On the shape of the spectrum of cosmic-rays accelerated inside superbubbles

Supernova remnants are believed to be a major source of cosmic-rays in the Galaxy. As their progenitors are commonly found clustered in OB associations, one has to consider the possibility of collective effects in the acceleration process. In this work we investigate the shape of the spectrum of protons accelerated inside the superbubbles blown around clusters of massive stars. To do so we embed semi-analytical models of particle acceleration and transport inside Monte-Carlo simulations of OB associations timelines. We consider regular acceleration at the shock front of supernova remnants, as well as stochastic re-acceleration and escape occurring between the shocks. We observe that particle spectra, although strongly intermittent, get a distinctive shape resulting from a competition between acceleration and escape: they are harder at the lowest energies and softer at the highest energies. The momentum at which this spectral break occurs depends on a single dimensionless parameter, which we evaluate for a selection of objects. The behaviour of a superbubble regarding acceleration depends on the magnetic turbulence: if B is low the superbubble is simply the host of a collection of individual supernovae shocks, but if B is high enough it acts as a global accelerator, producing distinctive spectra ? which has important implications on the high-energy emission from these objects.