Reliable estimation of the column density in Smoothed Particle
  Hydrodynamic simulations
  We describe a simple method for estimating the vertical column density in
Smoothed Particle Hydrodynamics (SPH) simulations of discs. As in the method of
Stamatellos et al. (2007), the column density is estimated using pre-computed
local quantities and is then used to estimate the radiative cooling rate. The
cooling rate is a quantity of considerable importance, for example, in
assessing the probability of disc fragmentation. Our method has three steps:
(i) the column density from the particle to the mid plane is estimated using
the vertical component of the gravitational acceleration, (ii) the "total
surface density" from the mid plane to the surface of the disc is calculated,
(iii) the column density from each particle to the surface is calculated from
the difference between (i) and (ii). This method is shown to greatly improve
the accuracy of column density estimates in disc geometry compared with the
method of Stamatellos. On the other hand, although the accuracy of our method
is still acceptable in the case of high density fragments formed within discs,
we find that the Stamatellos method performs better than our method in this
regime. Thus, a hybrid method (where the method is switched in regions of large
over-density) may be optimal.
