as described in the BATSE and GBM Spectroscopy Catalogs.

We have checked the effect of the spectral parameter $\alpha$ by performing three identical simulations, varying only the $\alpha$ value in three steps between $-0.66$ and $-1.2$, bracketing the most likely fitted value of $-1$ from the distributions reported in the various spectroscopy catalogs. For the three values simulated, there was no change in the mean value of [$E_{\rm peak}$]{} for the resulting distributions to well within the uncertainty on the mean: $177 \pm 2$ keV (these and following simulations are based on $E_{\rm hi} = 3000$ keV and $E_{\rm lo} = 50$ keV). This is expected, as [$E_{\rm peak}$]{} does not formally depend upon $\alpha$ (nor does it depend upon the Band parameter $\beta$, as long as $\beta < -2$). As discussed above, we do not present any simulations of Band functions, so the evolution of $\beta$ is not considered here.

![Fitted [$E_{\rm peak}$]{} values from an example pulse simulation shows that the input $-1$ power law spectral evolution is recovered after signal-to-noise temporal binning and forward-folding with the detector response. In this case, the pulse parameters were 3 s rise time, 11 s decay time, scaled to 40,000 count/s 