A Tight $L_{\rm p, iso}-E'_{\rm p}-\Gamma_0$ Correlation of Gamma-Ray Bursts

We select a sample of 34 gamma-ray bursts (GRBs) whose $\Gamma_0$ values are derived with the onset peaks observed in the afterglow lightcurves (except for GRB 060218 whose $\Gamma_0$ is estimated with its radio data), and investigate the correlations among $\Gamma_0$, the isotropic peak luminosity ($L_{\rm iso}$), and the peak energy ($E_{\rm p,z}$) of the $\nu f_\nu$ spectrum in the cosmological rest frame. An analysis of pair correlations among these observables well confirms the results reported by the previous papers. More interestingly, a tight correlation among $L_{\rm iso}$, $E_{\rm p,z}$, and $\Gamma_0$ is found from a multiple regression analysis, which takes the form of $L_{\rm iso} \propto E_{\rm p,z}^{1.34\pm 0.14} \Gamma_0^{1.32\pm 0.19}$ or $E_{\rm p,z} \propto L_{\rm iso}^{0.55\pm 0.06}\Gamma_0^{-0.50\pm 0.17}$. Nine other GRBs whose $\Gamma_0$ are derived via the pair production opacity constraint also follow such a correlation. Excluding GRB 060218, the $L_{\rm iso}-E_{\rm p,z}-\Gamma_0$ correlation is valid, and it even holds in the jet co-moving frame. However, GRB 060218 deviates the $L^{'}_{\rm iso}-E^{'}_{\rm p}$ relation of typical GRBs in the jet co-moving frame with $3\sigma$. We argue that the $L_{\rm iso} - E_{\rm p, z} - \Gamma_0$ correlation may be more physical than the $L_{\rm iso} - E_{\rm p,z}$ correlation, since physically the relationship between the observed $L_{\rm iso}$ and $E_{\rm p,z}$ not only depends on radiation physics, but also depends on the bulk motion of the jet. We explore the possible origins of this correlation and discuss its physical implications for understanding GRB jet composition and radiation mechanism.