Three-dimensional properties of a coronal shock and the longitudinal distribution of its related solar energetic particles

Context. The widespread longitudinal distribution of solar energetic particles (SEPs) is influenced by magnetic connectivity from the observers to coronal mass ejection (CME)-driven shocks. This connectivity determines shock properties encountered by magnetic-field lines, which in turn regulate the initial particle injection and acceleration efficiency. Aims. We aim to investigate the relationship between the spatial–temporal evolution of shock properties and the longitudinal dependence of SEP intensities and spectra. Methods. The shock parameters, including the normal speed, oblique angles, compression ratio, and Alfvén Mach number, were derived by combining a steady-state solar-wind simulation with the three-dimensional (3D) reconstruction of the shock surface based on multi-view observations. We compared the local shock parameters at the magnetic connecting points with in situ proton intensities and peak spectra to establish the link between shock evolution and SEP characteristics. Results. The shock nose consistently exhibited higher particle-acceleration efficiency with the largest normal speed, compression ratio, and supercritical Alfvén Mach number, while the flanks showed delayed transition to supercritical Alfvén Mach number with weaker efficiency. The earliest and most rapid proton enhancement of STEREO-B correlated with efficient shock acceleration and prompt magnetic connectivity to the shock. Spectral analysis revealed that proton energy spectra were consistent with the relativistic diffusive shock acceleration (DSA) estimations. Conclusions. The initial shock acceleration began at about 1.4 ∼ 5 R ⊙ and caused the widespread longitudinal SEP distribution. The longitudinal dependence of SEP intensity and spectral variations arise from the combined influence of 3D shock properties, magnetic connectivity, and particle transport processes. The agreement between in situ proton indices and relativistic DSA estimations supports DSA in this SEP event and provides insights into the early-stage acceleration at the source region.