180$^\circ$ Rotations in the Polarization Angle for Blazars

Rotations of the electric vector position angle (EVPA) in blazars are often close to an integral multiple of 180$^\circ$. There are multiple examples of this in the literature, and our analysis here, of the optical polarization data from the RoboPol monitoring program, strengthens the evidence by showing that $n\pi$ rotations occur more frequently than expected by chance. We explain this with a model consisting of two polarized emission components: a "jet" that is constant in time, and a "burst" that is variable. The EVPA of the combination is $\rm EVPA_{jet}$ at both the beginning and the end of the burst, so the net rotation across the burst must be $n\pi$. Examples are analyzed on the Stokes plane, where the winding number for the Stokes vector of the combination gives the value of $n$. The main conclusion is that the EVPA rotation can be much larger than the physical rotation of the emission region around the axis of the jet, but this requires the EVPAs of the jet and the burst to be nearly orthogonal. A shock-in-jet calculation by Zhang et al. can provide a physical model for our toy model, and in addition automatically gives the needed orthogonality. The model is illustrated with data on OJ287 published by Myserlis et al., and we suggest that the large rapid EVPA rotation seen there might be a phase effect and not representative of a physical rotation.