Through the Heliospheric Lens: Directional Deflection of High-Energy Cosmic-Ray Electrons and Positrons

TL;DR

This study analyzes how the large-scale heliospheric magnetic field deterministically bends high-energy cosmic-ray electrons and positrons, developing a modular back-tracing framework to quantify the heliospheric contribution to arrival directions. By comparing Parker spiral, DQCS-based topologies, and spiral-augmented variants with tilt/waviness of the heliospheric current sheet, the authors map energy- and direction-dependent deflections and provide practical thresholds, $E_{ m crit}( heta_{ m inst})$, to decide when heliospheric bending can be neglected. Key findings show that most bending accumulates in the inner tens of AU and that deflection scales approximately as $\langle\Delta\theta\rangle \propto 1/E$ with substantial model- and solar-cycle-dependent normalization, while charge-sign effects are strongest at sub-TeV energies and wash out at multi-TeV scales. The work offers actionable guidance for CRE anisotropy analyses and source matching, stressing the need to bracket results with a small set of field configurations and to consider down-weighting HCS-affected regions at low energies. It lays a foundation for end-to-end, calendar-resolved predictions by incorporating time dependence, turbulence, and a geomagnetic leg in future extensions, thereby enabling robust interpretation of CRE directional measurements across solar conditions.

Abstract

We investigate how the large-scale heliosphere alters the arrival directions of high-energy cosmic-ray electrons and positrons and ask if and when this "heliospheric lens" can be ignored for anisotropy and source-association studies - an especially timely topic given, for instance, the persistent cosmic-ray positron fraction and its unknown origin. Using a modular back-tracing framework, we explore a set of widely used magnetic-field descriptions-from a Parker spiral baseline to more structured configurations that include latitudinal wind contrasts, Smith-Bieber-type azimuthal strengthening, and tilted or wavy heliospheric current sheets. We model the deterministic deflections of high-energy cosmic-ray electrons and positrons (CREs) induced by large-scale heliospheric magnetic-field structures using a back-tracing approach. Our results apply to CREs above tens of GeV, where diffusion, convection, and adiabatic energy losses play a subdominant role; these processes are neglected in the present study and will be addressed in future work. Across these models the picture is consistent: most bending is accumulated within the inner tens of astronomical units and decreases rapidly with energy. Field choices and solar-cycle geometry set the overall normalization, with stronger spiral winding or a more highly tilted current sheet producing larger deflections at the same energy. Differences between electrons and positrons are most apparent at lower energies, where drift histories and current-sheet encounters diverge, and become increasingly small at multi-TeV energies. [...]

Figures (9)

• Figure 1: (a): Fractional numerical error on the sky--averaged deflection for the Parker unipolar reference model at fixed outer radius $r_{\rm stop}=50$ AU. We show $(D - D_\infty)/D_\infty$ as a function of the number of integration steps per gyration for four representative energies ($E = 50, 500, 5\times10^3$, and $5\times10^4$ GeV), where $D_\infty$ is estimated from the highest–resolution run (${\rm steps/gyro}=3200$). All curves are computed with 64 deterministic sky directions. The shaded horizontal bands mark target tolerances of $|\Delta D|/D_\infty <0.5\%$ (dark band) and $<1.5\%$ (light band). For ${\rm steps/gyro}\gtrsim 1300$ the fractional deviation for all energies lies well within the $1.5\%$ band, demonstrating stable convergence of the integrator at the resolutions adopted for our production runs. (b): mean deflection versus outer boundary $r_{\rm stop}$ at fixed resolution. Configuration: DQCS$+$spiral with a wavy HCS ($m=1$, tilted–sheet polarity), $N=100$ random sky directions per point. Error bars are omitted for clarity; trends are robust to the number of rays.
• Figure 2: Baseline Parker spiral with $B_{r}\xspace(1\,\mathrm{AU}\xspace)=5$ nT, $V_{\mathrm{sw}}\xspace=400$ km s$^{-1}$, $\Omega_\odot=2.865\times10^{-6}$ rad s$^{-1}$. Each panel shows the mean heliospheric deflection $\langle\Delta\theta\rangle$ (solid line) and the 16--84% percentile band (shaded) over 100 isotropically sampled arrival directions, as a function of energy from 10 GeV to 30 TeV. Note that at energies $\lesssim$10 GeV), the effective dispersion is limited by our back-tracing cutoff at 50 AU, as discussed in the text.
• Figure 3: Sky maps of the heliospheric deflection angle (degrees; Mollweide projection, $60\times120$ grid) for the baseline Parker spiral with $B_{r}\xspace(1\,\mathrm{AU}\xspace)=5$ nT, $V_{\mathrm{sw}}\xspace=400$ km s$^{-1}$, and unipolar polarity. Each panel shows the angular deviation between the local arrival direction at $1$ AU and the asymptotic direction at $r_{\rm stop}=50$ AU for electrons, computed direction–by–direction with identical numerical settings (steps-per-gyro $=80$). Color scales are per–panel and labeled in degrees -- note especially the significantly different scale for the bottom, right panel. Note that the maps are displayed in heliocentric ecliptic coordinates, with the $(0,0)$ direction corresponding to the observer’s line of sight (i.e., the Sun–Earth axis), see the text for details.
• Figure 4: Mean heliospheric deflection and 16--84% ranges for six representative configurations. All runs use identical numerics (100 sky directions per energy, $r_{\rm stop}=50$ AU, steps-per-gyro $=80$) and the same normalization of $\langle|B_r(1\,\mathrm{AU}\xspace)|\rangle$ and $V_{\rm sw}$.
• Figure 5: Overlay of mean deflection curves for the Parker baseline, DQCS(core), and the DQCS + spiral family (flat HCS, Smith--Bieber, and wavy HCS variants including $m=2$). Same normalization and numerics as in Fig. \ref{['fig:model_comparisons_bands']}.