The three-parameter correlations about optical plateaus of gamma-ray bursts

Well-sampled optical light curves of 50 gamma-ray bursts (GRBs) with plateau features are compiled from the literature. By empirical fitting, we obtained the parameters of the optical plateaus, such as the decay slopes ($\alpha_{\rm 1}$ and $\alpha_{\rm 2}$), the break times ($T_{\rm b}$), and the corresponding optical fluxes ($F_{\rm b}$) at the break times. The break time of optical plateaus ranges from tens of seconds to $10^6$ seconds, with a typical value about $10^4$ seconds. We have calculated the break luminosity, and it mainly ranges from $10^{44}$ erg $s^{-1}$ to $10^{47}$ erg $s^{-1}$, which is generally two or three orders of magnitude less than the corresponding break luminosity of the X-ray afterglow plateaus. We reanalyzed the optical plateaus and also found that a significantly tighter correlation exists when we added the isotropic equivalent energy of GRBs $E_{\rm \gamma,iso}$ into the $L_{\rm b,z}-T_{\rm b,z}$ relation. The best fit correlation is obtained to be $L_{\rm b,z}\propto T_{\rm b,z}^{-0.9}E_{\rm \gamma,iso}^{0.4}$. We next explored the possible correlations among $L_{\rm b,z}$, $T_{\rm b,z}$ and $E_{\rm p,i}$, and found there is also a tight correlation between them, which takes the form of $L_{\rm b,z}\propto T_{\rm b,z}^{-0.9}E_{\rm p,i}^{0.5}$. We argue that these two tight $L_{\rm b,z}-T_{\rm b,z}-E_{\rm \gamma,iso}$ and $L_{\rm b,z}-T_{\rm b,z}-E_{\rm p,i}$ correlations are more physical, and it may be directly related to radiation physics of GRBs. The tight correlations are possible to be used as standard candles.