Radiative transfer in the refractive atmospheres of very cool white dwarfs

We consider the problem of radiative transfer in stellar atmospheres where the index of refraction departs from unity and is a function of density and temperature. We present modified Feautrier and Lambda-iteration methods to solve the equation of radiative transfer with refraction in a plane parallel atmosphere. These methods are general and can be used in any problem with 1-D geometry where the index of refraction is a monotonically varying function of vertical optical depth. We present an application to very cool white dwarf atmospheres where the index of refraction departs significantly from unity. We investigate how ray curvature and total internal reflection affect the limb darkening and the pressure-temperature structure of the atmosphere. Refraction results in a much weakened limb darkening effect. We find that through the constraint of radiative equilibrium, total internal reflection warms the white dwarf atmosphere near the surface ($\tau \wig< 1$). This effect may have a significant impact on studies of very cool white dwarf stars.