Jet Break Time-Flux Density Relationship and Constraints on Physical Parameters of Gamma-Ray Burst Afterglows

We derive a relation between the flux density F_ν,j at the light-curve break of a gamma-ray burst (GRB) afterglow and the break time t_j. The break is due to the transition from the spherical-like to jet-like evolution of the afterglow, when the Lorentz factor of the jet equals the inverse of the initial half-opening angle, i.e., γ = 1/θ₀. We show that this relation indeed behaves as F_ν,j ∝ t among GRBs for the slow-cooling case, where p is the power-law index of electron distribution. A statistical analysis of the optical jet breaks of nine GRBs gives p = 2.10 ± 0.21, which is consistent with the shock acceleration theory. The value of p derived in this way is different from the observed temporal index α₂ (F_ν ∝ t) of the late-time light curve after t_j, which suffers several uncertainties from the unclear hydrodynamics of the sideways expansion and exhibits a large dispersion. Our results not only confirm that the remnants of GRBs are standard candles, but also provide the first evidence that the physical parameters of relativistic shocks are universal, with the favored values _e ~ 0.1 and _B ~ 10^-3.