Stellar versus Galactic: The intensity of cosmic rays at the evolving Earth and young exoplanets around Sun-like stars

Energetic particles, such as stellar cosmic rays, produced at a heightened rate by active stars (like the young Sun) may have been important for the origin of life on Earth and other exoplanets. Here we compare, as a function of stellar rotation rate ($\Omega$), contributions from two distinct populations of energetic particles: stellar cosmic rays accelerated by impulsive flare events and Galactic cosmic rays. We use a 1.5D stellar wind model combined with a spatially 1D cosmic ray transport model. We formulate the evolution of the stellar cosmic ray spectrum as a function of stellar rotation. The maximum stellar cosmic ray energy increases with increasing rotation i.e., towards more active/younger stars. We find that stellar cosmic rays dominate over Galactic cosmic rays in the habitable zone at the pion threshold energy for all stellar ages considered ($t_*=0.6-2.9\,$Gyr). However, even at the youngest age, $t_*=0.6\,$Gyr, we estimate that $\gtrsim\,80$MeV stellar cosmic ray fluxes may still be transient in time. At $\sim1\,$Gyr when life is thought to have emerged on Earth, we demonstrate that stellar cosmic rays dominate over Galactic cosmic rays up to $\sim$4$\,$GeV energies during flare events. Our results for $t_*=0.6\,$Gyr ($\Omega = 4\Omega_\odot$) indicate that $\lesssim$GeV stellar cosmic rays are advected from the star to 1$\,$au and are impacted by adiabatic losses in this region. The properties of the inner solar wind, currently being investigated by the Parker Solar Probe and Solar Orbiter, are thus important for accurate calculations of stellar cosmic rays around young Sun-like stars.