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

Abstract

Abstract We revisit the relation between the variance of three-dimensional (3D) density ( σ ρ 2 ) and that of the projected two-dimensional (2D) column density ( σ Σ 2 ) in turbulent media, which is of great importance in obtaining turbulence properties from observations. Earlier studies showed that σ Σ / Σ 0 2 / σ ρ / ρ 0 2 = R , where Σ/Σ0 and ρ/ρ 0 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, σ Σ / Σ 0 2 / σ ρ / ρ 0 2 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 σ Σ / Σ 0 2 / σ ρ / ρ 0 2 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 σ Σ / Σ 0 2 against σ ρ / ρ 0 2 , 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.