We report the first detection of galactic spiral structure by means of thermal emission from magnetically aligned dust grains. Our 89 $mu$m polarimetric imaging of NGC 1068 with the High-resolution Airborne Wideband Camera/Polarimeter (HAWC+) on NASAs Stratospheric Observatory for Infrared Astronomy (SOFIA) also sheds light on magnetic field structure in the vicinity of the galaxys inner-bar and active galactic nucleus (AGN). We find correlations between the 89 $mu$m magnetic field vectors and other tracers of spiral arms, and a symmetric polarization pattern as a function of the azimuthal angle arising from the projection and inclination of the disk field component in the plane of the sky. The observations can be fit with a logarithmic spiral model with pitch angle of $16.9^{+2.7}_{-2.8}$$^{circ}$ and a disk inclination of $48pm2^{circ}$. We infer that the bulk of the interstellar medium from which the polarized dust emission originates is threaded by a magnetic field that closely follows the spiral arms. Inside the central starburst disk ($<1.6$ kpc), the degree of polarization is found to be lower than for far-infrared sources in the Milky Way, and has minima at the locations of most intense star formation near the outer ends of the inner-bar. Inside the starburst ring, the field direction deviates from the model, becoming more radial along the leading edges of the inner-bar. The polarized flux and dust temperature peak $sim 3-6$ NE of the AGN at the location of a bow shock between the AGN outflow and the surrounding interstellar medium, but the AGN itself is weakly polarized ($< 1$%) at both 53 and 89 um.
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