  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 these
three kinds of bursts are the same phenomenon. A key feature found by HETE-2 is
that the density of bursts is roughly constant per logarithmic interval in
burst fluence S_E and observed spectral peak energy Ep_obs, and in
isotropic-equivalent energy Eiso and rest frame spectral peak energy Epeak. In
this paper, we explore a unified jet model of all three kinds of bursts, using
population synthesis simulations of the bursts and detailed modeling of the
instruments that detect them. We show that both a variable jet opening-angle
model in which the emissivity is a constant independent of the angle relative
to the jet axis and a universal jet model in which the emissivity is a
power-law function of the angle relative to the jet axis can explain the
observed properties of GRBs reasonably well. However, if one tries to account
for the properties of all three kinds of bursts in a unified picture, the extra
degree of freedom available in the variable jet opening-angle model enables it
to explain the observations reasonably well while the power-law universal jet
model 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 times
less than has been thought, and that most GRBs have very small jet opening
angles (~ half a degree). It also implies that there are ~ 10^4 - 10^5 more
bursts 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 Type
Ic core collapse supernovae.
