A Comprehensive Study of Gamma-Ray Burst Optical Emission: III. Brightness Distributions and Luminosity Functions of Optical Afterglows

Abstract

We continue our systematic statistical study on optical afterglow data of gamma-ray bursts (GRBs). We present the apparent magnitude distributions of early optical afterglows at different epochs (t= 10^2 s, t = 10^3 s, and 1 hour) for the optical lightcurves of a sample of 93 GRBs (the global sample), and for sub-samples with an afterglow onset bump or a shallow decay segment. For the onset sample and shallow decay sample we also present the brightness distribution at the peak time t_{p} and break time t_{b}, respectively. All the distributions can be fit with Gaussian functions. We further perform Monte Carlo simulations to infer the luminosity function of GRB optical emission at the rest-frame time 10^3 seconds, t_{p}, and t_{b}, respectively. Our results show that a single power-law luminosity function is adequate to model the data, with indices -1.40+/-0.10, -1.06+/- 0.16, and -1.54\+/- 0.22, respectively. Based on the derived rest-frame 10^3 s luminosity function, we generate the intrinsic distribution of the R-band apparent magnitude M_{R} at the observed time 10^{3} seconds post trigger, which peaks at M_{R}=22.5 mag. The fraction of GRBs whose R-band magnitude is fainter than 22 mag, and 25 mag and at the observer time 10^3 seconds are ~63% and ~25%, respectively. The detection probabilities of the optical afterglows with ground-based robotic telescopes and UVOT onboard {Swift} are roughly consistent with that inferred from this intrinsic M_{R} distribution, indicating that the variations of the dark GRB fraction among the samples with different telescopes may be due to the observational selection effect, although the existence of an intrinsically dark GRB population cannot be ruled out.