# Optical depth in polarised Monte Carlo radiative transfer

## TL;DR

This paper explores how optical depth in polarised Monte Carlo radiative transfer differs from traditional measures due to dichroism, emphasizing the need for accurate calculations in complex astrophysical environments.

## Contribution

It provides a detailed analysis of the dichroic optical depth, contrasting it with extinction and total optical depth, and offers an efficient algorithm for its calculation in polarised radiative transfer.

## Key findings

- Dichroic optical depth can differ from extinction or total optical depth by several ten percent.
- In dichroic media, optical depth is more complex and cannot be approximated by simpler measures.
- An efficient algorithm for calculating dichroic optical depth is proposed.

## Abstract

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.