The Astrophysical Journal (Jan 2024)
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 ( ${\sigma }_{\rho }^{2}$ ) and that of the projected two-dimensional (2D) column density ( ${\sigma }_{{\rm{\Sigma }}}^{2}$ ) in turbulent media, which is of great importance in obtaining turbulence properties from observations. Earlier studies showed that ${\sigma }_{{\rm{\Sigma }}/{{\rm{\Sigma }}}_{0}}^{2}/{\sigma }_{\rho /{\rho }_{0}}^{2}={ \mathcal R }$ , where Σ/Σ _0 and ρ / ρ _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, ${\sigma }_{{\rm{\Sigma }}/{{\rm{\Sigma }}}_{0}}^{2}/{\sigma }_{\rho /{\rho }_{0}}^{2}$ 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 ${ \mathcal R }$ 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 ${\sigma }_{{\rm{\Sigma }}/{{\rm{\Sigma }}}_{0}}^{2}/{\sigma }_{\rho /{\rho }_{0}}^{2}$ 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 $N{\sigma }_{{\rm{\Sigma }}/{{\rm{\Sigma }}}_{0}}^{2}$ against ${\sigma }_{\rho /{\rho }_{0}}^{2}$ , 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.
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