Optical depth in polarised Monte Carlo radiative transfer
  Context: The Monte Carlo method is the most widely used method to solve
radiative transfer problems in astronomy, especially in a fully general 3D
geometry. A crucial concept in any Monte Carlo radiative transfer code is the
random generation of the next interaction location. In polarised Monte Carlo
radiative transfer with aligned non-spherical grains, the nature of dichroism
complicates the concept of optical depth. Aims: We investigate in detail the
relation between optical depth and the optical properties and density of the
attenuating medium in polarised Monte Carlo radiative transfer codes that take
into account dichroic extinction. Methods: Based on solutions for the radiative
transfer equation, we discuss the optical depth scale in polarised radiative
transfer with spheroidal grains. We compare the dichroic optical depth to the
extinction and total optical depth scale. Results: In a dichroic medium, the
optical depth is not equal to the usual extinction optical depth, nor to the
total optical depth. For representative values of the optical properties of
dust grains, the dichroic optical depth can differ from the extinction or total
optical depth by several ten percent. A closed expression for the dichroic
optical depth cannot be given, but it can be derived efficiently through an
algorithm that is based on the analytical result corresponding to elongated
grains with a uniform grain alignment. Conclusions: Optical depth is more
complex in dichroic media than in systems without dichroic attenuation, and
this complexity needs to be considered when generating random free path lengths
in Monte Carlo radiative transfer simulations. There is no benefit in using
approximations instead of the dichroic optical depth.
