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

We revisit the relation between the variance of three-dimensional (3D) density ( sigma rho 2 ) and that of the projected two-dimensional (2D) column density ( sigma Sigma 2 ) in turbulent media, which is of great importance in obtaining turbulence properties from observations. Earlier studies showed that sigma Sigma/Sigma 02/sigma rho/rho 02=R , where Sigma/Sigma 0 and rho/rho 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, sigma Sigma/Sigma 02/sigma rho/rho 02 is proportional to similar 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 sigma Sigma/Sigma 02/sigma rho/rho 02 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 Sigma/Sigma 02 against sigma rho/rho 02 , 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.