11institutetext: Institute for Particle Physics and Astrophysics, ETH Zurich,
Wolfgang Pauli Strasse 17, CH-8093 Zurich
11email<EMAIL_ADDRESS>
# A model grid for the reflected light from transition disks

J. Ma 11 H.M. Schmid , 11

(Received –; accepted –)

###### Abstract

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 $\chi$, and an angular wall
height $\alpha$. Dust scattering parameters are the single scattering albedo
$\omega$, the Henyey-Greenstein scattering phase function with the asymmetry
parameter $g$, and the maximal fractional polarization $p_{\rm max}$ 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 $\overline{I}/I_{\star}$, azimuthal polarization
$\overline{Q}_{\varphi}/I_{\star}$, the polarization aligned with the apparent
disk axes $\overline{Q}/I_{\star}$, the quadrant polarization parameters
$Q_{xxx}$ and $U_{xxx}$, the disk-averaged fractional polarization $\langle
p_{\varphi}\rangle$ or $\langle p_{Q}\rangle$, but also the front-to-back
intensity ratio $I_{180}/I_{000}$ or the maximum fractional scattering
polarization ${\rm max}(p_{\varphi})$.

Results. The results of our simple disk models reproduce well the measurements
for $I/I_{\star}$, $Q_{\varphi}/I_{\star}$, and $\langle p_{\varphi}\rangle$
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 $\langle p_{\varphi}\rangle$ or ${\rm max}(p_{\varphi})$ depend
predominantly on the dust-scattering parameters $\omega$ and $g$; for disks
with well-defined inclination, ratios of the quadrant polarization parameter
depend mainly on the scattering asymmetry $g$ and the wall slope $\chi$;
wavelength dependencies of $I/I_{\star}$ and $Q_{\varphi}/I_{\star}$ can
mostly be attributed to the wavelength dependence of the dust scattering
parameters $\omega(\lambda)$, $g(\lambda)$, and $p_{\rm max}(\lambda)$; and
the ratio between the scattered and thermal light of the disk roughly
constrains the disk reflectivity ${\cal{R}}$ and the single scattering albedo
of the dust $\omega$.

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.

###### Key Words.:

circumstellar disks – dust scattering – radiative transfer – polarization –
transition disks

##  1 Introduction

Dust is an important ingredient of the planet formation in protoplanetary
disks. Dust particles can be observed over a large wavelength range as strong
absorbers, scatterers, or emitters of radiation and the data show a large
diversity of geometric morphologies of these dust tracers for different disks,
and even within a given disk. For example, in the millimeter range one can see
the thermal emission of the large dust particles settled in the cold disk
midplane; in the infrared (IR) range we see the thermal emission of the dust
illuminated and heated by the central star; and in the visual and near-IR
spectral range we can observe the scattered light from the small dust
particles located at the disk surface. To understand the evolution of dust
particles in protoplanetary disks and their role in the planet-formation
process we need to investigate these different dust components and combine the
complementary information from all these observations (e.g., Andrews 2015).

This work simulates with model calculations the photometric and polarimetric
properties of the stellar light scattered by the inner wall of transitions
disks. The simulations should be useful for comparisons with high-resolution
and high-contrast observations and provide better constraints on the dust
particle properties and geometric parameters of disks. The calculated disk
models adopt a dust scattering phase matrix which depends on the particle
albedo, scattering asymmetry parameter, and the induced polarization by the
scattering particles. These parameters are indicative of the particle size
distribution and the presence of specific dust types such as high-albedo icy
grains, low-albedo carbon-rich particles, or porous aggregates (e.g., Draine &
Lee 1984; Shen et al. 2009; Kolokolova & Kimura 2010; Min et al. 2016; Tazaki
et al. 2019, 2021). Such a characterization will improve our understanding of
the evolution of the dust in protoplanetary disks and the resulting
composition of the forming planets.

Transition disks are a special class of protoplanetary disks characterized by
a large, strongly depleted central cavity (e.g., Calvet et al. 2005; Espaillat
et al. 2014). They are relatively easy targets for observation of the
scattered light with high-resolution observations because they produce a
strong signal at the inner wall of an extended disk $\gtrapprox 10$ AU which
can be separated for systems in nearby star-forming regions with a distance of
$\approx 100-150$ pc. For example, the intensity of the scattered light has
been measured for several disks by the Hubble Space Telescope (e.g., Grady et
al. 2005; Ardila et al. 2007; Mulders et al. 2013; Debes et al. 2013).
Particularly well suited for the disk detection are adaptive optics (AO)
instruments at large ground-based telescopes equipped with imaging
polarimeters, which allow the scattered and therefore polarized radiation of
the disk to be separated from the very strong but typically unpolarized
radiation of the central star (e.g., Schmid 2022 in press). For this reason,
there exist many observations of the polarized flux from disks taken with
differential polarimetric imaging (DPI) (e.g., Quanz et al. 2011; Hashimoto et
al. 2012; Rapson et al. 2015; Stolker et al. 2016; Monnier et al. 2017; Garufi
et al. 2017; Avenhaus et al. 2018). If the instrumental polarization and image
convolution effects can be calibrated, then these data provide quantitative
measurements of the polarized intensity of the disk. In favorable cases, both
the intensity and the polarized intensity can be derived (e.g., Perrin et al.
2009; Monnier et al. 2019; Hunziker et al. 2021; Tschudi & Schmid 2021). Such
measurements are particularly suitable for comparisons with model calculations
for the investigation of dust scattering properties.

Model calculations for geometrically thin disks that treat multiple scattering
and polarization by dust were first presented by Whitney & Hartmann (1992)
long before such disks could be imaged. More popular were calculations for
extended dusty envelopes as observed with seeing-limited observations (e.g.,
Bastien & Menard 1988; Whitney & Hartmann 1993; Fischer et al. 1994). Disk
modelling obtained a strong boost from progress in high-resolution imaging,
which revealed that many protoplanetary disks are geometrically thin and show
structures such as cavities, gaps, and spirals. Many simulated disk images for
the scattered light were presented to prove that observed disk morphologies
are in good qualitative agreement with the proposed three-dimensional disk
geometry (e.g., Pinte et al. 2008; Murakawa 2010; Dong et al. 2012; de Juan
Ovelar et al. 2013; Jang-Condell & Turner 2013; Pohl et al. 2015; Dong et al.
2016; Monnier et al. 2017; Jang-Condell 2017). In these studies, the dust
scattering properties were usually not the focus and the fractional
polarization or the wavelength dependence of the scattered light was not
investigated. There exist only a few disk studies where the dependence of the
scattering polarization and intensity on dust parameters are investigated.
Systematic radiative transfer calculations are presented by Tazaki et al.
(2019), who, for a fixed disk geometry, model the dependencies between the
disk-integrated color of the scattered intensity and polarized intensity, and
the disk averaged fractional polarization in order to constrain —with
observations— the size and porosity of dust particles. This work has the same
goal but we aim to provide a more general overview of the dependencies of
observational parameters on the dust scattering parameters and on the disk
geometry.

In this paper, we present a grid of simple transition disk models that
describe the disk geometry with three parameters, the slope $\chi$, the
angular height $\alpha$ of the inner disk wall, and the disk inclination $i$.
The dust scattering properties are defined by the three parameters, single
scattering albedo $\omega$, scattering asymmetry $g$, and induced fractional
polarization $p_{\rm max}$ for a scattering angle of 90∘.

Such simple models are ideal for exploring the dependencies of the
polarization and intensity of the scattered radiation for a large range of
dust-scattering parameters, for particles with very low or very high albedos,
with isotropic or strongly forward-scattering phase functions, and particles
producing low or high fractional polarization. The results of this model grid
should include roughly the range of observed intensity and polarization for
transition disks in order to constrain the disk parameters. This investigation
paves the way to identifying observational parameters that have significant
diagnostic potential for the analysis of the geometry of transition disks and
the scattering particles. The calculations will also clarify how the models
should be improved in order to better determine the properties of the observed
disks.

