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

We revisit the relation between the variance of three-dimensional (3D) density (σ²ρ) and that of the projected two-dimensional (2D) column density (σ²Σ) in turbulent media, which is of great importance in obtaining turbulence properties from observations. Earlier studies showed that σ²_Σ / Σ₀/σ²_ρ / ρ₀ = R, where Σ/Σ₀ and ρ/ρ₀ are 2D column and 3D volume densities normalized by their mean values, respectively. The factor 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 R can be expressed in terms of N, the number of independent eddies along the line of sight. To be specific, σ²_Σ / Σ₀/σ²_ρ/ρ₀ is proportional to ~ 1/N, due to the averaging effect arising from independent eddies along the line of sight. Second, we show that the factor 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 σ²_Σ / Σ₀/σ²_ρ / ρ₀ 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 Nσ²_Σ / Σ₀ against σ²_ρ / ρ₀, 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.