Quadrant polarization parameters for the scattered light of circumstellar disks
Authors: Schmid H. M. 
Published: September 2021

Context. Modern imaging polarimetry provides spatially resolved observations for many circumstellar disks and quantitative results for the measured polarization which can be used for comparisons with model calculations and for systematic studies of disk samples.
Aims. This paper introduces the quadrant polarization parameters Q000, Q090, Q180, Q270 for Stokes Q and U045, U135, U225, U315 for Stokes U for circumstellar disks and describes their use for the polarimetric characterization of the dust in debris disks.
Methods. We define the quadrant polarization parameters Qxxx and Uxxx and illustrate their properties with measurements of the debris disk around HR 4796A from Milli et al. (2019, A&A, 626, A54).. We calculate quadrant parameters for simple models of rotationally symmetric and optically thin debris disks and the results provide diagnostic diagrams for the determination of the scattering asymmetry of the dust. This method is tested with data for HR 4796A and compared with detailed scattering phase curve extractions in the literature.
Results. The parameters Qxxx and Uxxx are ideal for a well-defined and simple characterization of the azimuthal dependence of the polarized light from circumstellar disk because they are based on the “natural” Stokes Q and U quadrant pattern produced by circumstellar scattering. For optically thin and rotationally symmetric debris disks the quadrant parameters normalized to the integrated azimuthal polarization Qxxx∕Qϕ and Uxxx∕Qϕ or quadrant ratios like Q000∕Q180 depend only on the disk inclination i and the polarized scattering phase function fϕ(θ) of the dust, and they do not depend on the radial distribution of the scattering emissivity. Because the disk inclination i is usually well known for resolved observations, we can derive the shape of fϕ(θ) for the phase angle range θ sampled by the polarization quadrants. This finding also applies to models with vertical extensions as observed for debris disks. Diagnostic diagrams are calculated for all normalized quadrant parameters and several quadrant ratios for the determination of the asymmetry parameter g of the polarized Henyey-Greenstein scattering phase function fϕ(θ, g). We apply these diagrams to the measurement of HR 4796A, and find that a phase function with only one parameter does not reproduce the data well. We find a better solution with a three-parameter phase function fϕ(θ, g1, g2, w), but it is also noted that the well-observed complex disk of HR 4796A cannot be described in full detail with the simple quadrant polarization parameters.
Conclusions. The described quadrant polarization parameters are very useful for quantifying the azimuthal dependence of the scattering polarization of spatially resolved circumstellar disks illuminated by the central star. They provide a simple test of the deviations of the disk geometry from axisymmetry and can be used to constrain the scattering phase function for optically thin disks without detailed model fitting of disk images. The parameters are easy to derive from observations and model calculations and are therefore well suited to systematic studies of the dust scattering in circumstellar disks.