Superbubbles as Galactic PeVatrons: The Potential Role of Rapid Second-Order Fermi Acceleration

Context: The origin of Galactic cosmic rays is still a mystery, in particular the sources and acceleration mechanism for cosmic rays with energies up to or beyond a PeV. Recently LHAASO has and H.E.S.S have shown that two gamma-ray sources associated with superbubbles created by young massive stellar clusters are likely PeVatrons. This has renewed the interest in the cosmic-ray acceleration processes in superbubbles. Aims: To study the possibility and conditions under which second-order Fermi acceleration can accelerate particles beyond PeV energies in superbubbles. Methods: An analytical equation is derived for the maximum energy a cosmic-ray particle can obtain as a function of acceleration duration and size. The maximum energy depends critically on the diffusion coefficient D and the Alfven velocity, V_A. The analytical solutions for the acceleration time scale shows that second-order Fermi acceleration can be just as efficient as diffusive shock acceleration, when comparable relevant velocities are used -- i.e. the Alfven velocity or shock velocity. Ultimately, the maximum energy is determined by the diffusive escape of the cosmic rays, which depends on the size and diffusion coefficient. This limits the maximum energy to a few PeV for superbubble radii of ~50 pc. The probable values for the diffusion coefficient and Alven speed are studied for two likely PeVatron regions, HESS J1646-458 associated with Westerlund 1, and the Cygnus Cocoon, associated Cyg OB2. Results: It is shown that within a typical stellar cluster time scale of 1-5 Myr cosmic rays can be accelerated to > 10^15 eV, provided that V_A >300 km/s, and the diffusion coefficient is D~10^26 cm^2s at 100 TeV. This suggests that second-order Fermi acceleration in superbubbles should be considered as a possible source of Galactic cosmic rays up to, or beyond a PeV.