Variable Eddington Factor in a Relativistic Plane-Parallel Flow

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We examine the Eddington factor in an optically thick, relativistic flow accelerating in the vertical direction. % When the gaseous flow is radiatively accelerated and there is a velocity gradient, there also exists a density gradient. The comoving observer sees radiation coming from a closed surface where the optical depth measured from the observer is unity. Such a surface, called a one-tau photo-oval, is elongated in the flow direction. In general, the radiation intensity emitted by the photo-oval is non-uniform, and the photo-oval surface has a relative velocity with respect to the position of the comoving observer. Both effects introduce some degree of anisotropy in the radiation field observed in the comoving frame. As a result, the radiation field observed by the comoving observer becomes anisotropic, and the Eddington factor must deviate from the usual value of 1/3. Thus, the relativistic Eddington factor generally depends on the optical depth τ and the velocity gradient du/dτ, u being the four velocity. % In the case of a plane-parallel vertical flow, we obtain the shape of the photo-oval and calculate the Eddington factor in the optically thick regime. We found that the Eddington factor f is well approximated by f(τ, du/dτ) = 1/3 (1/u du/dτ). % This relativistic variable Eddington factor can be used in various relativistic radiatively-driven flows.