TY - JOUR

T1 - Random walks and effective optical depth in relativistic flow

AU - Shibata, Sanshiro

AU - Tominaga, Nozomu

AU - Tanaka, Masaomi

PY - 2014/5/20

Y1 - 2014/5/20

N2 - We investigate the random walk process in relativistic flow. In the relativistic flow, photon propagation is concentrated in the direction of the flow velocity due to the relativistic beaming effect. We show that in the pure scattering case, the number of scatterings is proportional to the size parameter ξ ≡ L/l 0 if the flow velocity β ≡ v/c satisfies β/Γ ≫ ξ-1, while it is proportional to ξ2 if β/Γ ≪ ξ-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 τa is considerably smaller than the optical depth for scattering τs (τa/τs ≪ 1) and the flow velocity satisfies , then the effective optical depth is approximated by τ* ≃ τa(1 + β)/β. Furthermore, we perform Monte Carlo simulations of radiative transfer and compare the results with the analytic expression for the number of scatterings. The analytic expression is consistent with the results of the numerical simulations. The expression derived in this study can be used to estimate the photon production site in relativistic phenomena, e.g., gamma-ray burst and active galactic nuclei.

AB - We investigate the random walk process in relativistic flow. In the relativistic flow, photon propagation is concentrated in the direction of the flow velocity due to the relativistic beaming effect. We show that in the pure scattering case, the number of scatterings is proportional to the size parameter ξ ≡ L/l 0 if the flow velocity β ≡ v/c satisfies β/Γ ≫ ξ-1, while it is proportional to ξ2 if β/Γ ≪ ξ-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 τa is considerably smaller than the optical depth for scattering τs (τa/τs ≪ 1) and the flow velocity satisfies , then the effective optical depth is approximated by τ* ≃ τa(1 + β)/β. Furthermore, we perform Monte Carlo simulations of radiative transfer and compare the results with the analytic expression for the number of scatterings. The analytic expression is consistent with the results of the numerical simulations. The expression derived in this study can be used to estimate the photon production site in relativistic phenomena, e.g., gamma-ray burst and active galactic nuclei.

KW - gamma-ray burst: general

KW - radiative transfer

KW - relativistic processes

KW - scattering

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U2 - 10.1088/2041-8205/787/1/L4

DO - 10.1088/2041-8205/787/1/L4

M3 - Article

AN - SCOPUS:84900444242

SN - 2041-8205

VL - 787

JO - Astrophysical Journal Letters

JF - Astrophysical Journal Letters

IS - 1

M1 - L4

ER -