Analytical Solutions of Relativistic Plane-Parallel Flows.

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Relativistic radiative transfer in a relativistic plane-parallel flow is examined in the fully special relativistic treatment. Under the assumption of a constant flow speed, we obtain simple analytical solutions of the relativistic radiative transfer equation for relativistic plane-parallel flows in the cases of linear-flow and two-stream approximations. In both cases, the solutions exhibit exponential behavior on the optical depth. In addition, the optical depth τ in the exponential term is replaced by the apparent optical depth Γτ, where Γ is a function of the flow speed v (= βc); for example, in the case of the linear-flow approximation, Γ = γβ, γ being the Lorentz factor. This modification of the apparent/effective optical depth is an essential property of the relativistic radiative transfer. Furthermore, in the nonrelativistic regime, the radiative intensity in the comoving frame increases as the optical depth increases, whereas it approaches a constant value with the optical depth in the relativistic regime because of the exponential term. The radiation energy density in the comoving frame also approaches a constant value, while the radiative flux becomes zero as the optical depth increases. Such behavior of the radiative quantities in the comoving frame, which originates from the exponential term, is also an essential property of the relativistic radiative transfer.

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