A Population View of the Cosmic-Ray Knee: The Role of Variance in Supernova Maximum Rigidities

Abstract

The broad shape of the Galactic cosmic-ray knee challenges source models in which all supernova remnants share a nearly universal, sharp maximum rigidity. We investigate whether the knee can instead arise as a population effect, produced by source-to-source variations in the maximum energy of Galactic supernova remnants. We derive the population-averaged spectrum for sources with sharp individual cutoffs and distributed $E_{\max}$, showing that it is given by an underlying propagated power law multiplied by the survival probability of the cutoff distribution. A lognormal distribution of $E_{\max}$ naturally produces a smooth, continuously curving knee, while a power-law tail gives an approximately constant post-knee steepening. We then connect the lognormal width to supernova-remnant physics through maximum-energy scalings with explosion energy and ambient density, finding that the expected variance is mainly driven by the spread in explosion energies. Fitting the measured proton spectrum with a two-component lognormal-cutoff model, we find that the PeV component requires $\sigma_{\log_{10}E_{\max}}\simeq 0.24$. This width is substantially smaller than the variance expected for the full Galactic remnant population, indicating that the PeV component must originate from a more restricted and comparatively homogeneous subset of remnants. Our results show that the knee can be understood as the gradual exhaustion of a heterogeneous population of PeV-capable supernova remnants, without requiring a universal maximum rigidity.