2001ApJ...557..727V

We discuss the probability distribution function (PDF) of column density resulting from density fields with lognormal PDFs, applicable to isothermal gas (e.g., probably molecular clouds). For magnetic and nonmagnetic numerical simulations of compressible, isothermal turbulence forced at intermediate scales ((1)/(4) of the box size), we find that the autocorrelation function (ACF) of the density field decays over relatively short distances compared to the simulation size. We suggest that a ``decorrelation length'' can be defined as the distance over which the density ACF has decayed to, for example, 10% of its zero-lag value, so that the density ``events'' along a line of sight can be assumed to be independent over distances larger than this, and the central limit theorem should be applicable. However, using random realizations of lognormal fields, we show that the convergence to a Gaussian is extremely slow in the high-density tail. As a consequence, the column density PDF is not expected to exhibit a unique functional shape, but to transit instead from a lognormal to a Gaussian form as the ratio η of the column length to the decorrelation length (i.e., the number of independent events in the cloud) increases. Simultaneously, the variance of the PDF decreases. For intermediate values of η, the column density PDF assumes a nearly exponential decay. For cases with a density contrast of 10⁴, as found in intermediate-resolution simulations, and expected from giant molecular clouds (GMCs) to dense molecular cores, the required value of η for convergence to a Gaussian is at least a few hundred, or, for 10⁶, several thousand. We then discuss the density power spectrum and the expected value of η in actual molecular clouds, concluding that they are uncertain since they may depend on several physical parameters. Observationally, our results suggest that η may be inferred from the shape and width of the column density PDF in optically thin line or extinction studies. Our results should also hold for gas with finite-extent power-law underlying density PDFs, which should be characteristic of the diffuse, nonisothermal neutral medium (with temperatures ranging from a few hundred to a few thousand degrees). Finally, we note that for η≳100, the dynamic range in column density is small (less than a factor of 10), but this is only an averaging effect, with no implication on the dynamic range of the underlying density distribution.