2017ApJ...851..113G

Derived here is a more conceptually intuitive means of interpreting magnetic-field diagnostics from circularly polarized lines in a wide array of astrophysical applications. The method applies to individual "Stokes V" profile snapshots and complements standard Zeeman Doppler imaging techniques by providing the explicit form of the averaging kernel for the magnetic field that the polarization diagnostic is sensitive to. This new perspective centers on the antiderivative, or cumulative integral with respect to frequency, of the Stokes V profile. The new approach would not yield different answers for magnetic field determinations, but rather presents a more directly conceptual means of understanding the connection between what is observed and what types of fields produce it. In particular, it elucidates how lateral and line-of-sight field gradients affect the Zeeman profile. This approach is especially useful when the Zeeman shift varies in a way that correlates with the Doppler shift, as then spectral resolution serves as a proxy for spatial imaging in each polarization snapshot. Hence, the perspective is particularly useful for rapidly rotating stars, hypersonic winds, galactic rotation, and large-amplitude turbulence, when the longitudinal field varies across the source or with depth. The approach also generates an improved unsigned mean-field diagnostic that suffers less polarity cancellation than the commonly used center-of-gravity diagnostic. Reduced cancellation produces a better estimate of the field magnitude in toroidal, spotty, or dipolar fields, and a complementary comparison with the current unsigned diagnostic can help characterize the degree of field polarity reversal concealed within integrated diagnostics.