The rigidity-dependent delay time of the galactic cosmic ray modulation with respect to the open solar magnetic flux

¹ National Key Laboratory of Deep Space Exploration/School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China
² Department of Physics and Astronomy, University of Hawaii at Manoa, Honolulu, HI 96822, USA
³ Department of Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109, USA
⁴ Institute of Experimental and Applied Physics, Christian-Albrechts-University, Kiel, Germany

^★ Corresponding author: jnguo@ustc.edu.cn

Received: 9 September 2025
Accepted: 19 October 2025

Abstract

Context. Studying the transport of galactic cosmic rays (GCRs) is crucial for understanding the space radiation environment and large-scale heliospheric structures. Various numerical, observational, and theoretical studies have demonstrated that GCR fluxes are modulated by the interplanetary magnetic field (IMF), which evolves with the solar cycle. However, there are still open questions on how different modulation processes, and their dependence on the IMF, impact the GCR transport in the heliosphere. In particular, we still do not fully understand how GCR time variations lag behind solar activity changes, referred to as GCR delay time in this study.

Aims. We aim to parameterize the GCR delay time with respect to several solar activity indices and determine how this delay changes with particle rigidity, thereby contributing to a better understanding of GCR modulation in the heliosphere.

Methods. Based on long-term GCR observations with the SOlar and Heliospheric Observatory (SOHO) telescope, the Interplanetary Monitoring Platform-8 (IMP-8), and the Alpha Magnetic Spectrometer (AMS-02), we used the force-field approximation to derive an analytical formula for estimating the GCR modulation delay. We then applied information theory to quantify the GCR modulation delay innovatively and employed Monte Carlo methods to evaluate its uncertainty.

Results. Consistent with previous findings, we confirm GCRs have a longer delay time for qA < 0 than qA > 0, where q is the GCR particle charge and A = 1 (or −1) if the solar magnetic field is predominantly outward (or inward) at the solar north pole. For protons with a rigidity of 0.8 GV or higher, the modulation delay time gradually decreases from 7–12 months to 2–3 months as rigidity increases and then remains constant, which can be explained by the finite propagation speed of solar activity information within the heliosphere.

Conclusions. We formulate a rigidity-dependent expression for the GCR modulation delay using the force-field approximation and assess its applicability through observational analysis grounded in information theory. These findings offer new insights into the heliospheric transport of GCRs.