The cosmic-ray knee and ensuing spectrum seen as a consequence of Bell's self-magnetized SNR shock acceleration process

Bell's version of diffusive shock acceleration in supernova remnants, in which the (highly contorted) magnetic field is mainly self-generated by the accelerated particles, seems to explain the sharpness of the knee in the cosmic ray spectrum (an effect of the limitation of acceleration in the source), despite contributions from very many supernovae; and the expected knee has a sharp drop near 3 PV rigidity for each nuclear species, and then a lower rounded shoulder extending towards 100 PV contributed by some of the type II SNRs. It is shown that the allparticle flux determined by very different techniques in air shower observations defines a very precise spectrum, below 10¹⁹ eV, and this spectrum is discussed in terms of separate nuclear components of Galactic and extragalactic origin. An extragalactic component with similar source spectrum to the Galactic component is favoured, but a helium/hydrogen mixture and a pure proton flux from the extragalactic sources have different attractions, and offer different tests through composition, even below 10¹⁸ eV.