Random Walks and Effective Optical Depth in Relativistic Flow

Abstract

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 parameterwhere 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 τ a is considerably smaller than the optical depth for scattering τ s (τ a /τ s ≪ 1) and the flow velocity satisfies β ≫ 2τ a /τ s , the effective optical depth is approximated by τ * ≃ τ a (1 + β)/β. 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.

Photospheric Emission From Collapsar Jets in 3d Relativistic Hydrodynamics

“…We have checked that our result depends only weakly on the injection radius as long as τ ≫ 1.4 The photon production site is located at regions with much higher optical depth(Beloborodov 2013;Vurm et al 2013;Shibata et al 2014;Vurm & Beloborodov 2015).5 Since our calculation is in 3D, the emission depends not only on the observer angle, but also on the azimuthal angle. However, the dependence is not strong.…”

Black hole accretion disc impacts

“…Based on these estimates, it is clear that for estimating bolometric luminosity, Thomson opacity is the determining factor, and the outflow does not turn optically thin until ξ 300. Effective optical depth does in principle determine the initial spectrum of emitted photons, but the value of τe must be calculated by taking into account the bulk motion of the plasma (Shibata et al 2014). This will be further discussed in the Section 3.…”

“…After the last emission, the photons are scattered by the electrons several times, but as noted in Section 2.3, the spectrum is not appreciably comptonized by this. For a careful analysis, as noted in Shibata et al (2014), the effective optical depth τe needs to be modified for even mildly relativistic bulk velocities. The principal reason is that while the scattering events are isotropic in the flow rest frame, they are nonisotropic in the observer frame.…”

Identical algorithm of radiative transfer across ultrarelativistic shock in different inertial frames

“…Although the observed spectra are characterized by a broken power-law shape [11], these properties have not been reproduced accurately in the numerical works of the relativistic hydrodynamics. Analytical and numerical solutions for the equation of radiative transfer in relativistic flows can be obtained by various methods [12,13]; in particular, some authors have developed the method to solve the problem statistically by Monte Carlo (MC) technique [14,15], which has been adopted for the problem in the context of radiative transport in supernova explosions [16,17]. Since some observations indicate spectra with a thermal component [18], radiative transport for thermal photons produced by photospheric emission is regarded as the feasible model, and the feature of the spectra in GRBs was interpreted by overlapping thermal spectra for some components of various angles of photons escaped from the photosphere due to different observed time [19].…”