The following section describes our model calculations and the parameters used
for our model grid. Section 3 gives an overview of the reflectivity, the
polarized reflectivity, and the dust absorption of a plane-parallel surface
for the adopted dust-scattering parameters. The dependencies of the scattered
radiation from transition disks on the model parameters are described in Sect.
4, and in Sect. 5 we present an investigation of the diagnostic potential of
the presented results for the characterization of the dust and the disk
geometry with observations. In Sect. 6, we summarize our results and assess
the limitations of our interpretation of observational results stemming from
the simplicity of our model simulations compared to real disks which are
typically much more complex.

##  2 Model description

Figure 1: Examples of the $Q_{\varphi}$ image for the bright walls in the
transition disks around RX J1604.3-2130 and HD 100453 (R-band observations
with SPHERE/ZIMPOL taken from the ESO data archive).

In many transition disks, narrow ring-like emission from the inner wall
surrounding the large central cavity dominates the scattered light, as shown
in Fig. 1 for a pole-on disk and an inclined disk with spiral features and
shadows from a hot, unresolved dust disk located close to the star. Detailed
descriptions can be found for RXJ1604.3-2130 (e.g., Mayama et al. 2012), HD
100453 (e.g., Benisty et al. 2017), HD 142527 (e.g., Canovas et al. 2013), PDS
70 (e.g., Keppler et al. 2018), HD 169142 (e.g., Bertrang et al. 2018), HD
34700A (Monnier et al. 2019) and others. The bright signal in polarized
intensity from these inner walls can be measured with relatively high
precision. Often, it is also easy to derive a brightness ratio between the
front and back sides of the disks, and our models help to understand such
ratio measurements. For some strongly illuminated disks, it is even possible
to extract the intensity of the scattered light and derive the fractional
polarization of the scattered light (e.g., Monnier et al. 2019; Hunziker et
al. 2021; Tschudi & Schmid 2021). Based on these observational data, it seems
reasonable to simulate the frequently observed ring-like disk morphology of
transition disks with a simple geometry for the bright inner disk wall in
order to capture the main parameter dependencies for the reflected intensity
and polarization with our model simulations.

###  2.1 Disk geometry

Figure 2: Geometry of the transition disk model. Panel (a) Sketch of the
rotationally symmetric, inclined disk model with inclination $i$, wall slope
$\chi,$ and angular height of the disk wall $\alpha$ . Panel (b) Corresponding
vertical disk cross-section with the midplane disk distance to the star $r$
and the height $h$ for the surface point $S(r,h)$, and radii $r_{0}$ and
$r_{w}$ for the inner radius and the rim radius of the wall, respectively.
Emitted photons from the star are characterized by the angular direction
$\vartheta$, while $\theta_{0}$ and $\theta$ are the polar incidence and
emergence angles from surface point $S$ for the special case of scattering in
the principal plane. Figure 3: Plane parallel surface element $S(r,h)$ in
polar coordinates. Photons always enter in the principal plane
$\phi_{0}=0^{\circ}$, as indicated by the red arrow in the red dotted plane,
and are assumed to escape at the same point $S$ in an upward direction with
angle $(\theta,\phi)$, as shown by the blue arrow in the blue dotted plane.

Our transition disk model is defined according to Fig. 2(a) by an inclined,
rotationally symmetric geometry with an inner wall with a vertical cross-
section as shown in Fig. 2(b). The height of the disk surface $h(r)$ from the
disk midplane as a function of the radius $r$ is defined by an inner radius
$r_{0}$ and a wall with a constant slope $\chi$ or

$h(r)=\pm(r-r_{0})\,{\rm tan}\,\chi\quad{\rm for}\quad r_{0}\leq r\leq
r_{w}\,,$ (1)

where $r_{w}$ is the midplane radius of the wall rim with an angular height
$\alpha$ with respect to the disk midplane. There must be $\alpha<\chi$ for an
inner disk wall with a finite radial extent. The disk surface for $r>r_{w}$
has a slope smaller than $\alpha$ and is therefore not illuminated by the
star. The radius of the wall rim is defined by
$r_{w}=r_{0}(1-\tan\alpha/\tan\chi)^{-1}$ and the corresponding wall height is
$h_{w}=h(r_{w})=\pm r_{w}\,{\rm tan}\,\alpha$. The positive and negative signs
stand for the wall section above and below the disk midplane, referred to as
the upper and lower part, respectively.

The illuminated wall is described by the two parameters $\chi$ and $\alpha$,
while $r_{0}$ is only a scaling factor for the disk size. For a given angular
height $\alpha$, the same relative amount of stellar radiation interacts with
the disk independent of $\chi$. For our model grid, we choose a broad range
for the wall slopes $\chi$ from 12.5∘ to 82.5∘. For the wall height $\alpha$,
we consider values of between $5^{\circ}$ and $20^{\circ}$ but use mostly just
$\alpha=10^{\circ}$ ($h_{w}/r_{w}=0.176$) because many results for the
reflected intensity and polarized intensity simply scale as
$\propto\sin\alpha\propto\alpha$ for $\alpha<20^{\circ}$. This comfortably
covers the typical range of transition disks (Takami et al. 2014; Woitke et
al. 2019; Avenhaus et al. 2018).

The disk can be seen by an observer under different inclinations from pole-on
$i=0^{\circ}$ to edge-on $i=90^{\circ}$. The illuminated back wall or inner
wall of the far side in our disk images is always up from the star and the
inner wall on the near side or front wall is below the star as if we are
looking onto the disk. This also defines the apparent major and minor disk
axes for inclined disks, which are horizontal and vertical, respectively.

Depending on the inclination $i$ and the wall slope $\chi,$ the upper front
side wall and the upper and lower back side walls are visible, partially
visible, or not visible. The upper back side wall is fully visible near the
minor axis for $i<90^{\circ}-\alpha$, and can even be visible or partly
visible for slightly larger $i$.

The front side wall is only visible if $i<90^{\circ}-\chi$ and it disappears
for $i=90^{\circ}-\chi$, while the lower backside wall starts to become
(partially) visible for $i>90^{\circ}-\chi$. The lower backside disappears
behind the disk for nearly edge-on configurations $i\gtrapprox
90^{\circ}-\alpha$. The exact borderline depends on the unconstrained outer
$r>r_{w}$ disk geometry.

###  2.2 Illumination and surface reflectivity

Each disk element is treated like a plane-parallel, optically thick surface
where photons are scattered until they escape or are absorbed. It is assumed
that the photons enter and escape at the same point $S(r,h)$; see Fig. 3. The
plane-parallel surface is described by a polar coordinate system with the
surface normal as polar axis, where the incoming photon propagates in a
perpendicular plane, often also called the principal scattering plane, with
the polar incidence angle $\theta_{0}$ and the azimuthal incidence angle
$\phi_{0}=0^{\circ}$ (e.g., Chandrasekhar 1960; van de Hulst 1980). The
incidence angle $\theta_{0}(r)$ at point $S$ for a disk with given $r_{0}$ and
$\chi$ is

$\theta_{0}=90^{\circ}-\chi+\arctan\frac{(r-r_{0})\tan\chi}{r}\,.$ (2)

The incident angle is $\theta_{0}=90^{\circ}-\chi$ at the midplane $r=r_{0}$
and gradually increases along the wall slope until it reaches
$\theta_{0}=90^{\circ}-\chi+\alpha$ at the rim $r=r_{w}$.

