Exploration of Cosmic Ray Acceleration in Protostellar Accretion Shocks and A Model for Ionization Rates in Embedded Protoclusters

We construct a model for cosmic ray acceleration from protostellar accretion shocks and calculate the resulting cosmic ray ionization rate within star-forming molecular clouds. We couple a protostar cluster model with an analytic accretion shock model to calculate the cosmic ray acceleration from protostellar surfaces. We present the cosmic ray flux spectrum from keV to GeV energies for a typical low-mass protostar. We find that at the shock surface the spectrum follows a power-law trend across 6 orders of magnitude in energy. After attenuation, the spectrum at high energies steepens, while at low energies it is relatively flat. We calculate the cosmic ray pressure and cosmic ray ionization rate from relativistic protons at the protostellar surface and at the edge of the core. We present the cosmic ray ionization rate for individual protostars as a function of their instantaneous mass and final mass. The protostellar cosmic ray ionization rate is $\zeta \approx 0.01 - 1$ s$^{-1}$ at the accretion shock surface. However, at the edge of the core, the cosmic ray ionization rate drops substantially to between $\zeta \approx 10^{-20}$ to $10^{-17}$ s$^{-1}$. There is a large spatial gradient in the cosmic ray ionization rate, such that inner regions may experience cosmic ray ionization rates larger than the often assumed fiducial rate, $\zeta = 3\times10^{-17}$ s$^{-1}$. Finally, we calculate the cosmic ray ionization rate for protostellar clusters over 5 orders of magnitude of cluster size. We find that clusters with more than approximately 200 protostars produce a higher cosmic ray ionization rate within their natal cloud than the fiducial galactic value.