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Relativistic Variable Eddington Factor in a Relativistic Plane-Parallel Flow

We examine the behavior of the variable Eddington factor for a relativistically moving radiative flow in the vertical direction. We adopt the "one-tau photo-oval" approximation in the comoving frame. Namely, the comoving observer sees radiation coming from a closed surface where the optical depth measured from the observer is unity; such a surface is called a one-tau photo-oval. In general, the radiative intensity emitted by the photo-oval is non-uniform and anisotropic. Furthermore, the photo-oval surface has a relative velocity with respect to the comoving observer, and therefore, the observed intensity suffers from the Doppler effect and aberration. In addition, the background intensity usually depends on the optical depth. All of these introduce the anisotropy to the radiation field observed by the comoving observer. As a result, the relativistic Eddington factor $f$ generally depends on the optical depth $τ$, the four velocity $u$, and the velocity gradient $du/dτ$. % In the case of a plane-parallel vertical flow, we found that the relativistic variable Eddington factor $f$ generally decreases as the velocity gradient increases, but it increases as the velocity increases for some case. When the comoving radiation field is uniform, it is well approximated by $3f \sim 1/[ 1+ ({16}/{15})(-{du}/{γdτ}) +(-{du}/{γdτ})^{1.6-2} ]$. When the radiation field in the inertial frame is uniform, on the other hand, it is expressed as $f = (1+3β^2)/(3+β^2$). These relativistic variable Eddington factors can be used in various relativistic radiatively-driven flows, such as black-hole accretion flows, relativistic astrophysical jets and outflows, and relativistic explosions like gamma-ray bursts.