A model grid for the reflected light from transition disks

Context. The dust in protoplanetary disks is an important ingredient in planet formation and can be investigated with model simulations and quantitative imaging polarimetry of the scattered stellar light. Aims. This study explores circumstellar disks with calculations for the intensity and polarization of the reflected light. We aim to describe the observable radiation dependencies on parameters in order to constrain the dust scattering properties and the disk geometry. Methods. The photon scattering and absorption by the disk are calculated with a Monte Carlo method for a grid of simple, rotationally symmetric models approximated at each point by a plane–parallel dusty atmosphere. The adopted geometry is described by a strongly illuminated inner wall of a transition disk with inclination i, a constant wall slope χ, and an angular wall height a. Dust scattering parameters are the single scattering albedo ω, the Henyey–Greenstein scattering phase function with the asymmetry parameter ɡ, and the maximal fractional polarization pmax induced by the scattering. First, the results for the reflectivity, the polarized reflectivity, and the fractional polarization of a plane–parallel surface element are calculated as functions of the incidence angle and the escape direction of the photons and as functions of the scattering parameters. Integration over all escape directions yields the surface albedo and the fraction of radiation absorbed by the dust. Second, disk images of the reflected intensity and polarization are calculated, and the appearance of the disk is described for various parameter combinations. The images provide many quantitative radiation parameters for a large range of model calculations, which can be compared to observations. These include the disk integrated intensity ${{\bar I} \mathord{\left/ {\vphantom {{\bar I} {{I_ \star }}}} \right. \kern-\nulldelimiterspace} {{I_ \star }}}$, azimuthal polarization ${{\bar Q} \mathord{\left/ {\vphantom {{\bar Q} {{I_ \star }}}} \right. \kern-\nulldelimiterspace} {{I_ \star }}}$, the polarization aligned with the apparent disk axes ${{\bar Q} \mathord{\left/ {\vphantom {{\bar Q} {{I_ \star }}}} \right. \kern-\nulldelimiterspace} {{I_ \star }}}$, the quadrant polarization parameters Qxxx and Uxxx, the disk-averaged fractional polarization 〈pφ〉 or 〈pQ〉, but also the front-to-back intensity ratio I180/I000 or the maximum fractional scattering polarization тах(pφ). Results. The results of our simple disk models reproduce well the measurements for I/I⋆,Qφ/I⋆, and 〈pφ〉 reported for well-observed transition disks. They describe the dependencies of the scattered radiation on the disk geometry and the dust scattering parameters in detail. Particularly strong constraints on disk properties can be obtained from certain diagnostic quantities: for example the fractional polarization 〈pφ〉 or тах(pφ) depend predominantly on the dust-scattering parameters ω and pmax; for disks with well-defined inclination, ratios of the quadrant polarization parameter depend mainly on the scattering asymmetry ɡ and the wall slope χ; wavelength dependencies of I/I✶ and Qφ/I✶ can mostly be attributed to the wavelength dependence of the dust scattering parameters ω(λ), ɡ(λ), and pmmах(λ); and the ratio between the scattered and thermal light of the disk roughly constrains the disk reflectivity R and the single scattering albedo of the dust ω. Conclusions. This computational investigation of the scattered radiation from transition disks shows well-defined dependencies on model parameters and the results can therefore be used as a diagnostic tool for the analysis of quantitative measurements, specifically in constraining or even determining the scattering properties of the dust particles in disks. Collecting and comparing such information for many systems is required to understand the nature of the scattering dust in planet-forming disks