Reinterpretation of the Fermi acceleration of cosmic rays in terms of the ballistic surfing acceleration in shocks

It is shown that the first order Fermi acceleration of cosmic rays, based on a concept of ions reflected by shocks, is equivalent to the ballistic surfing acceleration (BSA) by the convection electric field. Despite big differences in the physics of the processes involved, both models lead to the same expression for the energy gain of a particle after one encounter with the shock, and consequently to the same power-law distribution of the cosmic ray energy spectrum after many encounters. BSA accelerates ions continuously from superthermal energies of 100eV up to very high energies observed in the cosmic ray spectrum. It is shown that the ‘knee’ observed in the spectrum at energy of 5×10¹⁵ eV could correspond to ions with gyroradius comparable to the size of shocks in supernova remnants.
It has been established that thermalization, heating, and energization of ions and electrons in collisionless shocks are related to the following plasma processes:

BSA – ballistic surfing acceleration
SWE –  stochastic wave energization
TTT – transit time thermalization
QAH – quasi adiabatic heating

References:
[1] K. Stasiewicz, MNRAS. 527, L71 (2023), Origin of flat-top electron distributions at the Earth’s bow shock, https://doi.org/10.1093/mnrasl/slad146
[2] K. Stasiewicz, MNRAS. 524, L50 (2023), Transit time  thermalization  and the stochastic wave energization of ions  in quasi-perpendicular shocks,  https://doi.org/10.1093/mnrasl/slad071
[3] K. Stasiewicz, B. Eliasson, MNRAS 520, 3238 (2023), Electron heating mechanisms at the bow shock - revisited with Magnetospheric Multiscale measurements, https://doi.org/10.1093/mnras/stad361