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

Abstract

We revisit the relation between the variance of three-dimensional (3D) density ( σρ2 ) and that of the projected two-dimensional (2D) column density ( σΣ2 ) in turbulent media, which is of great importance in obtaining turbulence properties from observations. Earlier studies showed that σΣ/Σ02/σρ/ρ02=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, σΣ/Σ02/σρ/ρ02 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 σΣ/Σ02/σρ/ρ02 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 NσΣ/Σ02 against σρ/ρ02 , 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.