Quantifying the Significance of the Magnetic Field from Large-Scale
  Cloud to Collapsing Core: Self-Similarity, Mass-to-Flux Ratio and Star
  Formation Efficiency
  Dust polarization observational results are analyzed for the high-mass star
formation region W51 from the largest parent cloud ($\sim$ 2~pc, JCMT) to the
large-scale envelope ($\sim$ 0.5~pc, BIMA) down to the collapsing core e2
($\sim$ 60~mpc, SMA). Magnetic field and dust emission gradient orientations
reveal a correlation which becomes increasingly more tight with higher
resolution. The previously developed polarization - intensity gradient method
(Koch et al. 2012) is applied in order to quantify the magnetic field
significance. This technique provides a way to estimate the local magnetic
field force compared to gravity without the need of any mass or field strength
measurements, solely making use of measured angles which reflect the
geometrical imprint of the various forces. All three data sets clearly show
regions with distinct features in the field-to-gravity force ratio. Azimuthally
averaged radial profiles of this force ratio reveal a transition from a field
dominance at larger distances to a gravity dominance closer to the emission
peaks. Normalizing these profiles to a characteristic core scale points toward
self-similarity. Furthermore, the polarization intensity-gradient method is
linked to the mass-to-flux ratio, providing a new approach to estimate the
latter one without mass and field strength inputs. A transition from a
magnetically supercritical to a subcritical state as a function of distance
from the emission peak is found for the e2 core. Finally, based on the measured
radius-dependent field-to-gravity force ratio we derive a modified star
formation efficiency with a diluted gravity force. Compared to a standard
(free-fall) efficiency, the observed field is capable of reducing the
efficiency down to 10\% or less.
