Cosmic Ray Acceleration in the Presence of Losses

Abstract

Diffusive Fermi acceleration on hydromagnetic shock fronts is a fairly slow process: most scatterings are energetically neutral, only those across the velocity jump yield a first order acceleration. Thus energy losses of cosmic rays (CR) due to e.g. adiabatic cooling, ionising, Coulomb−, or nuclear collisions, bremsstrahlung, synchrotron radiation, should play a role in limiting the acceleration, since most shocks are coupled with a loss region (HII regions, SNR's, galactic density wave, stellar wind shocks). In addition, for this process to be a local one in the galaxy, the magnetic irregularities must either be excited by the accelerated CR's or produced by a downstream source. This implies a finite wave build-up time during the shock life time. Nevertheless, in the loss free case, the time-asymptotic amplification is independent of the mean free path λ (or the diffusion coefficient κ) which only appears in the spatial scale for the CR intensity. To investigate the effects of losses, the CR diffusion equation is amended by a simple loss term f/τ, with an (energy dependent) loss time τ, f being the CR momentum distribution. For τ spatially homogeneous, a distributed source is required. Then acceleration is only effective if X ≡ 4κ/VS2·τ ≲ 1, i.e. if the acceleration time tacc = 4κ/VS2 is smaller than τ, on either side of the shock. As the Figure shows, also the spatial intensity profile is modified. Wave excitation in dense clouds is prohibitive. Even in a “warm” (T ≃ 104K) intercloud medium shock speeds VS ≳ 3×107 cm/sec are required to accelerate mildly relativistic particles. Waves from an upstream source (star) inside clouds should frictionally dissipate after distances L ≲ 5×1014 cm « λ (10 MeV) if a solar wind scaling is adopted. Thus, presumably in such a case there is no acceleration of stellar or shock-injected particles through a stellar wind shock by scattering within the cloud, but possibly by reflection from beyond the cloud, the condition being 1/τ · VS ≲ 1, where 1 is the linear cloud size. Diffusive approach from outside to a standing shock appears to be very energy-selective, even in a loss free medium.