The Relation Between Variances of a 3D Density and Its 2D Column Density Revisited

TL;DR

This paper revisits the relationship between 3D density variance and 2D column density variance in turbulent media, providing new insights into how the number of independent eddies and cloud dimensions affect this relation, with implications for observational turbulence analysis.

Contribution

It introduces a new formulation expressing the variance ratio in terms of the number of eddies, N, and clarifies how cloud dimensions influence this relation, enhancing interpretation of observational data.

Findings

Variance ratio inversely proportional to N

Dimension differences modify the variance ratio expression

N times column density variance correlates with 3D density variance

Abstract

We revisit the relation between the variance of three-dimensional (3D) density ($\sigma^{2}_{\rho}$) and that of the projected two-dimensional (2D) column density ($\sigma^{2}_{\Sigma}$) in turbulent media, which is of great importance in obtaining turbulence properties from observations. Earlier studies showed that $\sigma^{2}_{\Sigma / \Sigma_{0}}/\sigma^{2}_{\rho / \rho_{0}} = \mathcal{R}$, where $\Sigma/\Sigma_0$ and $\rho/\rho_0$ are 2D column and 3D volume densities normalized by their mean values, respectively. The factor $\mathcal{R}$ depends only on the density spectrum for isotropic turbulence in a cloud that has similar dimensions along and perpendicular to the line of sight. Our major findings in this paper are as follows. First, we show that the factor $\mathcal{R}$ can be expressed in terms of $N$, the number of independent eddies along the line of sight. To be specific, $\sigma^{2}_{\Sigma / \Sigma_{0}}/\sigma^{2}_{\rho/\rho_{0}}$ is proportional to $\sim 1/N$, due to the averaging effect arising from independent eddies along the line of sight. Second, we show that the factor $\mathcal{R}$ needs to be modified if the dimension of the cloud in the line-of-sight direction is different from that in the perpendicular direction. However, if we express $\sigma^{2}_{\Sigma / \Sigma_{0}}/\sigma^{2}_{\rho / \rho_{0}}$ in terms of $N$, the expression remains same even in the case the cloud has different dimensions along and perpendicular to the line of sight. Third, when we plot $N\sigma^{2}_{\Sigma / \Sigma_{0}}$ against $\sigma^{2}_{\rho / \rho_{0}}$, two quantities roughly lie on a single curve regardless of the sonic Mach number, which implies that we can directly obtain the latter from the former. We discuss observational implications of our findings.