HAWC+/SOFIA Polarimetry in L1688: Relative Orientation of Magnetic Field
  and Elongated Cloud Structure
  We present a study of the relative orientation between the magnetic field and
elongated cloud structures for the $\rho$ Oph A and $\rho$ Oph E regions in
L1688 in the Ophiuchus molecular cloud. Combining inferred magnetic field
orientation from HAWC+ 154 $\mu$m observations of polarized thermal emission
with column density maps created using Herschel submillimeter observations, we
find consistent perpendicular relative alignment at scales of $0.02$ pc
($33.6"$ at $d \approx 137$ pc) using the histogram of relative orientations
(HRO) technique. This supports the conclusions of previous work using Planck
polarimetry and extends the results to higher column densities. Combining this
HAWC+ HRO analysis with a new Planck HRO analysis of L1688, the transition from
parallel to perpendicular alignment in L1688 is observed to occur at a
molecular hydrogen column density of approximately $10^{21.7}$ cm$^{-2}$. This
value for the alignment transition column density agrees well with values found
for nearby clouds via previous studies using only Planck observations. Using
existing turbulent, magnetohydrodynamic simulations of molecular clouds formed
by colliding flows as a model for L1688, we conclude that the molecular
hydrogen volume density associated with this transition is approximately
$\sim10^{4}$ cm$^{-3}$. We discuss the limitations of our analysis, including
incomplete sampling of the dense regions in L1688 by HAWC+.
