Statistical analysis of solar energetic particle rise times using Earth and Mars observations and constraints on particle transport parameters

¹ Deep Space Exploration Laboratory/School of Earth and Space Sciences, University of Science and Technology of China, Hefei, PR China
² Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China
³ National Key Laboratory of Space Environment Interaction with Matters, Lanzhou Institute of Physics, Lanzhou, China

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 31 March 2026
Accepted: 25 April 2026

Abstract

Context. The propagation of solar energetic particles (SEPs) in interplanetary space is modulated by turbulence caused by the solar wind, which significantly affects the particle diffusion and energy evolution through scattering processes. However, traditional analyses based on absolute flux measurements face inherent difficulties in separating the effects of particle acceleration at the source from subsequent transport processes. In contrast, the temporal features of SEPs, such as the onset and peak times, are less affected by these uncertainties and are therefore more suitable for exploring the transport process of SEPs.

Aims. We establish a statistical relation between the rise time of SEP events at different energies using multi-satellite observations at Earth and Mars. The results are used to further invert the turbulence parameters affecting the transport of SEPs based on a parallel diffusion model, thereby revealing the characteristics and evolutionary trend of turbulence at different radial distances.

Methods. Using data from SOHO/ERNE and Tianwen-1/MEPA between November 2020 and March 2025, we selected 75 SEP events at 1 AU and 58 events near Mars. For each energy range of each event, the onset time was determined by a linear fitting method, and the peak time was extracted using a sliding median filter combined with Savitzky-Golay smoothing methods. The difference between the onset time and the peak time subsequently determines the SEP rise time. Together with the theoretical relation of rise time and its dependence on energy derived from the pure diffusion equation, we compared the statistical behavior of the rise time between Earth and Mars.

Results. Despite uncertainties introduced by the selection criteria of SEP events, the rise time of SEP events, Δt in minutes, follows a clear power-law relation with energy (Δt ∝ E^η), with η = −0.368 ± 0.058 averaged for SEP events near Earth and η = −0.302 ± 0.031 averaged for events near Mars. The flatter power-law shape at Mars suggests that the dependence of the rise time on energy weakens with increasing radial distance from the Sun. Based on these empirical relations, we constrain the rigidity dependence of the parallel mean free path of SEPs within the parallel diffusion model. Based on the simple assumption of a pure diffusion model, our results indicate that turbulence scattering of energetic particles at Mars approaches a rigidity-independent regime, reflecting an evolutionary trend of turbulence toward a dissipation-dominated state from Earth to Mars.