Exact analytical models of the streaming instability driven by intense neutrino beams

Detailed properties of the neutrino streaming instability in a plasma are investigated for two particular choices of beamed neutrino distributions that permit exact analytical solutions to the dispersion relation. The dispersion relation is solved analytically, and it is shown that the maximum instability growth rate scales with powers κ of the Fermi constant GF lower than 2 (1 ≤ κ ≤ 4/3). For conditions present in type II supernovae, the neutrinos are sufficiently collimated for the neutrino streaming instability to operate.