Probing Three-Dimensional Magnetic Fields: I -- Polarized Dust Emission
  The plane-of-the-sky component of interstellar magnetic fields can be traced
in two dimensions using polarized dust emission. Its potential to access
three-dimensional magnetic fields, including the inclination of the magnetic
fields relative to the line-of-sight, is crucial for a variety of astrophysical
problems. Based on the statistical features of the polarization fraction and
the averaged POS Alfv\'en Mach number $\overline{M_{\rm A}}_{,\bot}$, we
present a new method for estimating the inclination angle. The magnetic field
fluctuations raised by MHD turbulence are taken into account in our method. By
using synthetic dust emission from 3D MHD turbulence simulation, we show that
the fluctuations are preferentially perpendicular to the mean magnetic field
and magnify the depolarization effect. We analytically and numerically derive
that the polarization fraction is characterized by both the inclination angle
and magnetic field fluctuations. We propose and demonstrate that the mean
inclination angle over a region of interest can be calculated from the
polarization fraction in a strongly magnetized reference position, where
$\overline{M_{\rm A}}_{,\bot}^2\ll1$. We test the new method in sub-Alfv\'enic,
trans-Alfv\'enic, and moderately super-Alfv\'enic situations ($0.4\lesssim
M_{\rm A}\lesssim1.2$) and show that it recovers the mean inclination angle
well. This method is extended to determine the inclination angle distribution
over the sub-regions of a cloud.
