Velocity-Dependent Eddington Factor in Relativistic Radiative Flow

Abstract

We propose a variable Eddington factor, depending on the {\it flow velocity} $v$, for the relativistic radiative flow, whose velocity becomes of the order of the speed of light. When the gaseous flow is radiatively accelerated up to the relativistic regime, the velocity gradient becomes very large in the direction of the flow. As a result, the radiative diffusion may become {\it anisotropic} in the comoving frame of the gas. Hence, in a flow that is accelerated from subrelativistic to relativistic regimes, the Eddington factor should be different from 1/3 even in the diffusion limit. As a simple form, the velocity-dependent Eddington factor may be written as $f(\beta) = 1/3+(2/3)\beta$, where $\beta=v/c$. Using the velocity-dependent Eddington factor, we can solve the rigorous equations of the relativistic radiative flow accelerated up to the relativistic speed. We also propose a generalized form for a variable Eddington factor as a function of the optical depth $\tau$ as well as the flow velocity: %$f(\tau, \beta) = {1/3} + {2/3} % \frac{1+(\tau+1)\beta}{1+\tau+\beta}$ $f(\tau, \beta) = 1/3 + (2/3) [{1+(\tau+1)\beta}]/({1+\tau+\beta})$ for a spherically symmetric case. The velocity-dependent Eddington factor 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.