The incident flux $F_{0}$ at point $S(r,h)$ per surface area is given by the
stellar spectral luminosity $L_{\lambda}$, the distance to the star and the
incidence angle

$F_{0}(S(r,h))=\frac{L_{\lambda}}{4\pi(r^{2}+h^{2}(r))}\,\cos\theta_{0}\,.$
(3)

The distribution of the reflected radiation is described by the polar $\theta$
and azimuthal $\phi$ angles

  * •

$I(\theta_{0},\phi_{0}=0^{\circ},\theta,\phi)$ for the intensity, and

  * •

$Q(\theta_{0},\phi_{0}=0^{\circ},\theta,\phi)$ and
$U(\theta_{0},\phi_{0}=0,\theta,\phi)$ for the polarized intensity.

In our models, we only consider the Stokes parameters $Q$ and $U$ for linear
polarization and disregard scattering processes producing circular
polarization. $Q$ and $U$ for the escaping photons are defined in a plane
perpendicular to the photon propagation direction given by a blue arrow in
Fig. 3. Thereby, the positive $Q$-direction is also perpendicular to the
vertical blue plane with orientation $\phi$ and $\phi+180^{\circ}$, while the
negative $Q$-direction is in this plane. The positive and negative
$U$-directions are rotated with respect to $Q$ by $45^{\circ}$. (see Schmid
1992). For the principal plane $\phi=0^{\circ}$ or $180^{\circ}$, the
integrated linear polarization for many multiple scattered photons escaping
from point $S(r,h)$ is, for symmetry reasons, either perpendicular or parallel
or $\pm Q$ to the principal plane, while $U=0$.

###  2.3 Dust scattering

The reflectivity at a given point on the disk surface is treated as photon
random walk with a Monte Carlo code. The parameters describing the interaction
with the dust are the single scattering albedo $\omega$, a scattering
asymmetry parameter $g$, and the polarization $p_{\rm max}$ introduced by
scattering with an angle of 90∘. The scattering albedo is defined as

$\omega=\dfrac{\sigma_{\rm sca}}{\sigma_{\rm sca}+\kappa_{\rm abs}},$ (4)

where $\sigma_{\rm sca}$ and $\kappa_{\rm abs}$ stand for scattering and
absorption cross-sections, respectively. The scattering albedo $\omega$ is
defined for $[0,1]$ and $\omega=1$ represents a purely scattering particle
while $\omega=0$ is a purely absorbing particle without scattering.

The angular distribution of the scattered photons is described by the Henyey-
Greenstein (HG) phase function (Henyey & Greenstein 1941):

$F_{\rm
HG}(\theta_{s},g)=\frac{1}{4\pi}\dfrac{1-g^{2}}{(1+g^{2}-2g\cos\theta_{s})^{\frac{3}{2}}}\,,$
(5)

where $\theta_{s}$ is the scattering angle with respect to the incident
direction. The asymmetry parameter $g\in]-1,1[$ describes the predominant
scattering direction $g=\langle\cos\theta\rangle$, where $g=0$ stands for
isotropic scattering, $g>0$ means more forward scattering, and $g<0$ more
backward scattering.

The fractional polarization introduced by the dust scattering $p_{\rm
sca}(\theta_{S})$ is approximated by an angle dependence which is identical to
Rayleigh scattering but with a scaling factor $p_{\rm max}\leq 1:$

$p_{\rm sca}(\theta_{s},p_{\rm max})=p_{\rm
max}\dfrac{cos^{2}\theta_{s}-1}{\cos^{2}\theta_{s}+1},$ (6)

which accounts for depolarization effects of dust scattering when compared to
the dipole-type Rayleigh scattering. Hereafter, we refer to the polarized
scattering phase function

$F_{\rm HGpol}(\theta_{s},g,p_{\rm max})=F_{\rm HG}(\theta_{s},g)\cdot p_{\rm
sca}(\theta_{s},p_{\rm max})$ (7)

as the HGpol function. In this description of a single scattering event, the
angle distribution of the polarized intensity for unpolarized incoming light
depends only on $g$, and the maximum fractional polarization $p_{\rm max}$
introduced by scatterings with $\theta_{s}=90^{\circ}$ (e.g., Schmid 2021).
The polarization of the reflected light from a multiple scattering surface
changes significantly with $p_{\rm max}$, but the scattered intensity depends
only slightly on $p_{\rm max}$ as described for example by Chandrasekhar
(1960) for Rayleigh scattering. The adopted description for the scattering
phase function is simple and versatile, however it disregards more complex
features of dust scattering, such as a back-scattering maximum for
$\theta_{s}$ near $180^{\circ}$ as expected for larger particles or deviations
from the polarization function given in Eq. 6 as observed for the dust in the
debris disk HR 4796A (Milli et al. 2019; Arriaga et al. 2020).

###  2.4 Monte Carlo simulations

A widely used tool for calculations of the scattered intensity and the
scattering polarization are Monte Carlo simulations which follow the random
walk of photons (e.g., Witt 1977). In astronomy, many codes are available
which use similar procedures to those described here for the dust scattering
in circumstellar disks (e.g., Pinte et al. 2006; Min et al. 2009; Whitney et
al. 2013).

##### Scattering in a plane-parallel surface.

In a first step, we investigate the reflected intensity
$I(\theta_{0},0,\theta,\phi)$ and the Stokes parameters
$Q(\theta_{0},0,\theta,\phi)$, $U(\theta_{0},0,\theta,\phi)$ from a
homogeneous, semi-infinite, plane parallel dust atmosphere for different
incidence angles $\theta_{0}$ and dust parameters ($\omega,g,p_{\rm max}$). In
a rotationally symmetric disk, the stellar illumination of a surface element
always occurs in the principal plane and is described by the polar angle
$\theta_{0}$ and the azimuthal angle $\phi_{0}=0^{\circ}$. The incident
photons have a random polarization angle as expected for unpolarized incident
radiation. The vertical optical depths for the photon interactions $\tau_{z}$
are calculated from random distributions of the free path lengths $\delta\tau$
and the photon propagation direction. The vertical optical depth $\tau_{z}$ is
zero above the surface and is positive for deeper layers. After one or several
scattering events, the photon may escape in a direction $(\theta,\phi)$ with a
polarization angle $\gamma$.

Our Monte Carlo simulations are based on the code described in Schmid (1992),
which was compared thoroughly with analytic calculations for isotropic and
Rayleigh scattering (Chandrasekhar 1960; Abhyankar & Fymat 1970, 1971). To
this code, we added the Henyey-Greenstein-type dust scattering phase function
with polarization. The procedure for calculating the scattering angles
$(\theta_{s},\phi_{s})$ and the polarization angle change $\gamma_{s}$ of the
photon includes the following steps. First the polar scattering angle
$\theta_{s}$ is drawn from the probability distribution function ${\cal{P}}$
for the Henyey-Greenstein function using a rejection method:

${\cal{P}}(\theta_{s})=\dfrac{1}{4\pi}\int_{0}^{\theta_{s}}\dfrac{1-g^{2}}{(1+g^{2}-2g\cos\theta)^{\frac{3}{2}}}\sin\theta
d\theta$ (8)

Then we draw an evenly distributed random number $p_{0}$ in the interval
$[0,1]$ and compare it to $p_{max}$. If $p_{0}>p_{max}$, then we adopt non-
polarizing scatterings and therefore equally distributed azimuthal scattering
angles $\phi_{s}$ and polarization angle $\gamma_{s}$. Otherwise, $\phi_{s}$
and $\gamma_{s}$ are evaluated with a probability distribution as for Rayleigh
scattering, as described by Schmid (1992). The escaping photons are sampled in
direction bins defined by the central angles $\theta_{i}=2.5^{\circ}+i\cdot
5^{\circ}$ and $\phi_{j}=j\cdot 10^{\circ}$. One bin covers the solid angle
$\Omega_{ij}=|\sin\theta_{i}|\cdot(\pi/18)\cdot(\pi/36)$. The reflectivity
$I(\theta,\phi)$ in a given bin is the ratio between the collected photons
$N_{ij}$ and the incident photons $N_{\rm tot}$ scaled by the bin size
$I=N_{ij}/(N_{\rm tot}\Omega_{ij})$. The relative statistical errors are
$(N_{ij})^{-1/2}$ and they are large for small $\theta_{i}$ or small solid
angle bins which collect fewer photons.

