We perform Monte Carlo simulations to study E _ { p } – E _ { iso } correlation in the context of a multiple subjet model ( or inhomogeneous jet model ) for gamma-ray bursts ( GRBs ) , X-ray–rich GRBs ( XRRs ) , and X-ray flashes ( XRFs ) .
For a single subjet , we find that E _ { p } \propto { E _ { iso } } ^ { 0.4 } for large viewing angles .
For the multiple subjet model in which all the subjets have the same intrinsic properties , off-axis events show E _ { p } \propto { E _ { iso } } ^ { a } with 0.4 < a < 0.5 .
If the intrinsic properties of the subjets are distributed so that on-axis emission of each subjet follows a correlation E _ { p } \propto { L _ { iso } } ^ { 1 / 2 } , we obtain the Amati correlation ( E _ { p } \propto { E _ { iso } } ^ { 1 / 2 } ) over three orders of magnitude in E _ { p } .
Although the scatter around the Amati correlation is large in the simulation , the results are consistent with the observed properties of GRBs with known redshifts and the BASTE GRBs with pseudo redshifts derived from the lag-luminosity correlation .
We also calculate the event rates , the redshift distributions , and the T _ { 90 } duration distributions of GRBs , XRRs , and XRFs which can be detected by HETE-2 , assuming that the source redshift distribution is in proportion to the cosmic star formation rate .
It is found that the event rates of three classes are comparable , that the average redshift of the XRRs is a little larger than those of the GRBs and the XRFs , and that short XRRs arise when a single subjet is viewed off-axis or viewed on-axis with slightly high redshift .