Effect of optical depth on the resonance accumulation. The time evolution of the L_z distribution is shown with various initial optical depths τ₀ and a fixed particle radius R = 0.001 (corresponds to 200 m for Chariklo’s rings). Upper row: Mass anomaly of µ = 0.05; lower row: µ = 0.03. Because R is fixed, different values of τ₀ correspond to different numbers of particles. The orange curve shows the evolution of the optical depth at the resonance zone, calculated from the number of particles with |a − a₀| < W_res (axis labels on the right), while the green curve indicates the excess radial velocity dispersion at the same region (full scale corresponds to a five-fold dispersion compared to unperturbed dispersion). The dashed lines indicate the estimated τ_lim: for τ_res < τ_lim the excitation of resonance perturbations (indicated by the growing oscillations of c_res) starts, either initially in the case τ₀ ≲ t_lim, or after a sufficient drop in τ_res caused by the initial viscous diffusion. The latter cases are indicated by green arrows and provide an estimate of τ_lim. Based on the simulations τ_lim ≈ 0.06 and 0.025 for µ = 0.05 and 0.03, respectively.