A Computational Study of the Loaded Layer-Cake Model for the Cosmic Ray Spectra

Understanding the origin of cosmic rays requires models that can connect their sources, transport, and interactions to the spectra observed at Earth. The Loaded Layer-Cake Model (LLCM) is one such framework, combining time-dependent acceleration, diffusion, and spallation processes within supernova shocks, stellar wind shells, and OB-superbubble regions. The model contains three characteristic zones. In the acceleration zone, a common high rigidity spectral slope of −7/3 ± 0.02 dominates across cosmic ray species. In the interaction zone, a Kolmogorov spectrum of magnetic field irregularities drives a slope change of 1/3. Finally, in the bubble zone, lightning-induced magnetic wave-fields lead to a slope change of 5/3. After integration, the model produces expressions that can be directly compared to all AMS-02 cosmic ray nuclei data of Z = 1–26, and recent fits show that the LLCM successfully reproduces these spectra. Since all low rigidity and also all high rigidity spectral shapes have to be the same for primary nuclei, there remains ambiguity as regards number of contributing OB associations, and their contribution. To reduce the number of parameters, we incorporate observable source properties, such as size and density, to better anchor the model in physical structure. Quantities such as magnetic field strength and density vary across regions; therefore, averaged values, together with LLCM predictions, are used to provide a representative description of their effects within the framework. These refinements aim to make the connection between OB associations and the cosmic rays they produce more direct and predictive while assessing the accuracy of a physically grounded LLCM. Fitting to Fe strongly supports the conclusion that OB associations are the primary sources of Galactic cosmic rays, with several likely contributing. Notably, within this framework, the Scorpius–Centaurus superbubble emerges as the dominant contributor.