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$\alpha_{\rm 2}$),thebreaktimes($T_{\rm b}$), and the corresponding optical fluxes ($F_{\mathrm{b}}$) 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 $E_{\gamma ,\mathrm{i}\mathrm{s}\mathrm{o}}$ into the $L_{\rm b,z}-T_{\rm b,z}$relation\mathrm{.}Thebestfitcorrelationisobtainedtobe$L_{\rm b,z}\propto T_{\rm b,z}^{-0.9}E_{\rm \gamma,iso}^{0.4}$. We next explored the possible correlations among $L_{\mathrm{b},\mathrm{z}}$, $T_{\mathrm{b},\mathrm{z}}$ and $E_{\mathrm{p},\mathrm{i}}$, and found there is also a tight correlation between them, which takes the form of $L_{\mathrm{b},\mathrm{z}}\propto T_{\mathrm{b},\mathrm{z}}^{-0.9}{E}_{\mathrm{p},\mathrm{i}}^{0.5}$. We argue that these two tight $L_{\mathrm{b},\mathrm{z}}-T_{\mathrm{b},\mathrm{z}}-E_{\gamma ,\mathrm{i}\mathrm{s}\mathrm{o}}$ 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.