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

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 Emax, showing that it is given by an underlying propagated power law multiplied by the survival probability of the cutoff distribution. A lognormal distribution of Emax 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 σlog10Emax≃ 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.