Radiative Transfer and Limb Darkening of Accretion Disks

TL;DR

This paper analytically investigates how the emergent intensity and limb darkening or brightening effects in accretion disks vary with optical depth, considering cases with and without internal heating.

Contribution

It provides a detailed analytical analysis of radiative transfer and limb darkening in thin accretion disks across different optical depths and heating conditions.

Findings

Limb darkening occurs at large optical depths.

Limb brightening occurs at small optical depths with internal heating.

Emergent intensity becomes isotropic at small optical depths without heating.

Abstract

Transfer equation in a geometrically thin accretion disk is reexamined under the plane-parallel approximation with finite optical depth. Emergent intensity is analytically obtained in the cases with or without internal heating. For large or infinite optical depth, the emergent intensity exhibits a usual limb-darkening effect, where the intensity linearly changes as a function of the direction cosine. For small optical depth, on the other hand, the angle-dependence of the emergent intensity drastically changes. In the case without heating but with uniform incident radiation at the disk equator, the emergent intensity becomes isotropic for small optical depth. In the case with uniform internal heating, the limb brightening takes place for small optical depth. We also emphasize and discuss the limb-darkening effect in an accretion disk for several cases.