##### Scattering in a circumstellar disk

In our Monte Carlo simulations of the circumstellar disks, the photons are
released with random polarization orientations from an isotropically emitting
central star with a polar angle distribution
$D(\vartheta)\propto\cos{\vartheta}$. For our calculations, $D(\vartheta)$ is
restricted to the range $\vartheta\in[0,\alpha]$, where interaction with the
inner disk wall will occur. The photon then penetrates the disk at point
$S(r,h(r))$ where dust scattering or absorption can occur as described above
for the plane-parallel surface. The incidence angle
$\theta_{0}(r,h)=90^{\circ}-\chi+\vartheta$, radial distance from the star
$r=r_{0}\sin^{2}\chi(1+\cot\chi\cot(\chi-\vartheta)),$ and $h(r)$ (Eq. 1)
follow from the adopted disk parameters $r_{0}$, $\chi$, and $\alpha$.

The direction and polarization angle of the photon escaping from $S(r,h)$ are
defined by $(\theta,\phi,\gamma)$ for the plane-parallel surface. Because the
adopted disk is rotationally symmetric, these parameters define the azimuthal
angle $\varphi$ of the surface region and the apparent inclination of the disk
for which this escaping photon can be observed. In other words, each escaping
photon characterized by $(r,\theta,\phi,\gamma)$ in a given disk geometry
$(r_{0},\chi,\alpha)$ can be converted into a photon escaping from the
cylindrical coordinates $r,h,\varphi$ with a polar inclination angle $i$, or
from a corresponding Cartesian point $x(i),y(i)$ in the sky plane image
corresponding to a disk seen under inclination $i$. Also, the polarization
direction of the photon must be rotated from $\gamma$ in the plane parallel
system to $\gamma^{\prime}$ in the system of the observer. The trigonometric
formulas for these transformations are straightforward but remain complex.

For the resulting intensity $I(x,y,i)$ of the scattered radiation, the photons
are added together for each $(x(i),y(i))$ bin using inclination bins with a
width of $5^{\circ}$ defined by the central angles $i_{k}=2.5^{\circ}+k\cdot
5^{\circ}$. For the Stokes $Q$ and $U$ parameters, the direction of the
polarization for each photon $\gamma^{\prime}_{j}$ must also be taken into
account according to

$Q=\sum_{j}\cos(2\gamma^{\prime}_{j})\quad{\rm and}\quad
U=\sum_{j}\sin(2\gamma^{\prime}_{j})\,.$ (9)

After the calculation of the random walk of sufficiently large numbers of
photons, the distribution of the scattered intensity $I(x,y,i)$ and the
polarized intensities $Q_{(}x,y,i)$ and $Q_{(}x,y,i)$ for the reflected light
from the disk can be established for each inclination bin.

Various tests for the Monte Carlo code were carried out, including simulations
and comparisons with analytic results for disks with a perfect Lambert
surface, analyzing disks with isotropic scattering, or the polarization
obtained for single scattered photons for $p_{\rm max}=1$, and many more.

##  3 Calculations for a plane parallel surface

The reflectivity $I(\theta,\phi)$ and the Stokes parameters $Q(\theta,\phi)$
and $U(\theta,\phi)$ of the reflected radiation for a given point $S(r,h)$ on
the disk wall depend on the illumination angle $\theta_{0}$ and the dust
scattering parameters $\omega$, $g,$ and $p_{\rm max}$. This section gives a
brief overview of the scattered light from a plane parallel atmospheric
surface to support the discussion and interpretation of the results of the
scattered radiation from an entire disk. Previous results for Henyey-
Greenstein scattering in a semi-infinite atmosphere are described in Dlugach &
Yanovitskij (1974) and are tabulated in van de Hulst (1980), but without
considering the scattering polarization. Appendix A provides an overview of
the parameter dependencies for the reflected intensity and polarization for an
atmospheric surface with dust scattering as described by Eqs. (4) to (7).

Figure 4: Reflection in the principal plane of the 0.5-model. Panel (a):
Reflected intensity $I(\theta_{\rm pp})$ and polarized intensity
$P(\theta_{\rm pp})$ for the 0.5-model ($\theta_{0}=57.5^{\circ}$,
$\omega=0.5$, $g=0.5$, $p_{\rm max}=0.5$) are shown in red. The analytic
solution and simulations of $I_{\rm iso}(\theta)$ for the reflectivity of a
surface with isotropically scattering particles $g=0$ with
$\theta_{0}=57.5^{\circ}$, $\omega=0.5$, $p_{\rm max}=0.5$ is shown in black.
The analytic result $I_{\rm Lam}(\theta)\propto\cos\theta$ for a Lambertian
surface with a surface albedo $A_{S}=0.156$ is in blue. Panel (b): Fractional
Stokes parameter $q(\theta_{\rm pp})$ for the 0.5-model.

###  3.1 The reference model (”0.5-model”).

To simplify our discussion we introduce a reference model with fixed incidence
angle $\theta_{0}=57.5^{\circ}$ and dust scattering parameters $\omega=0.5$,
$g=0.5$, and $p_{\rm max}=0.5$, which we also refer to as the 0.5-model. The
selected dust scattering values are quite typical for interstellar dust grains
for wavelengths of around $1\leavevmode\nobreak\ \mu$m (Draine 2003). Appendix
A.2 presents results for other model parameters and compares them to this
reference case.

Further, we focus our description on the reflection in the principal plane
which is the scattering plane perpendicular to the surface. For this plane, we
can plot one-dimensional functions for $I(\theta_{\rm pp})$ (see Fig. 4), but
also $Q(\theta_{\rm pp})$ or the fractional polarization $p(\theta_{\rm pp})$
and other parameters. The angle $\theta_{\rm pp}$ for the escaping radiation
in the principal plane is defined for the range $-90^{\circ}$ to
$+90^{\circ}$, where $\theta_{\rm pp}=+\theta$ for $\phi=0^{\circ}$ and
$\theta_{\rm pp}=-\theta$ for $\phi=180^{\circ}$, which are for $\theta_{0}\gg
0^{\circ}$ the backward reflection side and the forward reflection side,
respectively.

The scattering in the principal plane describes for an axisymmetric
circumstellar disk the reflected intensity and polarization for the front and
back side along the minor axis. Of course, the $\phi$-dependence of the
reflectivity is important for other regions of the disk and this is discussed
in Appendix A.1 for the reference model.

##### The reflected intensity in the principal plane.

The reflected intensity of the 0.5-model in the principal plane $I(\theta_{\rm
pp})$ is compared in Fig. 4 with the analytic solution $I_{\rm iso}(\theta)$
for isotropic $(g=0)$ scattering with $\omega=0.5$ and
$\theta_{0}=57.5^{\circ}$ as calculated by Chandrasekhar (1960) and with our
Monte Carlo code, and $I_{\rm Lam}(\theta)$ for a gray, Lambertian surface
with surface albedo $A_{S}$. The intensity distributions for $I_{\rm iso}$ and
$I_{\rm Lam}$ are rotationally symmetric with respect to the surface normal
and produce no scattering polarization. The surface albedo $A_{S}$ of the
Lambert model was set to 0.156, because this yields the same surface albedo
(equivalent to the term reflectivity ${\cal{R}}$) as the $\omega=0.5$
isotropic scattering model.

