Velocity-Dependent Eddington Factor in Relativistic Radiative Flow

We propose a variable Eddington factor, depending on the {\it flow velocity}$v$, for the relativistic radiative flow, whose velocity becomes of the orderof the speed of light. When the gaseous flow is radiatively accelerated up tothe relativistic regime, the velocity gradient becomes very large in thedirection of the flow. As a result, the radiative diffusion may become {\itanisotropic} in the comoving frame of the gas. Hence, in a flow that isaccelerated from subrelativistic to relativistic regimes, the Eddington factorshould be different from 1/3 even in the diffusion limit. As a simple form, thevelocity-dependent Eddington factor may be written as $f(\beta) =1/3+(2/3)\beta$, where $\beta=v/c$. Using the velocity-dependent Eddingtonfactor, we can solve the rigorous equations of the relativistic radiative flowaccelerated up to the relativistic speed. We also propose a generalized formfor a variable Eddington factor as a function of the optical depth $\tau$ aswell 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. Thevelocity-dependent Eddington factor can be used in various relativisticradiatively-driven flows, such as black-hole accretion flows, relativisticastrophysical jets and outflows, and relativistic explosions like gamma-raybursts.Comment: 7 pages, 3 figures, PASJ 58 (2006), No 2, in pres