The Relation Between Variances of a 3D Density and Its 2D Column Density Revisited
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2024
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| author | Yoon, Heesun Cho, Jungyeon |
| author_facet | Yoon, Heesun Cho, Jungyeon |
| contents | We revisit the relation between the variance of three-dimensional (3D) density ($σ^{2}_ρ$) and that of the projected two-dimensional (2D) column density ($σ^{2}_Σ$) in turbulent media, which is of great importance in obtaining turbulence properties from observations. Earlier studies showed that $σ^{2}_{Σ/ Σ_{0}}/σ^{2}_{ρ/ ρ_{0}} = \mathcal{R}$, 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, $σ^{2}_{Σ/ Σ_{0}}/σ^{2}_{ρ/ρ_{0}}$ is proportional to $\sim 1/N$, 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 $σ^{2}_{Σ/ Σ_{0}}/σ^{2}_{ρ/ ρ_{0}}$ 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 $Nσ^{2}_{Σ/ Σ_{0}}$ against $σ^{2}_{ρ/ ρ_{0}}$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_17022 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Relation Between Variances of a 3D Density and Its 2D Column Density Revisited Yoon, Heesun Cho, Jungyeon Astrophysics of Galaxies We revisit the relation between the variance of three-dimensional (3D) density ($σ^{2}_ρ$) and that of the projected two-dimensional (2D) column density ($σ^{2}_Σ$) in turbulent media, which is of great importance in obtaining turbulence properties from observations. Earlier studies showed that $σ^{2}_{Σ/ Σ_{0}}/σ^{2}_{ρ/ ρ_{0}} = \mathcal{R}$, 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, $σ^{2}_{Σ/ Σ_{0}}/σ^{2}_{ρ/ρ_{0}}$ is proportional to $\sim 1/N$, 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 $σ^{2}_{Σ/ Σ_{0}}/σ^{2}_{ρ/ ρ_{0}}$ 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 $Nσ^{2}_{Σ/ Σ_{0}}$ against $σ^{2}_{ρ/ ρ_{0}}$, 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. |
| title | The Relation Between Variances of a 3D Density and Its 2D Column Density Revisited |
| topic | Astrophysics of Galaxies |
| url | https://arxiv.org/abs/2406.17022 |