Relativistic Variable Eddington Factor

TL;DR

This paper derives a relativistic variable Eddington factor in radiative flows, showing it depends on velocity and its gradient, and introduces the concept of a one-tau photo-oval affecting radiation anisotropy.

Contribution

The paper analytically derives a relativistic Eddington factor considering velocity gradients and introduces the concept of a one-tau photo-oval affecting radiation fields.

Findings

Eddington factor depends on velocity gradient and flow velocity.

The shape of the optical depth surface becomes an oval (photo-oval).

Analytic expression for Eddington factor in optically thick flows.

Abstract

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.