The forward scattering dust of the 0.5-model plotted in Fig. 4 produces a
clear asymmetry with respect to $\theta_{\rm pp}=0^{\circ}$ and for most
viewing angles much less reflected flux than isotropic scattering, because
photons are scattered predominantly further ”down” into deep layers where
photon absorptions are frequent (Mulders et al. 2013). A Lambertian surface
behaves like $I_{\rm Lam}(\theta)\propto\cos\theta$ and this is equivalent to
a surface brightness (SB), which is independent from the observers viewing
angle $\theta$. The isotropic scattering case $I_{\rm iso}(\theta_{\rm pp})$
in Fig. 4 produces a higher surface brightness for large angles
$\theta>60^{\circ}$ when compared to $\theta<60^{\circ}$. For the forward
scattering dust of the 0.5-model, there exists a strong difference between the
”bright” forward scattering direction $\theta_{\rm pp}<-45^{\circ}$ and the
”faint” backward scattering direction $\theta_{\rm pp}>45^{\circ}$. For
example, the intensity ratio between $I(-67.5^{\circ})$ and $I(+67.5^{\circ})$
is about a factor of 4.

##### The polarization in the principal plane.

In the principal scattering plane, the fractional polarization of the
reflected radiation can be described by the fractional Stokes parameter
$q=Q/I;$ because of reasons of symmetry, the reflected radiation can only be
polarized perpendicular or parallel to this plane. Therefore, the fractional
polarization is $p(\theta_{\rm pp})=|q(\theta_{\rm pp})|$ and $u(\theta_{\rm
pp})=0$. The polarization $q(\theta_{\rm pp})$ shows a strong maximum
perpendicular to the scattering plane around $\theta_{\rm pp}=-30^{\circ}$
(Fig. 4(b)). This angle for the maximum polarization $\theta_{\rm pp}^{\rm
maxp}$ is about $90^{\circ}$ away from the incidence angle
$\theta_{0}=57.5^{\circ}$ and represents the polarization dependence for the
adopted Rayleigh-like phase curve for the fractional polarization (Eq. 6). The
polarization of the reflected light is dominated by photons escaping after the
first scattering event, while the polarization of multiple scattered photons
is randomized and therefore they add mainly to the ”unpolarized” intensity and
lower the fractional polarization. The $q(\theta_{\rm
pp})=q(\theta,0^{\circ}:180^{\circ})$ curve shows a weakly negative section
around the back-scattering directions $\theta_{\rm pp}=50^{\circ}-80^{\circ}$.
This is a well-known higher order scattering effect (e.g., van de Hulst 1980),
which for dust (or haze) scattering in Jupiter and Titan leads to strong
”negative” limb polarization effects (Schmid et al. 2011; McLean et al. 2017;
Bazzon et al. 2014).

Also included in Fig. 4(a) is the polarized flux $P(\theta_{\rm
pp})=p(\theta_{\rm pp})\times I(\theta_{\rm pp})$ of the reflected radiation.
The maximum of $P(\theta_{\rm pp})$ lies between the maxima of $p(\theta_{\rm
pp})$ and $I(\theta_{\rm pp})$. High-contrast observations of disks with
differential polarimetric imaging provide often only the signal for the
polarized flux $P(\theta_{\rm pp})$ which differs very significantly from the
reflected intensity of a Lambert surface or an isotropically scattering
atmosphere, and even from the reflected intensity of a surface with Henyey-
Greenstein scattering particles.

###  3.2 Scattering parameter dependencies

The 0.5-model represents only one combination of model parameters, and
Appendix A provides a more detailed description of the dependencies of the
intensity $I(\theta,\phi)$ and polarizations such as $Q(\theta,\phi)$,
$U(\theta,\phi),$ and $p(\theta,\phi)$ on the incidence angle $\theta_{0}$,
the scattering albedo $\omega$, asymmetry $g$, and polarization $p_{\rm max}$.
Here, we summarize some key results.

The single scattering albedo $\omega$ has a very important impact on the
amount of reflected intensity. A low single scattering albedo significantly
reduces the multiple scattered photons, which contribute more ”randomized” or
unpolarized reflected light, while the strongly polarizing single scattered
photons are less suppressed.
The scattering asymmetry parameter $g$ can greatly change the angular
distribution of the reflected intensity $I(\theta_{\rm pp})$. The reflected
intensity also depends significantly on $g$ because the incoming stellar
photons are scattered for large $g$ predominantly into deeper layers where
absorption is very likely. Interestingly, the fractional polarization of the
scattered radiation $p(\theta_{\rm pp})$ depends only slightly on the
scattering asymmetry parameter.
The parameter $p_{\rm max}$ of the polarization phase function has of course a
strong effect on fractional polarization $p(\theta,\phi)$ of the reflected
radiation but only a small impact on the reflected intensity.

Figure 5: Integrated reflectivity ${\cal{R}}$ for different asymmetry
parameters $g$ as functions of $\omega$ (panel a) and $\theta_{0}$ (panel b).

##### Absorption and surface albedo.

A very interesting value from the plane parallel surface calculations is the
integrated reflectivity ${\cal{R}}(\theta_{0},\omega,g)$ or surface albedo
according to

${\cal{R}}=\frac{1}{F_{0}}\int_{0}^{90^{\circ}}\int_{0}^{360^{\circ}}I(\theta,\phi)\sin\theta\,{\rm
d}\phi{\rm d}\theta\,.$ (10)

This integrated reflectivity ${\cal{R}}$ does not depend on the viewing
directions $\theta$ and $\phi$ and also the small dependence on $p_{\rm max}$
can be neglected. This allows us to characterize the reflectivity of a surface
of dust described by the HG scattering phase function in a simple way by
${\cal{R}}(\theta_{0},\omega,g)$. Values of ${\cal{R}}$ for the models
discussed in this section are also included in Table 5.
The radiation absorbed by the dust is directly related to the reflectivity
${\cal{R}}$ by the relation

${\cal{K}}(\theta_{0},\omega,g)=1-{\cal{R}}(\theta_{0},\omega,g)\,.$ (11)

The total radiation energy absorbed by the dust, integrated for all
wavelengths, can be roughly estimated from the thermal IR emission of the
dust. Therefore, ${\cal{R}}$ is an important parameter for the energy budget
of circumstellar disks.

The dependence of the integrated reflectivity ${\cal{R}}$ on the single
scattering albedo $\omega$, the asymmetry parameter $g,$ and the incidence
angle $\theta_{0}$ is shown in Fig. 5. The reflectivity is high for small $g$
(and would be very high for negative $g$ not considered in this work), for
grazing incidence $\theta_{0}\rightarrow 90^{\circ}$, and in particular for
high scattering albedo $\omega$. For $\omega=1$, there is no absorption
${\cal{K}}=0$ and all radiation is reflected from the surface ${\cal{R}}=1$.
The reflectivity for the 0.5-model is at a level of 9 %, indicating that about
$\approx 90\leavevmode\nobreak\ \%$ or most of the radiation falling onto this
type of dust surface is absorbed.

##  4 Calculations for transition disk models

The geometry of the simple three-dimensional, rotational symmetric disk models
used in this work are described by three parameters, namely disk inclination
$i$, angular wall height $\alpha$, and wall slope $\chi$. The scattering at
each point $S(r,h)$ is treated as for a plane-parallel surface.

