The relation between the true and observed fractal dimensions of
  turbulent clouds
  Observations of interstellar gas clouds are typically limited to
two-dimensional (2D) projections of the intrinsically three-dimensional (3D)
structure of the clouds. In this study, we present a novel method for relating
the 2D projected fractal dimension ($\mathcal{D}_{\text{p}}$) to the 3D fractal
dimension ($\mathcal{D}_{\text{3D}}$) of turbulent clouds. We do this by
computing the fractal dimension of clouds over two orders of magnitude in
turbulent Mach number $(\mathcal{M} = 1-100)$, corresponding to seven orders of
magnitude in spatial scales within the clouds. This provides us with the data
to create a new empirical relation between $\mathcal{D}_{\text{p}}$ and
$\mathcal{D}_{\text{3D}}$. The proposed relation is
$\mathcal{D}_{\text{3D}}(\mathcal{D}_{\text{p}}) = \Omega_1 erfc ( \xi_1
erfc^{-1}[ (\mathcal{D}_{\text{p}} - \mathcal{D}_{\text{p,min}})/\Omega_2 ] +
\xi_2 ) + \mathcal{D}_{\text{3D,min}}$, where the minimum 3D fractal dimension,
$\mathcal{D}_{\text{3D,min}} = 2.06 \pm 0.35$, the minimum projected fractal
dimension, $\mathcal{D}_{\text{p,min}} = 1.55 \pm 0.13$, $\Omega_1 = 0.47 \pm
0.18$, $\Omega_2 = 0.22 \pm 0.07$, $\xi_1 = 0.80 \pm 0.18$ and $\xi_2 = 0.26
\pm 0.19$. The minimum 3D fractal dimension, $\mathcal{D}_{\text{3D,min}} =
2.06 \pm 0.35$, indicates that in the high $\mathcal{M}$ limit the 3D clouds
are dominated by planar shocks. The relation between $\mathcal{D}_{\text{p}}$
and $\mathcal{D}_{\text{3D}}$ of molecular clouds may be a useful tool for
those who are seeking to understand the 3D structures of molecular clouds,
purely based upon 2D projected data and shows promise for relating the physics
of the turbulent clouds to the fractal dimension.
