Extension of the Bjorken Formula for Relativistic Heavy Ion Collisions

For relativistic heavy ion collisions, the Bjorken formula is very useful for the estimation of the initial energy density once an initial time $\tau_0$ is specified. However, the formula is only valid when $\tau_0$ is much bigger than the finite time it takes for the two nuclei to cross each other, therefore it cannot be trusted at low energies, below $\sqrt{s_{_{\rm NN}}} \sim 50$ GeV for central Au+Au collisions. In this study we extend the Bjorken formula by including a time profile for the initial energy production. Analytical solutions for the formed energy density as functions of time are derived for several time profiles, and we find that at low energies they are much less sensitive to the uncertainty of the formation time. Compared to the Bjorken formula, the maximum energy density are much lower while the width of the energy density time evolution is much bigger at low energies. Comparisons with results from a multi-phase transport confirm the key features of these solutions. This work thus provides a general model for the initial energy production of relativistic heavy ion collisions, especially at low energies.