Cosmic Ray Acceleration by a Versatile Family of Galactic Wind Termination Shocks

There are two distinct breaks in the cosmic ray (CR) spectrum: the so-called "knee" around $3 \times 10^{15}$ eV and the so-called "ankle" around $10^{18}$ eV. Diffusive shock acceleration (DSA) at supernova remnant (SNR) shock fronts is thought to accelerate galactic CRs to energies below the knee, while an extragalactic origin is presumed for CRs with energies beyond the ankle. CRs with energies between $3 \times 10^{15}$ and $10^{18}$ eV, which we dub the "shin," have an unknown origin. It has been proposed that DSA at galactic wind termination shocks, rather than at SNR shocks, may accelerate CRs to these energies. This paper uses the galactic wind model of Bustard et al. (2016) to analyze whether galactic wind termination shocks may accelerate CRs to shin energies within a reasonable acceleration time and whether such CRs can subsequently diffuse back to the galaxy. We argue for acceleration times on the order of 100 Myrs rather than a few billion years, as assumed in some previous works, and we discuss prospects for magnetic field amplification at the shock front. Ultimately, we generously assume that the magnetic field is amplified to equipartition. This formalism allows us to obtain analytic formulae, applicable to any wind model, for CR acceleration. Even with generous assumptions, we find that very high wind velocities are required to set up the necessary conditions for acceleration beyond $10^{17}$ eV. We also estimate the luminosities of CRs accelerated by outflow termination shocks, including estimates for the Milky Way wind.