Does The Addition of a Duration Improve the L_iso - E_peak Relation For   Gamma-Ray Bursts?
  Firmani et al. proposed a new Gamma Ray Burst (GRB) luminosity relation that showed a significant improvement over the L_iso-E_peak relation. The new proposed relation simply modifies the E_peak value by multiplying it by a power of T_0.45, where T_0.45 is a particular measure of the GRB duration. We begin by reproducing the results of Firmani for his 19 bursts. We then test the Firmani relation for the same 19 bursts except that we use independently measured values for L_iso, T_0.45, and E_peak, and we find that the relation deteriorates substantially. We further test the relation by using 60 GRBs with measured spectroscopic redshifts, and find a relation that has a comparable scatter as the original L_iso-E_peak relation. That is, a much larger sample of bursts does not reproduce the small scatter as reported by Firmani et al. Finally, we investigate whether the Firmani relation is improved by the use of any of 32 measures of duration in place of T_0.45. The quality of each alternative duration measure is evaluated with the root mean square of the scatter between the observed and fitted logarithmic Liso values. Although we find some durations yield slightly better results than T_0.45, the differences between the duration measures are minimal. We find that the addition of a duration does not add any significant improvement to the L_iso-E_peak relation. We also present a simple and direct derivation of the Firmani relation from both the L_iso-E_peak and Amati relations. In all we conclude that the Firmani relation neither has an independent existence nor does it provide any significant improvement on previously known relations that are simpler. 
Andrew C. Collazzi and Bradley E. Schaefer