Radial dependence of solar energetic particle peak fluxes and fluences

Context. We present a list of solar energetic particle (SEP) events detected by instruments on board the Solar and Heliospheric Observatory (SOHO), Parker Solar Probe (PSP), and Solar Orbiter between 2021 and 2023. The investigation focuses on identifying the peak flux and the fluence of SEP events in four energy ranges from 10.5 to 40 MeV, as observed by PSP or Solar Orbiter at heliospheric distances shorter than 1 AU and by SOHO at the Sun-Earth L1 Lagrangian point.Aims. Based on the data from these events, we conduct a statistical analysis to study the radial dependence of the SEP proton peak flux and fluence at different energies.Methods. We identified 42 SEP events with enhanced proton flux that were observed simultaneously by at least two out of three spacecraft (SOHO, PSP, and Solar Orbiter). These events were further selected based on a criterion of a difference smaller than a 30° difference in longitudinal separation between the magnetic footpoints of the two spacecraft. For the selected events, we used a linear interpolation method to compute the proton peak flux and fluence in four energy ranges and quantified their radial dependence as a function of R^{α, where R is the radial distance of the observer from the Sun.Results. The peak flux and fluence of the SEP events display the following radial dependence: The average values of α across all energies range between about −3.7 and −2 for the peak fluxes and between −2.7 and −1.4 for the fluences. We also obtained the energy dependence of |α|, which decreases with increasing energy. Additionally, based on theoretical functions, we find that the SEP source and transport parameters may have a significant impact on α(E), and the measurement-derived |α(E)| values and their distribution fall within the range of theoretical predictions.Conclusions. (1) Despite the uncertainties arising from the low statistics and the longitudinal influence, the radial dependence of the peak flux agrees with the upper limit R−3 predicted by previous studies. (2) The radial dependence on the fluence R−2 tends to be weaker than the radial decay of the peak flux. (3) As the proton energy increases, the proton mean free path increases, and the adiabatic cooling effect modifies the proton energy. As a result, the peak flux and fluence decay more significantly with increasing radial distance for lower-energy particles.}