TreeCol: a novel approach to estimating column densities in
  astrophysical simulations
  We present TreeCol, a new and efficient tree-based scheme to calculate column
densities in numerical simulations. Knowing the column density in any direction
at any location in space is a prerequisite for modelling the propagation of
radiation through the computational domain. TreeCol therefore forms the basis
for a fast, approximate method for modelling the attenuation of radiation
within large numerical simulations. It constructs a HEALPix sphere at any
desired location and accumulates the column density by walking the tree and by
adding up the contributions from all tree nodes whose line of sight contributes
to the pixel under consideration. In particular when combined with widely-used
tree-based gravity solvers the new scheme requires little additional
computational cost. In a simulation with $N$ resolution elements, the
computational cost of TreeCol scales as $N \log N$, instead of the $N^{5/3}$
scaling of most other radiative transfer schemes. TreeCol is naturally
adaptable to arbitrary density distributions and is easy to implement and to
parallelize. We discuss its accuracy and performance characteristics for the
examples of a spherical protostellar core and for the turbulent interstellar
medium. We find that the column density estimates provided by TreeCol are on
average accurate to better than 10 percent. In another application, we compute
the dust temperatures for solar neighborhood conditions and compare with the
result of a full-fledged Monte Carlo radiation-transfer calculation. We find
that both methods give very similar answers. We conclude that TreeCol provides
a fast, easy to use, and sufficiently accurate method of calculating column
densities that comes with little additional computational cost when combined
with an existing tree-based gravity solver.
