Random walks and effective optical depth in relativistic flow

We investigate the random walk process in relativistic flow. In the relativistic flow, photon propagation is concentrated in the directions of the flow velocity due to relativistic beaming effect. We show that, in the pure scattering case, the number of scatterings is proportional to the size parameter $\xi\equiv L/l_0$ if the flow velocity $\beta\equiv v/c$ satisfies $\beta/\Gamma\gg \xi^{-1}$, while it is proportional to $\xi^2$ if $\beta/\Gamma\ll \xi^{-1}$ where $L$ and $l_0$ are the size of the system in the observer frame and the mean free path in the comoving frame, respectively. We also examine the photon propagation in the scattering and absorptive medium. We find that, if the optical depth for absorption $\tau_{\rm a}$ is considerably smaller than the optical depth for scattering $\tau_{\rm s}$ ($\tau_{\rm a}/\tau_{\rm s} \ll 1$) and the flow velocity satisfies $\beta\gg \sqrt{2\tau_{\rm a}/\tau_{\rm s}}$, the effective optical depth is approximated by $\tau_*\simeq\tau_{\rm a}(1+\beta)/\beta$. Furthermore, we perform Monte Carlo simulations of radiative transfer and compare the results with the analytic expression for the number of scattering. The analytic expression is consistent with the results of the numerical simulations. The expression derived in this Letter can be used to estimate the photon production site in relativistic phenomena, e.g., gamma-ray burst and active galactic nuclei.