Collimated and spinning fireballs for ultra-relativistic jets: long vs short Gamma-ray bursts by angular momentum and mass ratio

In this study, we investigate the gravitational collapses of rotating stellar systems accounting for Gamma-Ray Burst jet progenitors. Based on the virial theorem of hadron collisional relaxations and Newtonian slow-rotating approximation, we analyze the conversion of gravitational binding energy into kinetic energy of hadrons, whose collisions produce photons and electron-positron pairs forming fireballs. Our qualitative analysis implies that rotation effects collimated and spinning fireballs with nontrivial angular momenta along the propagating direction, thus making ultra-relativistic jets. Results reveal the possible trends that the fireball becomes more collimated and the jet angle decreases as the total angular momentum and mass ratio $J/M$ of the slow-rotating collapsing core increases. Discussing the extrapolation of these trends to fast-rotating collapsing systems, we speculate that the ratio $J/M$ should be a key quantity for differentiating long bursts (massive core collapses) from short bursts (binary coalescence). We derive the intrinsic correlations of collimated fireball quantities that should be imprinted on a large sample of observed GRB data as empirical correlations.