A fundamental plane for long gamma-ray bursts with X-ray plateaus

A class of long Gamma-Ray Bursts (GRBs) presenting light curves with an extended plateau phase in their X-ray afterglows obeys a correlation between the rest frame end time of the plateau, $T_a$, and its corresponding X-ray luminosity, $L_{a}$, Dainotti et al. (2008). In this work we perform an analysis of a total sample of 176 {\it Swift} GRBs with known redshifts, exhibiting afterglow plateaus. By adding a third parameter, that is the peak luminosity in the prompt emission, $L_{peak}$, we discover the existence of a new three parameter correlation. The scatter of data about this plane becomes smaller when a class-specific GRB sample is defined. This sample of 122 GRBs is selected from the total sample by excluding GRBs with associated Supernovae (SNe), X-ray flashes and short GRBs with extended emission. {\bf With this sample the three parameter correlation identifies a GRB `fundamental plane'}. Moreover, we further limit our analysis to GRBs with lightcurves having good data coverage and almost flat plateaus, 40 GRBs forming our `gold sample'. The intrinsic scatter, $\sigma_{int}=0.27 \pm 0.04$, for the three-parameter correlation for this last subclass is more than twice smaller than the value for the $L_{a}-T_a$ one, making this the tightest three parameter correlation involving the afterglow plateau phase. Finally, we also show that a slightly less tight correlation is present between $L_{peak}$ and a proxy for the total energy emitted during the plateau phase, $L_a T_a$, confirming the existence of an energy scaling between prompt and afterglow phases.