Characteristics of Long Gamma-Ray bursts in the Comoving Frame

We compile a sample of 93 long gamma-ray bursts (GRBs) from Fermi satellite and 131 from Konus-Wind, which have measured redshifts and well determined spectra, and estimate their pseudo Lorentz factors (${\Gamma_0}$) using the tight ${L_{{\rm{iso}}}}$-${E_{\rm p}}$-${\Gamma_0}$ correlation. The statistical properties and pair correlations of temporal and spectral parameters are studied in the observer frame, rest frame and comoving frame, respectively. We find that the distributions of the duration, peak energy, isotropic energy and luminosity in the different frames are basically lognormal, and their distributions in the comoving frame are narrow, clustering around $T'_{\rm 90}\sim 4000$ s, $E'_{\rm p,c}\sim 0.7$ keV, $E'_{\rm iso,c} \sim 8\times10^{49}$ erg and $L'_{\rm iso,c}\sim 2.5\times10^{46}$ erg s$^{-1}$, where the redshift evolution effect has been taken into account. We also find that the values of ${\Gamma_0}$ are broadly distributed between few tens and several hundreds with median values $\sim 270$. We further analyze the pair correlations of all the quantities, and well confirm ${E_{{\rm{iso}}}}$-${E_{\rm p}}$, ${L_{{\rm{iso}}}}$-${E_{\rm p}}$, ${L_{{\rm{iso}}}}$-${\Gamma_0}$ and $E_{\rm{iso}}$-${\Gamma_0}$ relations, and find that the corresponding relations in the comoving frame do still exist, but have large dispersions. This suggests not only the well-known spectrum-energy relations are intrinsic correlations, but also the observed correlations are governed by the Doppler effect. In addition, the peak energies of long GRBs are independent of durations both in the rest frame and in the comoving frame. And there is a weak anticorrelation between the peak energy and Lorentz factor.