Relativistic Variable Eddington Factor

Abstract

Abstract We analytically derived a relativistic variable Eddington factor in relativistic radiative flow, and found that the Eddington factor depends on the 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 one-tau photo-oval. For a 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 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 a sufficiently optically thick linear regime, the Eddington factor is analytically expressed as $f (\tau, \beta, \frac{d\beta}{d\tau}) = \frac{1}{3} (1 + \frac{16}{15} \frac{d\beta}{d\tau})$, where $\tau$ is the optical depth and $\beta$ ($=v/c$) is the flow speed normalized by the speed of light. We also examined the linear and semi-linear regimes, and found that the Eddington factor generally depends both on the velocity and its gradient.