Magnetic Field Strength Maps for Molecular Clouds: A New Method Based on
  a Polarization - Intensity Gradient Relation
  Dust polarization orientations in molecular clouds often tend to be close to
tangential to the Stokes $I$ dust continuum emission contours. The magnetic
field and the emission gradient orientations, therefore, show some correlation.
A method is proposed, which -- in the framework of ideal magneto-hydrodynamics
(MHD) -- connects the measured angle between magnetic field and emission
gradient orientations to the total field strength. The approach is based on the
assumption that a change in emission intensity (gradient) is a measure for the
resulting direction of motion in the MHD force equation. In particular, this
new method leads to maps of position-dependent magnetic field strength
estimates. When evaluating the field curvature and the gravity direction
locally on a map, the method can be generalized to arbitrary cloud shapes. The
technique is applied to high-resolution ($\sim0\farcs7$) Submillimeter Array
polarization data of the collapsing core W51 e2. A tentative $\sim 7.7$~mG
field strength is found when averaging over the entire core. The analysis
further reveals some structures and an azimuthally averaged radial profile
$\sim r^{-1/2}$ for the field strength. Maximum values close to the center are
around $19$~mG. The currently available observations lack higher resolution
data to probe the innermost part of the core where the largest field strength
is expected from the method. Application regime and limitations of the method
are discussed. As a further important outcome of this technique, the local
significance of the magnetic field force compared to the other forces can be
quantified in a model-independent way, from measured angles only. Finally, the
method can potentially also be expanded and applied to other objects (besides
molecular clouds) with measurements that reveal the field morphology, as e.g.
Faraday rotation measurements in galaxies.
