Statistical Properties of a Polarization Vector's Ellipticity Angle

TL;DR

This paper analyzes the statistical properties of the ellipticity angle in polarized radio signals, providing formulas for its probability density, mean, and confidence limits, crucial for interpreting polarization measurements in astrophysics.

Contribution

It develops a comprehensive statistical framework for the ellipticity angle, including its probability density and bias correction, extending understanding beyond the well-studied position angle.

Findings

Probability density of EA is asymmetric and depends on intrinsic EA.

Standard deviation of EA varies inversely with signal-to-noise ratio.

Derived joint probability densities for polarization angles with fluctuating amplitudes.

Abstract

The orientation of a polarization vector on the Poincare sphere is defined by its position angle (PA) and ellipticity angle (EA). The radio emission from pulsars, magnetars, and fast radio bursts can be elliptically polarized, and measurements of the EA have become increasingly important in interpretations and models of their polarization. An in-depth understanding of the statistical properties of the measured polarization angles is a prerequisite to their detailed interpretation. While the statistics of the PA have been understood for some time, the statistics of the EA do not appear to be as well developed as those of the PA. The statistical properties of the EA are derived when the amplitude of the polarization vector is constant, to include its probability density, mean, standard deviation, and confidence limits. Similar to the PA, the standard deviation and confidence limits of the EA vary inversely with the polarization signal-to-noise ratio. However, unlike the PA, the probability density of the EA is generally asymmetric, its standard deviation and confidence limits are dependent upon the intrinsic value of the EA, and the measured EA is biased by the instrumental noise, particularly at low signal-to-noise ratios and large values of the intrinsic EA. General expressions for the joint probability density of the polarization angles and the probability density of the EA are also derived when the amplitude of the polarization vector fluctuates due to the superposition of incoherent modes of orthogonal polarization.