A. Kopp

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.

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.

On the propagation times and energy losses of cosmic rays in the heliosphere

[1] We present calculations of the propagation times and energy losses of cosmic rays as they are transported through the heliosphere. By calculating these quantities for a spatially 1D scenario, we benchmark our numerical model, which uses stochastic differential equations to solve the relevant transport equation, with known analytical solutions. The comparison is successful and serves as a vindication of the modeling approach. A spatially 3D version of the modulation model is subsequently used to calculate the propagation times and energy losses of galactic electrons and protons in different drift cycles. We find that the propagation times of electrons are longer than those of the protons at the same energy. Furthermore, the propagation times are longer in the drift cycle when the particles reach the Earth by drifting inward along the heliospheric current sheet. The calculated energy losses follow this same general trend. The energy losses suffered by the electrons are comparable to those of the protons, which is in contrast to the generally held perception that electrons experience little energy losses during their propagation through the heliosphere.

The effects of magnetic field modifications on the solar modulation of cosmic rays with a SDE-based model

Abstract A numerical model for the solar modulation of cosmic rays, based on the solution of a set of stochastic differential equations (SDEs), is used to illustrate the effects of modifying the heliospheric magnetic field, particularly in the polar regions of the heliosphere. SDE-based models are well suited for such studies so that new insights are gained. To this end, the differences in the modulation brought about by each of three choices for the heliospheric magnetic field, i.e. the unmodified Parker field, the Smith–Bieber modified field, and the Jokipii–Kota modified field, are studied as typical well-known cases. It is illustrated that although both these modifications change the Parker field satisfactorily in the polar regions of the heliosphere, the Smith–Bieber modification is more effective in reducing cosmic ray drift effects in these regions. The features of these two modifications, as well as the effects on the solar modulation of cosmic rays, are illustrated qualitatively and quantitatively. In particular, it is shown how the Smith–Bieber modified field is applied in a cosmic ray modulation model to reproduce observational proton spectra from the PAMELA mission during the solar minimum of 2006–2009. These SDE-based results are compared with those obtained in previous studies of this unusual solar minimum activity period and found to be in good qualitative agreement.

Cosmic-ray positrons from millisecond pulsars

Observations by the Fermi Large Area Telescope of gamma-ray millisecond pulsar light curves imply copious pair production in their magnetospheres, and not exclusively in those of younger pulsars. Such pair cascades may be a primary source of Galactic electrons and positrons, contributing to the observed enhancement in positron flux above ~10 GeV. Fermi has also uncovered many new millisecond pulsars, impacting Galactic stellar population models. We investigate the contribution of Galactic millisecond pulsars to the flux of terrestrial cosmic-ray electrons and positrons. Our population synthesis code predicts the source properties of present-day millisecond pulsars. We simulate their pair spectra invoking an offset-dipole magnetic field. We also consider positrons and electrons that have been further accelerated to energies of several TeV by strong intrabinary shocks in black widow and redback systems. Since millisecond pulsars are not surrounded by pulsar wind nebulae or supernova shells, we assume that the pairs freely escape and undergo losses only in the intergalactic medium. We compute the transported pair spectra at Earth, following their diffusion and energy loss through the Galaxy. The predicted particle flux increases for non-zero offsets of the magnetic polar caps. Pair cascades from the magnetospheres of millisecond pulsars are only modest contributors around a few tens of GeV to the lepton fluxes measured by AMS-02, PAMELA, and Fermi, after which this component cuts off. The contribution by black widows and redbacks may, however, reach levels of a few tens of percent at tens of TeV, depending on model parameters.

The millisecond pulsar contribution to the rising positron fraction

Pair cascades from millisecond pulsars (MSPs) may be a primary source of Galactic electrons and positrons that contribute to the increase in positron flux ab ove 10 GeV as observed by PAMELA and AMS−02. The Fermi Large Area Telescope (LAT) has increased the number of detected γ-ray MSPs tremendously. Light curve modelling furthermore favours abundant pair production in MSP magnetospheres, so that models of primary cosmic-ray positrons from pulsars should include the contribution from the larger numbers of MSPs and their potentially higher positron output per source. We model the contribution of Galactic MSPs to the terrestrial cosmic-ray electron / positron flux by using a population synthesis code to predict the source properties of presentday MSPs. We simulate pair spectra assuming an offset-dipole magnetic field which boosts pair creation rates. We also consider positrons and electrons th at have additionally been accelerated to very high energies in the strong intrabinary shocks in black widow (BW) and redback (RB) binary systems. We transport these particles to Earth by calculati ng their diffusion and the radiative energy losses they suffer in the Galaxy using a model. Our model particle flux increases for nonzero offsets of the magnetic polar caps. We find that pair casc ades from MSP magnetospheres contribute only modestly around a few tens of GeV to the measured fluxes. BW and RB fluxes may reach a few tens of percent of the observed flux up to a few Te V. Future observations should constrain the source properties in this case.

The Contribution of Millisecond Pulsars to the Galactic Cosmic-Ray Lepton Spectrum

Abstract Pulsars are believed to be sources of relativistic electrons and positrons. The abundance of detections of γ -ray millisecond pulsars by Fermi Large Area Telescope coupled with their light curve characteristics that imply copious pair production in their magnetospheres, motivated us to investigate this old pulsar population as a source of Galactic electrons and positrons and their contribution to the enhancement in cosmic-ray positron flux at GeV energies. We use a population synthesis code to predict the source properties (number, position, and power) of the present-day Galactic millisecond pulsars, taking into account the latest Fermi and radio observations to calibrate the model output. Next, we simulate pair cascade spectra from these pulsars using a model that invokes an offset-dipole magnetic field. We assume free escape of the pairs from the pulsar environment. We then compute the cumulative spectrum of transported electrons and positrons at Earth, following their diffusion and energy losses as they propagate through the Galaxy. Our results indicate that the predicted particle flux increases for non-zero offsets of the magnetic polar caps. Comparing our predicted local interstellar spectrum and positron fraction to measurements by AMS-02, PAMELA, and Fermi, we find that millisecond pulsars are only modest contributors at a few tens of GeV, after which this leptonic spectral component cuts off. The positron fraction is therefore only slightly enhanced above 10xa0GeV relative to a background flux model. This implies that alternative sources such as young, nearby pulsars and supernova remnants should contribute additional primary positrons within the astrophysical scenario.