Line Polarization of Molecular Lines at Radio Frequencies: The case of   DR21(OH)
  We present polarization observations in DR21(OH) from thermal dust emission at 3 mm and from CO J=1-0 line emission. The observations were obtained using the Berkeley-Illinois-Maryland Association array. Previous observations of this region at 1.3 mm for the polarized continuum emission, and also the CO J=2-1 polarized emission have been made before. Our continuum polarization results are consistent with those results. However, the direction of the linear polarization for the J=1-0 is perpendicular to that of the CO J=2-1 polarization. This unexpected result was explored by obtaining numerical solutions to the multilevel, radiative transfer equations for a gas with anisotropic optical depths. We find that in addition to the anisotropic optical depths, anisotropic excitation due to a source of radiation that is external to the CO is needed to understand the orthogonality in the directions of polarization. The continuum emission by dust grains at the core of DR21(OH) is sufficient to provide this external radiation. The CO polarization must arise in relatively low density (nH2 ~ 100 [cm^-3]) envelope gas. We infer B ~ 10 [micro Gauss] in this gas, which implies that the envelope is subcritical. 
P. C. Cortes and R. M. Crutcher and W. D. Watson