Untangling Magnetic Complexity in Protoplanetary Disks with the Zeeman Effect

With the recent advent of circular polarization capabilities at the Atacama Large Millimeter/submillimeter Array (ALMA), Zeeman effect measurements of spectral lines are now possible as a means to directly probe line-of-sight magnetic fields in protoplanetary disks (PPDs). We present a modeling study that aims to guide physical interpretation of these anticipated observations. Using a fiducial density structure based on a typical ringed disk, we simulate line emission for the hyperfine components of the CN $J = 1-0$ transition with the POLARIS radiative transfer code. Since the expected magnetic field and typical CN distribution in PPDs remain largely unconstrained, we produce models with several different configurations. Corresponding integrated Stokes $I$ and $V$ profiles and 0.4 km/s resolution, 1'' beam convolved channel maps are presented. We demonstrate that the emission signatures from toroidally dominated magnetic fields are distinguishable from vertically dominated magnetic field based on channel map morphology. Due to line-of-sight and beam cancellation effects, disks with toroidal $\boldsymbol{B}$-field configurations result in significantly diminished Stokes $V$ emission. Complex magnetic fields therefore render the traditionally used method for inferring line-of-sight magnetic field strengths (i.e., fitting the derivative of the Stokes $I$ to the Stokes $V$ profile) ambiguous, since a given intrinsic field strength can yield a variety of Stokes $V$ amplitudes depending on the magnetic field geometry. In addition, gas gaps can create structure in the integrated Stokes $V$ profile that might mimic magnetic substructure. This method should therefore be applied with caution in PPD environments, and can only confidently be used as a measure of magnetic field strength if the disk's magnetic field configuration is well understood.