  In this study, we investigate the homologous collapse dynamics of a slowly
rotating stellar nuclear core. We incorporate the virial theorem of hadron
collisional relaxations to analyze the conversion of gravitational potential
energy into kinetic energy of hadrons. Additionally, we consider the production
of photons and electron-positron pairs resulting from hadron collisions. Our
analysis reveals a gravo-thermal process wherein gravitational energy
transforms into energy associated with photons and pairs. The presence of axial
symmetric centrifugal potential reduces the gravitational potential energy
gain. The presence of axial symmetric centrifugal potential mitigates the gain
in gravitational potential energy. Consequently, we observe the formation of
highly energetic and opaque photon-pair jets, whose size, peak spectra, energy
and number densities are consistent with observed energetic features and
empirical relationships of Gamma-Ray Burst (GRB) progenitors. Furthermore, we
find that the angular momentum and mass ratio ($J/M$) of the collapsing core
and binary coalescence play crucial roles in determining the jet angle and the
differentiation between short and long bursts. Our findings contribute to a
deeper understanding of the mechanisms governing GRBs' progenitor systems.
