A Unified Jet Model of X‐Ray Flashes, X‐Ray–rich Gamma‐Ray Bursts, and Gamma‐Ray Bursts. I. Power‐Law–shaped Universal and Top‐Hat–shaped Variable Opening Angle Jet Models

HETE-2 has provided strong evidence that the properties of X-Ray Flashes(XRFs), X-ray-rich GRBs, and GRBs form a continuum, and therefore that thesethree kinds of bursts are the same phenomenon. A key feature found by HETE-2 isthat the density of bursts is roughly constant per logarithmic interval inburst fluence S_E and observed spectral peak energy Ep_obs, and inisotropic-equivalent energy Eiso and rest frame spectral peak energy Epeak. Inthis paper, we explore a unified jet model of all three kinds of bursts, usingpopulation synthesis simulations of the bursts and detailed modeling of theinstruments that detect them. We show that both a variable jet opening-anglemodel in which the emissivity is a constant independent of the angle relativeto the jet axis and a universal jet model in which the emissivity is apower-law function of the angle relative to the jet axis can explain theobserved properties of GRBs reasonably well. However, if one tries to accountfor the properties of all three kinds of bursts in a unified picture, the extradegree of freedom available in the variable jet opening-angle model enables itto explain the observations reasonably well while the power-law universal jetmodel cannot. The variable jet opening-angle model of XRFs, X-ray-rich GRBs,and GRBs implies that the energy Egamma radiated in gamma rays is ~ 100 timesless than has been thought, and that most GRBs have very small jet openingangles (~ half a degree). It also implies that there are ~ 10^4 - 10^5 morebursts with very small jet opening angles for every burst that is observable.If this is the case, the rate of GRBs could be comparable to the rate of TypeIc core collapse supernovae.Comment: 51 pages, 19 figures, accepted by ApJ; revised; condensed abstrac