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

The relationship between the magnetic field and the circular polarization of astrophysical maser radiation due to the Zeeman effect under idealized conditions is investigated when the Zeeman splitting is much smaller than the spectral linebreadth and when radiative saturation is significant. The description of the circular polarization, as well as inferences about the magnetic field from the observations, are clearest when the rate for stimulated emission is much less than the Zeeman splitting. The calculations here are performed in this regime, which is relevant for some (if not most) observations of astrophysical masers. We demonstrate that Stokes-$V$ is proportional to the Zeeman splitting and the fractional linear polarization is independent of the Zeeman splitting when the ratio of the Zeeman splitting to the spectral line breadth is small---less than about 0.1. In contrast to its behavior for ordinary spectral lines, the circular polarization for masers that are at least partially saturated does not decrease with increasing angle between the magnetic field and the line-of-sight until they are nearly perpendicular.