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 hydrodynamic simulations of discs. As in the method of Stamatellos et al., 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 et al. 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 et al. method performs better than our method in this regime. Thus, a hybrid method (where the method is switched in regions of large overdensity) may be optimal.