Issue 
A&A
Volume 553, May 2013



Article Number  L8  
Number of page(s)  6  
Section  Letters  
DOI  https://doi.org/10.1051/00046361/201321431  
Published online  17 May 2013 
Online material
Appendix A: Numerical effects on the DGMFs
Appendix A.1: Effect of sink particles
Simulation properties (adapted from FK13).
Sink particles (Federrath et al. 2010a) in the simulations accrete material into them after their creation and affect the density structure of their immediate surroundings in the simulation (and the DGMFs). In the following, we consider the effects of sink particles to the DGMFs.
As described in FK12, the sink particles are created on a certain, resolutiondependent volume density and always have a radius of 2.5 pixels in the native resolution of the simulation. It follows that the sink particles have a resolutiondependent minimum density, which can be converted further into a minimum mean column density. Sink particles are created when a series of collapse criteria are fulfilled (see FK12), and when the local volume density exceeds (A.1)where c_{s} is the isothermal speed of sound and r_{sink} the radius of the sink particle. It follows that the mean column density of a sink particle at the moment of its creation is (A.2)The sink particle properties are listed in Table A.1 for different physical resolutions.
The sink particle column densities listed in Table A.1 represent levels below which the DGMFs are not affected by sink particles, regardless of whether the sinks are removed or not. In the most conservative interpretation, the DGMFs are reliable only below these column density limits. Therefore, we use the upper limit of N(H_{2}) = 11 × 10^{21} cm^{2}, which is the sink particle column density for the ℳ_{s} = 10 simulations 512^{3} cells in size, in the analysis performed in this paper.
However, it is not at all certain that the DGMF shape immediately above N(H_{2})_{sink} is greatly affected by the sink particles. Above N(H_{2})_{sink}, there are linesofsight whose column density is higher than the sink particle column density, but the local volume densities do not reach high enough values for sink particles to form. In fact, these linesofsight are more numerous in the simulations compared to those that contain sinks, especially at early times when the overall SFE is low.
We dealt with sink particles in this work by disregarding the linesofsight affected by them directly from the simulation data. While this procedure, in principle, eliminates the effects of sink particles, it removes mass from high column densities and can bias the DGMF downwards (steepen it). Consequently, it is important to note that the flattening of the DGMFs seen in the simulations (see Sect. 3.1) at around N(H_{2}) ≈ 10−15 × 10^{21} cm^{2} cannot be due to sink particle treatment, any associated incompleteness would bias the determination downwards, not upwards.
We can quantify the incompleteness due to sink particle removal by comparing DGMFs derived with and without the elimination of sink particles. This experiment is shown in Fig. A.1, which shows the ratio of the DGMFs with and without the sink particle elimination as a function of column density. The plot is shown for the model in which the effect of sinks in the examined column density range is expected to be strongest, i.e., the solenoidal simulation with 256^{3} cell resolution. Higher resolution increases the sink particle column density (cf. Table A.1), and more compressive forcing increases the relative amount of high column densities, thereby reducing the error in the examined column density regime. The figure shows that the error due to incompleteness (i.e., removal of highcolumn densities) is less than 30% below N(H_{2}) ≲ 25 × 10^{21} cm^{2} for SFEs up to 10%.
In summary, it can be concluded that the DGMFs derived for ℳ_{s} = 10 simulations are unaffected by the sink particles (or by their removal) below the N(H_{2})_{sink} values. In addition, the error in the predicted DGMFs is less than 30% when the range up to N(H_{2}) ≈ 25 × 10^{21} cm^{2} is considered.
Appendix A.2: Effect of the simulation resolution
The simulations of FK12 are either 128^{3}, 256^{3}, 512^{3}, or 1024^{3} computational cells in size. In this work, we used all but those simulations that are 128^{3} cells in size. It is possible that the different computational resolutions used in the simulations affect the DGMFs, because especially high column densities are potentially better resolved by highresolution simulations. We examined the possible effect of the simulation resolution to the DGMFs by comparing the DGMFs of simulations that were run with the same physical parameters, but have different computational resolution.
Figure A.2 shows as an example a comparison of DGMFs derived for models #10 and #11 that are 256^{3} and 512^{3} cells in size, respectively. All other parameters are same in these two models. The DGMF of model #10 is in good agreement with that of model #11 below the sink particle column density, N(H_{2}) = 11 × 10^{21} cm^{2}. At higher column densities, the lower resolution simulation (#10) begins to underestimate the column densities slightly. However, it is still within 30% of the higher resolution one up to the column density of N(H_{2}) ≈ 25 × 10^{21} cm^{2}. We conclude that the effect of resolution is less than the uncertainty due to the projection effects in the column density range N(H_{2}) = 11 × 10^{21} cm^{2} and accurate to 70% level up to N(H_{2}) = 25 × 10^{21} cm^{2}.
Fig. A.1
Error (incompleteness) in the derived DGMFs due to removal of sink particles. The figure shows the ratio of DGMFs derived with and without sink particle removal as a function of column density for time steps up to SFE = 10%. The curves for t = 0 and SFE = 0% are indistinguishable from unity. The plot is shown for simulation #10 (ℳ_{s} = 10, 256^{3} cells in size, b = 1/3). The error in other ℳ_{s} = 10 models is expected to be smaller, because of the higher sink particle column density and more compressive turbulence driving. 

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Fig. A.2
Effect of simulation resolution to the DGMFs. The red line shows the DGMF of simulation #10 (256^{3} cells in size) divided by the DGMF of simulation #11 (512^{3} cells in size). The physical parameters of the two simulations are the same. The dashed lines show the DGMFs calculated for different projections of model #11 divided by the mean DGMF of model #11. 

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Appendix B: Illustration of column density PDFs
Fig. B.1
Column density PDFs of models #11 (b = 1/3) and #24 (b = 1), and the PDF of the Taurus molecular cloud. Both models have ℳ_{s} = 10 and B = 0 μG, and they are 512^{3} computational cells in size. The black histograms show the PDFs of model #11 at t = 0 (solid line) and SFE = 5% (dotted line). The blue line shows the PDF of model #24. The red line shows the PDF of Taurus from Kainulainen et al. (2009). The dynamic range of the Taurus PDF ends at about lnN(H_{2}) = 3.2. The black dashed line shows, for reference, a lognormal function. The PDFs in the range N(H_{2}) = 3−11 × 10^{21} cm^{2} can be described by a lognormal function, but also reasonably well by a powerlaw function (which would be a linear curve in the given presentation). 

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Figure B.1 show a comparison of the column density PDFs derived for models #11 and #24, and the PDF of the Taurus molecular cloud from Kainulainen et al. (2009). The higher relative amount of highcolumn density material predicted by fully compressive simulations (#24) causes a flatter PDF. In the column density range N(H_{2}) = 3−25 × 10^{21} cm^{2}, the PDF of simulation #11 is close to what is observed in Taurus. In this narrow range, the PDF is in reasonable agreement with either a lognormal function (shown for a reference in the figure) or a powerlaw function.
© ESO, 2013
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