The reliability of observational measurements of column density probability distribution functions

Abstract

Context. Probability distribution functions (PDFs) of column densities are an established tool to characterize the evolutionary state of interstellar clouds. Aims. Using simulations, we show to what degree their determination is affected by noise, line-of-sight contamination, field selection, and the incomplete sampling in interferometric measurements. Methods. We solve the integrals that describe the convolution of a cloud PDF with contaminating sources such as noise and line-ofsight emission, and study the impact of missing information on the measured column density PDF. In this way we can quantify the effect of the different processes and propose ways to correct for their impact to recover the intrinsic PDF of the observed cloud. Results. The effect of observational noise can be easily estimated and corrected for if the root mean square (rms) of the noise is known. For σ noise values below 40% of the typical cloud column density, N peak , this involves almost no degradation in the accuracy of the PDF parameters. For higher noise levels and narrow cloud PDFs the width of the PDF becomes increasingly uncertain. A contamination by turbulent foreground or background clouds can be removed as a constant shield if the peak of the contamination PDF falls at a lower column or is narrower than that of the observed cloud. Uncertainties in cloud boundary definition mainly affect the low-column density part of the PDF and the mean density. As long as more than 50% of a cloud is covered, the impact on the PDF parameters is negligible. In contrast, the incomplete sampling of the uv-plane in interferometric observations leads to uncorrectable PDF distortions in the maps produced. An extension of the capabilities of the Atacama Large Millimeter Array (ALMA) would allow us to recover the high-column density tail of the PDF, but we found no way to measure the intermediate-and low-column density part of the underlying cloud PDF in interferometric observations.

The shapes of column density PDFs

“…The last closed contour represents then the natural completeness limit of a cloud PDF, if above the nominal noise of the map and not affected by an unrelated cloud inside the same map. The concept of the last closed contour is not new in the literature; in particular regarding column density PDFs, Kainulainen et al (2011Kainulainen et al ( , 2013 and Ossenkopf et al (2016) point to it as the natural definition of completeness of a column density PDF. We note that for a real cloud, and depending on cloud structure, column density dynamic range, and map size, there may likely be more than one closed connected surface like the red connected surface in Fig.…”

“…Ossenkopf et al (2016) conclude that inferences from the PDF parameters can be wrong by large factors, in particular at the low column density end of the PDF, but these authors fall short of ruling out the log-normal peak of the PDF as an artifact of the observations of molecular clouds. The conclusion in Ossenkopf et al (2016) on the impact of survey boundaries was a call for surveys to cover at least 50% of a cloud complex. While this is trivial to do in a controlled numerical experiment, clouds have ill-defined boundaries and observationally it is simply not possible to determine 50% of an unknown cloud size.…”

“…Also recently, Ossenkopf et al (2016) investigated through simulations of synthetic clouds how the determination of PDFs is affected by noise, line-of-sight contamination, survey boundaries, and the incomplete sampling in interferometric measurements. Ossenkopf et al (2016) conclude that inferences from the PDF parameters can be wrong by large factors, in particular at the low column density end of the PDF, but these authors fall short of ruling out the log-normal peak of the PDF as an artifact of the observations of molecular clouds. The conclusion in Ossenkopf et al (2016) on the impact of survey boundaries was a call for surveys to cover at least 50% of a cloud complex.…”

Herschelobservations of the Galactic H ii region RCW 79

“…The PDFs do not differ much (see Schneider et al 2015b;Ossenkopf-Okada et al 2016, for resolution effects on PDFs), therefore we only present the PDF of the highresolution map here. We did not perform a background sub-7 To be consistent with other Herschel studies, we used the visual extinction Av instead of the hydrogen column density, linked by NH 2 = Av × 0.94 × 10 21 cm −2 mag −1 (Bohlin et al 1978).…”

How well does CO emission measure the H2mass of MCs?

“…the abundance variations and the optical depth do not play significant roles, the recovery of the true distribution might be hindered by detection noise, line of sight contamination, field selection and incomplete sampling in interferometric measurements (Ossenkopf et al 2016). …”