Chaotic Behavior of Trapped Cosmic Rays

TL;DR

This work demonstrates that chaotic trapping of cosmic-ray trajectories in a magnetically mirroring, bounded environment, inspired by heliospheric structures, can be quantitatively characterized using finite-time Lyapunov exponents. The authors introduce a toy magnetic bottle with time-dependent perturbations and compute FTLEs for many particle trajectories, revealing a robust power-law relation $\lambda_{FTLE} \sim t_{esc}^{-1.04}$ with escape time $t_{esc}$, persisting under perturbations. The results yield a physically interpretable classification of particle behaviors and generate sky maps showing how chaotic dynamics imprint distinct regions in arrival directions. The study connects chaotic transport to observable CR anisotropy, suggesting time variability driven by heliospheric dynamics and motivating future work with realistic heliospheric models to interpret IceCube/HAWC data. Overall, FTLE provides a flexible, transitory-measure of chaos applicable to cosmic-ray propagation in diverse magnetic structures beyond simple diffusion.

Abstract

Recent experimental results on the arrival direction of high-energy cosmic rays have motivated studies to understand their propagating environment. The observed anisotropy is shaped by interstellar and local magnetic fields. In coherent magnetic structures, such as the heliosphere, or due to magnetohydrodynamic turbulence, magnetic mirroring can temporarily trap particles, leading to chaotic behavior. In this work, we develop a new method to characterize cosmic rays' chaotic behavior in magnetic systems using finite-time Lyapunov exponents. This quantity determines the degree of chaos and adapts to transitory behavior. We study particle trajectories in an axial-symmetric magnetic bottle to highlight mirroring effects. By introducing time-dependent magnetic perturbations, we study how temporal variations affect chaotic behavior. We tailor our model to the heliosphere; however, it can represent diverse magnetic configurations exhibiting mirroring phenomena. Our results have three key implications. (1)Theoretical: We find a correlation between the finite-time Lyapunov exponent and the particle escape time from the system, which follows a power law that persists even under additional perturbations. This power law may reveal intrinsic system characteristics, offering insight into propagation dynamics beyond simple diffusion. (2)Simulation: Chaotic effects play a role in cosmic ray simulations and can influence the resulting anisotropy maps. (3)Observational: Arrival maps display areas where the chaotic properties vary significantly; these changes can be the basis for time variability in the anisotropy maps. This work lays the framework for studying the effects of magnetic mirroring of cosmic rays within the heliosphere and the role of temporal variability in the observed anisotropy.

Figures (11)

• Figure 1: The magnetic bottle field geometry used as toy model to study the behavior of particles trapped by the interstellar magnetic field draping around the heliosphere. On the left, the static magnetic field, and on the right, with the additional perturbation imitating the effects of solar cycles on the heliospheric magnetic field along its tail.
• Figure 2: On the left, the field profile along the axis of the magnetic bottle with the weak and strong perturbations at their maximum amplitude. On the right, the 3D view of a snapshot of the magnetic perturbation.
• Figure 3: Method for the injection of particles. We introduce our reference particles starting at position ($x_o$,$y_o$,$z_o$)=(100, 100, 500) and with initial momentum in the direction of the 768 pixels in the map, which correspond to each pixel in the HealPix grid with $nside = 8$ (Figure on the left modified from Gorski_99). This figure shows an example of a reference particle denoted by a red dot, and the light blue sphere represents the possible momenta directions. For each reference particle (red dot), we have a set of 10 particles injected randomly on the surface of a sphere of radius $r$ = 0.01. On the right, we have an example of a reference particle with one of the particles in its family (blue dot). These two particles are separated a distance of 0.01 and have identical momenta. The main idea behind this process is to inject particles with almost identical initial conditions to track how chaotic these trajectories are.
• Figure 4: Trajectories in the unperturbed system. Top Left: Transient particle with a final time of 33000. Top Right: Intermediate particle with a final time of 75402. Bottom Left: Irregular particle in the power-law behavior section with a final time of 295366. Bottom Right: Trapped particle with the maximum integration time. The transient and trapped particles do not display chaotic behavior, whereas the intermediate and power-law behavior particles are chaotic.
• Figure 5: Comparison between the behavior of two particles with different escape times. Top Panels: These correspond to a particle with an escape time $t_{esc}= 97251$. Bottom Panels: Particle with an escape time $t_{esc}= 5.7 \times 10^5$. Panels (a) and (d) show distance in phase space vs. time; panels (b) and (e) show distance in phase space at time $t + \Delta T$ over the distance at time $t$ as a function of time $t$; panels (c) and (f) display a histogram of the finite-time Lyapunov exponent $\lambda_{\tiny{FTLE}}$ and the corresponding fits denoted with black, red, and blue lines; see \ref{['sec:method']} for details on the Gaussian fits. Note that for the shorter trajectories (a), they stay with almost no separation for a short time and then diverge rapidly and leave the system right away. On the contrary, longer trajectories (d) take longer to start diverging and when they do, the process takes a longer time with intermediate periods of slower divergence before they are able to escape.