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

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.