We have carried out a detailed study to understand the observed energy
spectrum and composition of cosmic rays with energies up to ~10^18 eV. Our
study shows that a single Galactic component with subsequent energy cut-offs in
the individual spectra of different elements, optimised to explain the observed
spectra below ~10^14 eV and the knee in the all-particle spectrum, cannot
explain the observed all-particle spectrum above ~2x10^16 eV. We discuss two
approaches for a second component of Galactic cosmic rays -- re-acceleration at
a Galactic wind termination shock, and supernova explosions of Wolf-Rayet
stars, and show that the latter scenario can explain almost all observed
features in the all-particle spectrum and the composition up to ~10^18 eV, when
combined with a canonical extra-galactic spectrum expected from strong radio
galaxies or a source population with similar cosmological evolution. In this
two-component Galactic model, the knee at ~ 3x10^15 eV and the second knee at
~10^17 eV in the all-particle spectrum are due to the cut-offs in the first and
second components, respectively. We also discuss several variations of the
extra-galactic component, from a minimal contribution to scenarios with a
significant component below the ankle (at ~4x10^18 eV), and find that
extra-galactic contributions in excess of regular source evolution are neither
indicated nor in conflict with the existing data. Our main result is that the
second Galactic component predicts a composition of Galactic cosmic rays at and
above the second knee that largely consists of helium or a mixture of helium
and CNO nuclei, with a weak or essentially vanishing iron fraction, in contrast
to most common assumptions. This prediction is in agreement with new
measurements from LOFAR and the Pierre Auger Observatory which indicate a
strong light component and a rather low iron fraction between ~10^17 and 10^18
eV.Comment: Added Table 4; Published in A&A, 595 (2016) A33 (Highlight paper
