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.