The primary results of the model calculation are images for the scattered
light intensity $I(x,y)$ and Stokes polarization parameters $Q(x,y)$ and
$U(x,y)$. The central star is at $x_{0}=0$, $y_{0}=0$ and the $x$-axis is
aligned with the major axis of the inclined disk and the more distant part of
the minor axis is in the positive $y$-direction. The orientation of Stokes $Q$
and $U$ are aligned with the $x,y$-coordinates, and $Q$ is positive for
polarization in the $y$-direction.

For the polarization of a circumstellar disk, the polarized flux
$P=(Q^{2}+U^{2})^{1/2}$ can also be described by

$P=(Q_{\varphi}^{2}+U_{\varphi}^{2})^{1/2}\,,$ (12)

where $Q_{\varphi}$ and $U_{\varphi}$ are the azimuthal polarization
components defined with respect to the star according to the description of
Schmid et al. (2006) for the radial Stokes parameters $Q_{r}$, $U_{r}$, and
using $Q_{\varphi}=-Q_{r}$ and $U_{\varphi}=-U_{r}$. The angle of the
scattered polarization of a circumstellar disk is to first order in azimuthal
direction $Q_{\varphi}$, and $U_{\varphi}\approx 0$ is very small, meaning
that the polarized flux can be approximated by $P(x,y)\approx
Q_{\varphi}(x,y)$ and the fractional polarization by $p(x,y)\approx
p_{\varphi}(x,y)=Q_{\varphi}(x,y)/I(x,y)$. The $Q_{\varphi}(x,y)$ parameter
confers the advantage that it is not biased by the statistical noise of the
Monte Carlo calculations or the photon noise in observations.

To quantify the reflected radiation, we use reflected intensity
$\overline{I}/I_{\star}$, the Stokes $Q$ intensity $\overline{Q}/I_{\star}$,
and the azimuthally polarized intensity $\overline{Q}_{\phi}$, which are disk-
integrated values relative to the intensity of the central star. There is
always $\overline{U}=0$ in our models because of the adopted disk symmetry.
For the disk-averaged fractional polarization, we calculate the ratios
$\langle p_{\varphi}\rangle=\overline{Q}_{\varphi}/\overline{I}$ and $\langle
p_{Q}\rangle=\overline{Q}/\overline{I}$ which are flux-weighted averages.

In addition, we use the quadrant polarization parameters $Q_{000}$, $Q_{090}$,
$Q_{180}$, $Q_{270}$ for Stokes $Q$ and $U_{045}$, $U_{135}$, $U_{225}$,
$U_{315}$ for Stokes $U$ for the characterization of the azimuthal dependence
of the polarization. The indices give the position angle of the Stokes
quadrant for which the $Q$ or $U$ signal is integrated, where $Q_{000}$ is the
backside quadrant, $Q_{090}$ the left quadrant and so on as indicated in Fig.
6. Basic symmetry properties of the polarization quadrant parameters $Q_{xxx}$
and $U_{xxx}$ and relations with respect to Stokes $Q$ and $U$ or the
azimuthal polarization $Q_{\varphi}$ are described in (Schmid 2021).

###  4.1 Reference model (0.5-model) for transition disks

The geometry for the reference model for the transition disks is defined by a
wall slope of $\chi=32.5^{\circ}$ and an angular wall height of
$\alpha=10^{\circ}$. This corresponds to incidence angles of
$\theta_{0}(r_{0})=57.5^{\circ}$ near the midplane and
$\theta_{0}(r_{w})=67.5^{\circ}$ at the wall rim. For the dust, we again adopt
the scattering parameters $\omega=0.5$, $g=0.5,$ and $p_{\rm max}=0.5$, so
that the three-dimensional 0.5-model disk model corresponds precisely to the
plane parallel 0.5-model.

Along the disk minor axis, the scattering occurs in the principal plane and
the calculated reflected radiation is very similar to the results for the
scattering in plane-parallel surface atmosphere. For surface regions at other
disk azimuth angles, the reflectivities depend on $\theta$ and $\phi$ and
cannot be compared directly to the presented plane parallel surface results.
For regions near the major axis, the scattering angle $\theta_{s}$ is always
close to $90^{\circ}$ and therefore the polarization signal at these positions
is strong.

Figure 6: Images of radiation parameters for the 0.5-reference model for four
inclinations $i$. The dashed ellipses show $r_{0}=1$ in the disk midplane and
the dashed straight lines show the splitting for $Q_{xxx}$ and $U_{xxx}$. For
$p_{\varphi}$, the locations with $p_{\varphi}\geq\max(p_{\varphi})$ are
marked with gray transparent points. We note the changed color scale for the
weak and therefore noisy $U_{\phi}$ signal.

####  4.1.1 Images of the reference model for transition disks

The two-dimensional images $I(x,y)$, $Q_{\varphi}(x,y)$, $U_{\varphi}(x,y)$,
$p_{\varphi}(x,y)$, $Q(x,y)$, and $U(x,y)$ for the 0.5-model are shown in Fig.
6 for the inclinations $i=7.5^{\circ},32.5^{\circ},57.5^{\circ}$, and
$77.5^{\circ}$. For polar viewing angles $i\la 10^{\circ}$, the images for the
intensity $I(x,y)$, azimuthal polarization $Q_{\varphi}(x,y)$, and fractional
polarization $p_{\varphi}(x,y)$ are close to axisymmetric. For intermediate
$i\approx 20^{\circ}-50^{\circ}$, the disk intensity $I(x,y)$ shows a high
surface brightness for the front wall because of the strongly ($g=0.5$)
forward scattering HG function. For higher inclinations ($i\geq
57.5^{\circ}$), the front wall becomes invisible and the maximum brightness
moves to regions close to the major axis of the projected disk. For high
inclination $i=77.5^{\circ}$, only the scattering from the back wall remains
visible. For $i>57.5^{\circ}$, the lower side of the back wall also
contributes to the signal.

For higher inclination, the azimuthally polarized intensity $Q_{\varphi}(x,y)$
shows a reduced signal compared to $I(x,y)$ near the minor axis, where the
reflection is dominated by forward and backward scattering which produces only
little polarization. The relative level of $Q_{\varphi}(x,y)$ with respect to
$I(x,y)$ is shown in the $p_{\varphi}(x,y)$ images. For all plotted disk
inclinations, there is ${\rm max}(p_{\varphi}(x,y))\approx
30\leavevmode\nobreak\ \%$ because there is always a surface region with a
scattering angle close to $\theta_{S}\approx 90^{\circ}$ where the fractional
polarization of the scattered radiation is close to the maximum. The disk
location where $\theta_{S}\approx 90^{\circ}$ depends strongly on disk
inclination (Fig. 6) and therefore the photon escape angles $\theta,\,\phi$
also change, but this has only a small effect on the resulting ${\rm
max}(p_{\varphi})$ value (see Figs. 17 and 18).

The calculations show a weak but significant $U_{\varphi}(x,y)$ signal for
inclined disks (Fig. 6). This indicates that the position angle of the
scattering polarization can deviate (slightly) from a strictly azimuthal
orientation as a result of multiple scatterings. This can happen everywhere on
the disk except the minor axis, because the scattered light is not escaping in
the principal plane (see also Fig. 17). This effect was described by Canovas
et al. (2015), who report a relative $U_{\varphi}/Q_{\varphi}$ signal of up to
4.5 % for their optical thick disk model. The parameters of our ”0.5-reference
model” are different but our results qualitatively confirm the effects
described by Canovas et al. (2015), including the quite subtle distribution of
positive and negative $U_{\varphi}$ regions on the disk. For $i=32.5^{\circ}$,
the $U_{\varphi}$ signal is mostly negative on the left ($x<0$) and positive
on the right side ($x>0$) of the disk ring and the ratio
$U_{\varphi}/Q_{\varphi}$ for the right side is $+1.4\leavevmode\nobreak\ \%$.
The corresponding values for $i=7.5^{\circ}$, $57.5^{\circ}$, and
$77.5^{\circ}$ are $0.6\%$, $2.5\%,$ and $10\%$, respectively. For high
inclination there are clearly positive and negative $U_{\varphi}$ parts on one
disk side, and this more complex pattern is particularly strong for the lower
back-side surface visible in the $i=77.5^{\circ}$ case.

