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The relationship between the magnetic field and the circular polarization ofastrophysical maser radiation due to the Zeeman effect under idealizedconditions is investigated when the Zeeman splitting is much smaller than thespectral linebreadth and when radiative saturation is significant. Thedescription of the circular polarization, as well as inferences about themagnetic field from the observations, are clearest when the rate for stimulatedemission is much less than the Zeeman splitting. The calculations here areperformed in this regime, which is relevant for some (if not most) observationsof astrophysical masers. We demonstrate that Stokes-$V$ is proportional to theZeeman splitting and the fractional linear polarization is independent of theZeeman splitting when the ratio of the Zeeman splitting to the spectral linebreadth is small---less than about 0.1. In contrast to its behavior forordinary spectral lines, the circular polarization for masers that are at leastpartially saturated does not decrease with increasing angle between themagnetic field and the line-of-sight until they are nearly perpendicular.Comment: to be published in the Astrophysical Journal Letter

The Relationship between the Circular Polarization and the Magnetic Field for Astrophysical Masers with Weak Zeeman Splitting