotron spectral index larger than the recently determined radio index between 22 – 408 MHz in the direction of the galactic disk (Roger , 1999), and smaller than the radio index between 0.5 – 2000 MHz in the direction of the galactic poles (Peterson , 1999). Diffuse gamma-ray data appear to be ‘contaminated’ by Crab-like point sources, so that it is difficult to derive a consistent local interstellar spectrum (IS) for electrons in the 1 to 30 MeV range. Using a phenomenological approach, we show that the synchrotron spectral indices calculated from the best-fit IS of Strong (2000) - for a 3 μG, and 5 μG, interstellar magnetic field - agree well with the spectral indices calculated with their full propagation model in the frequency range of interest. This allowed us to introduce two adjusted IS, such that the model radio spectral index agrees with observations of the galactic disk- and polar approaches above and below 20 MHz. By adding the constraints expected from the heliospheric modulation of galactic electrons, we find that the IS obtained by the ‘galactic disk approach’ is marginally above the lower limit for a local IS set by Pioneer 10 electron data at ∼4 MeV and ∼16 MeV observed in the outer heliosphere. The ‘polar approach’ gives an IS which can be considered a reasonable local IS for cosmic ray electrons.

Modulation effects of anisotropic perpendicular diffusion on cosmic ray electron intensities in the heliosphere

The modulation of cosmic ray electrons provides a useful tool to study the diffusion tensor applicable to heliospheric modulation. Electron modulation responds directly to the assumed energy dependence of the diffusion coefficients below ∼500 MeV in contrast to protons which experience large adiabatic energy losses below this energy. As a result of this and because drifts become unimportant for electrons at these low energies, conclusions can be made about the appropriate diffusion coefficients. Using a modulation model, we illustrate the role of anisotropic perpendicular diffusion on electron modulation. In general, we find that perpendicular diffusion dominates electron modulation below ∼100 MeV. Enhancing it in the polar direction typically produced an increase in modulation for both the A > O (e.g., ∼1990 to ∼2000) and A < 0 (e.g., ∼1980 to ∼1990) solar magnetic polarity cycles. It also causes the radial dependence of the intensity to become more uniform throughout the heliosphere, and causes a significant reduction in the latitude dependence of the intensities at all radial distances, with the largest effects in the inner heliosphere and at low energies. This agrees with studies of cosmic ray protons, which suggest that perpendicular diffusion enhanced in the polar direction of the heliosphere is required in conventional drift models to explain the small latitudinal gradients observed for protons on board the Ulysses spacecraft. The role of enhanced perpendicular diffusion was further investigated by examining electron modulation as a function of the “tilt angle” α of the wavy current sheet. In general, a reduction occurred between the modulation differences caused by drifts as a function of α for both polarity cycles. This work illustrates that anisotropic perpendicular diffusion has profound effects on the modulation of galactic cosmic ray electrons during both polarity cycles.