$Q_{\varphi}$ and $U_{\varphi}$ are vector components and even a signal of
$|U_{\varphi}|/Q_{\varphi}\approx 10\leavevmode\nobreak\ \%$ means that the
polarization angle deviates only $6^{\circ}$ from the azimuthal direction and
$Q_{\varphi}\approx 0.99\leavevmode\nobreak\ (p\times I)$ and therefore we
disregard in the following description of the disk polarization the
$U_{\varphi}$ signal. However, if $U_{\varphi}$ could be measured accurately,
this would provide important information for circumstellar disks.
Nevertheless, the investigation of this topic is beyond the scope of this
paper.

Figure 6 also shows the disk-aligned Stokes $Q(x,y)$ and $U(x,y)$ polarization
for different $i$. The polarization quadrants $Q_{xxx}$ and $U_{xxx}$ are
indicated and they are used for the quantitative characterization of the
azimuthal dependence of the polarization according to Schmid (2021). For pole-
on disks, the absolute value is the same for all quadrants and the integrated
Stokes parameters $\overline{Q}$ and $\overline{U}$ are zero because positive
and negative quadrants cancel each other. For inclined disks, the polarization
quadrants show clear front–back asymmetries which are apparent in the flux
ratios for $Q_{180}/Q_{000}$ or $U_{135}/U_{045}$. These asymmetries are
caused by the scattering asymmetry $g$ and the disk geometry and are discussed
in the following subsection. For high-inclination systems, the $Q$-signal is
strongly concentrated in the positive $Q_{090}$ and $Q_{270}$ quadrants.

Figure 7: Radiation parameters for the 0.5-model as functions of disk
inclination $i$. Panel (a): Disk integrated intensity
$\overline{I}(i)/I_{\star}$, azimuthal polarization intensity
$\overline{Q}_{\varphi}(i)/I_{\star}$, and Stokes $Q$-intensity
$\overline{Q}(i)/I_{\star}$ expressed relative to the stellar intensity. Panel
(b): Disk-averaged fractional polarization $\langle p_{\varphi}(i)\rangle$ and
$\langle p_{Q}(i)\rangle$ of the scattered radiation, and the maximum
fractional azimuthal polarization measured in the disk image ${\rm
max}(p_{\varphi}(i,x,y))$.

####  4.1.2 Radiation parameters for the 0.5-model

The radiation parameters $\overline{I}(i)/I_{\star}$,
$\overline{Q}_{\phi}(i)/I_{\star}$, $\overline{Q}(i)/I_{\star}$, and
$Q_{xxx}(i)/I_{\star}$ or $U_{xxx}(i)/I_{\star}$ for our transition disk
models are first presented in detail in Fig. 7 for the reference case.
Relative quantities with respect to the intensity of the central star and as a
function of inclination are given because observations often provide such
relative measurements and a comparison with our models assumes implicitly that
the stellar emission is isotropic without additional absorption or emission
components being present like ”disturbing” hot dust between the star and the
observer or between the star and the disk.

We use as additional parameters the averaged fractional polarizations $\langle
p_{\varphi}(i)\rangle$ and $\langle p_{Q}(i)\rangle$ and the maximum value for
the fractional polarization ${\rm max}(p_{\varphi}(i,x,y))$ for the scattered
light. Calculated model values for $i=7.5^{\circ}$, $32.5^{\circ}$,
$57.5^{\circ}$, and $77.5^{\circ}$ are given in Table 1 for the reference disk
case (column 3) and many other transition disks with different model
parameters.

##### Disk-integrated intensity and polarized intensity.

According to Fig. 7(a), the relative disk intensity is for a pole-on
orientation at the level of $\overline{I}_{\rm obs}(0)/I_{\star}=2.9\%$. This
value is for $i<40^{\circ}$ almost independent of the inclination because the
drop in brightness for the back side for enhanced $i$ is compensated by the
forward scattering gain on the front side. For $i>40^{\circ}$, the bright
front side starts to disappear; but for $i>60^{\circ}$, the lower back side
wall below the disk midplane becomes visible and therefore the
$\overline{I}(i)/I_{\star}$ curve has a shoulder around $i=70^{\circ}$. For
large $i>70^{\circ}$, significant parts of the back-side wall are also hidden
by the unilluminated surface of the front-side disk, and there is a step drop
$\overline{I}(i)/I_{\star}\rightarrow 0$ for $i\rightarrow 90^{\circ}$.

The curve for the azimuthal polarization $\overline{Q}_{\phi}(i)/I_{\star}$
shows a steady decrease from the maximum at $i=0^{\circ}$ towards
$i=90^{\circ}$. For $i=0^{\circ}$ or pole-on disks, the scattering occurs
everywhere under the polarimetrically very favourable scattering angle of
about $\theta_{S}\approx 90^{\circ}$. For increasing inclination, the induced
polarization is reduced for disk areas on the front and back side, because
forward and backward scattering conditions produce less polarization. Only the
regions close to the major axis keep a strong polarization signal for
increased $i$ because there the scattering angle is $\theta_{S}\approx
90^{\circ}$ for all $i$.

For Stokes $\overline{Q}(i)/I_{\star}$ , the curve starts for $i=0^{\circ}$ at
zero, because the positive and negative quadrant contributions compensate each
other for pole-on disks. For larger $i,$ the negative $Q$-features near the
minor axis diminish and the two positive $Q$-features near the major axis
become dominant so that $\overline{Q}\rightarrow\overline{Q}_{\varphi}$ for
high inclination.

##### Fractional polarization of the scattered light.

Figure 7(b) shows the disk averaged fractional polarizations, which are the
ratios $\langle
p_{\varphi}(i)\rangle=\overline{Q}_{\varphi}(i)/\overline{I}(i)$ and $\langle
p_{Q}(i)\rangle=\overline{Q}(i)/\overline{I}(i)$ of the parameters given in
panel (a). The azimuthal polarization $\langle p_{\varphi}(i)\rangle$ is about
31% for pole-on disks, decreases for increasing $i$ like
$\overline{Q}_{\varphi}(i)$, and reaches a plateau around $i\approx
40^{\circ}$ when the strong intensity contributions from the front side start
to disappear. The plateau with $\langle p_{\varphi}(i)\rangle\approx 20\%$
extends to about $i\approx 60^{\circ}$ and the shape of this plateau depends
on the changing visibility and obscuration of the back wall. The fractional
Stokes Q polarization $\langle p_{Q}(i)\rangle$ is zero at $i=0^{\circ}$,
increases with inclination, and joins the curve $\langle
p_{\varphi}(i)\rangle$ for $i\ga 60^{\circ}$ , because
$\overline{Q}_{\varphi}$ and $\overline{Q}$ are dominated for high $i$ by the
signal of the two disk regions near the major axis.

The same plot includes also determinations for the maximal fractional
polarization of the disk ${\rm max}(p_{\varphi}(i,x,y))$ as measured in the
two-dimensional $p_{\varphi}$ images. Interestingly, this value is almost
constant around ${\rm max}(p_{\varphi}(i))\approx 33\leavevmode\nobreak\ \%$,
because for all inclinations there exist certain disk surface regions where
the scattering angle is close to $90^{\circ}$ (Fig. 6).

