Relativistic Variable Eddington Factor

We analytically derive a relativistic variable Eddington factor in the relativistic radiative flow, and found that the Eddington factor depends on the {\it velocity gradient} as well as the flow velocity. When the gaseous flow is accelerated and there is a velocity gradient, there also exists a density gradient. As a result, an unobstructed viewing range by a comoving observer, where the optical depth measured from the comoving observer is unity, is not a sphere, but becomes an oval shape elongated in the direction of the flow; we call it a {\it one-tau photo-oval}. For the comoving observer, an inner wall of the photo-oval generally emits at a non-uniform intensity, and has a relative velocity. Thus, the comoving radiation fields observed by the comoving observer becomes {\it anisotropic}, and the Eddington factor must deviate from the value for the isotropic radiation fields. % In the case of a plane-parallel vertical flow, we examine the photo-oval and obtain the Eddington factor. In the sufficiently optically thick linear regime, the Eddington factor is analytically expressed as $f (\tau, \beta, \frac{d\beta}{d\tau}) = {1/3} (1 + {16/15} \frac{d\beta}{d\tau})$, where $\tau$ is the optical depth and $\beta$ ($=v/c$) is the flow speed normalized by the speed of light. %i.e., the Eddington factor depends on the velocity gradient. We also examine the linear and semi-linear regimes, and found that the Eddington factor generally depends both on the velocity and its gradient.