##### Azimuthal dependence of the disk polarization.

The images for $Q_{\varphi}$, $Q,$ and $U$ in Fig. 6 show a very strong
dependence of the azimuthal distribution of the scattering polarization on the
disk inclination. For the characterization of the azimuthal distribution, we
use the quadrant polarization parameters $Q_{000}$, $Q_{090}$, $Q_{180}$,
$Q_{270}$, and $U_{045}$, $U_{135}$, $U_{225}$, and $U_{315}$(see Schmid
2021), which measure the Stokes $Q$ and Stokes $U$ polarization fluxes of the
disk in the individual positive and negative quadrants indicated in Fig. 6 for
$Q$ and $U$ and $i=7.5^{\circ}$. Axisymmetric disk models provide five
independent quadrant values $Q_{000}$, $Q_{090}$, $Q_{180}$, $U_{045}$, and
$U_{135}$, because some parameters are redundant, that is, $Q_{090}=Q_{270}$,
$U_{045}=-U_{315}$, and $U_{135}=-U_{225}$ , as a result of the disk symmetry.

Figure 7(c) shows the dependence of the normalized quadrant polarization
parameters $Q_{xxx}/I_{\star}$ and $U_{xxx}/I_{\star}$ on the disk inclination
for the reference model. For pole-on, axisymmetric disks all quadrant
parameters have the same value
$Q_{xxx}=U_{xxx}=\overline{Q}_{\varphi}/(2\pi)$. For higher $i,$ first the
backside quadrants $Q_{000}$ and $U_{045}$ weaken, because of inefficient
backward reflectivity for forward scattering dust and the unfavorable
scattering angle $\theta_{s}\gg 90^{\circ}$ for the production of strongly
polarized radiation. The signals in the front-side quadrants $Q_{180}$ and
$U_{135}$ remain stronger than for the backside quadrants, because they profit
from the strong ($g=0.5$) forward scattering effect. For larger inclination,
the illuminated front-side wall disappears as soon as $i$ approaches $i\approx
90^{\circ}-\chi$, which is $\approx 57.5^{\circ}$ for the reference model. For
larger $i,$ only regions near the major axis $Q_{090}$ and $U_{270}$ or
$U_{045}$ and $U_{315}$ have a favorable scattering angle $\theta_{s}\approx
90^{\circ}$, are still not hidden by the front side and produce a relatively
strong polarization signal.

###  4.2 Disk appearance for different model parameters

Figure 8: Reflected intensity $I(x,y)$ and azimuthal polarization
$Q_{\varphi}(x,y)$ for (a) the scattering asymmetry parameters
$g=0.0,0.25,0.5,$ and 0.75 and fixed parameters $\omega=0.5$, $p_{\rm
max}=0.5$, $i=32.5^{\circ}$, $\chi=32.5^{\circ}$, and $\alpha=10^{\circ}$. The
case $g=0.5$ is the reference model; (b) the disk wall slopes
$\chi=82.5^{\circ},57.5^{\circ},32.5^{\circ}$, and $12.5^{\circ}$. The case
$\chi=32.5^{\circ}$ is the reference model. The dashed ellipses in
$\chi=12.5^{\circ}$ show the outer edge of the disk; and (c) the disk wall
heights $\alpha=5^{\circ},10^{\circ},15^{\circ}$ and $20^{\circ}$ and fixed
parameters $\omega=0.5$, $g=0.5$, $p_{\rm max}=0.5$, $i=32.5^{\circ}$, and
$\alpha=10^{\circ}$. The case $\alpha=10^{\circ}$ is the reference model.

The model parameters $i$, $\chi$, and $\alpha$ for the disk geometry and
$\omega$, $g$, and $p_{\rm max}$ for the dust scattering all influence the
resulting intensity and polarization of the disk to a greater or lesser
degree. Figure 6 is a good example of the strongly changing geometric
appearance of a disk seen under different inclinations. The scattering albedo
$\omega$ strongly defines the scattered intensity and $p_{\rm max}$ the
fractional polarization, but the overall morphology of the disk images is not
greatly changed by these parameters. However, the disk appearance depends
significantly on the scattering asymmetry parameter $g$ and the geometry of
the disk wall described by the wall slope $\chi$ and wall height $\alpha$.

The scattering asymmetry $g$ can strongly change the relative brightness
between the front- and backside as illustrated in Fig. 8(a) for disks with
$i=32.5^{\circ}$. The wall slope is $\chi=32.5^{\circ}$ and therefore
$\theta_{0}$ between $57.5^{\circ}$ and $67.5^{\circ}$ and the back wall is
seen under a perpendicular viewing angle $\theta=0^{\circ}$, while the front
wall is seen for $\theta=-65^{\circ}$. Along the minor axis, the signal from
the reflected light corresponds at $r_{0}$ to the principal plane calculations
$I(\theta_{\rm pp})$ in Fig. 18. The reflectivity depends on the
$g$-parameters and for isotropic scattering $g=0,$ the back wall of the disk
is brighter in intensity $I$ and azimuthal polarization $Q_{\varphi}$ (Fig.
8(b)) if the flux is integrated radially along the entire wall surface as in
Fig. 18, where results are given for a unit surface area. Because the width of
the projected back wall is about twice as large as for the projected front
wall, the surface brightnesses for $I(x,y)$ and $Q_{\varphi}(x,y)$ are
comparable for the two sides for the $g=0$ case. However, the front side
clearly dominates for larger $g$ (see also Fig. 18) as observed in many disks.

The slope of the inner disk wall also has a very important impact on the
front-to-back brightness ratio as can be seen in Fig. 8(b). For inclined disks
$i=32.5^{\circ}$ and steep walls (e.g., $\chi=82.5^{\circ}$), the illuminated
inner wall of the disk is only visible on the back side, but not on the front
side because $i+\chi>90^{\circ}$. The front side is still barely visible for
$\chi=57.5^{\circ}$ ($i+\chi=90^{\circ}$) but is clearly visible for
moderately steep $\chi=32.5^{\circ}$ or flat $\chi=12.5^{\circ}$ disks. In
these cases, the front side is much brighter than the back side because of the
forward scattering. Indeed, flat disks with the same wall height of
$\alpha=10^{\circ}$ are very extended. This can be inferred from the panels in
the fourth column in Fig. 8(b) when comparing them with the reference model in
the third panel.

The angular height $\alpha$ defines the total radiation received by the disk
from the central star. A disk wall with a larger angular height will intercept
more photons from the star and the reflected radiation will be enhanced. A
higher wall height does not change the reflectivity for the inner disk part,
but adds reflecting surface regions further out. For angular heights
$\alpha\leq 20^{\circ}$ the reflected radiation $\overline{I}/I_{\star}$,
$\overline{Q}_{\phi}/I_{\star}$, $\overline{Q}/I_{\star}$ scales roughly
proportional to $\alpha$.

The strong brightness dependencies for the front side of the presented disk
models is partly an artificial effect caused by the adopted simple disk
geometry. Real disks have no sharp edge between a strongly illuminated inner
wall and a dark, nonilluminated (self-shadowed) flat disk surface, and
therefore the transition from one regime to the other will be much more
gradual and not an abrupt switch, as is seen for our model results. For this
reason, caution is advised when considering the model results for the
brightness of the steep inner wall on the front side of the disk for the
interpretation of observations.

Table 1: Numerical results for our grid of disk models

| Parm. | Ref. |  $\omega$-dep. |  $g$-dep. |  $p_{\rm max}$-dep. |  $\chi$-dep. |  $\alpha$-dep.
---|---|---|---|---|---|---|---
| $\omega$ |  $\omega_{r}$=$0.5$ | 0.20 | 0.80 | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$ | $\omega_{r}$
| $g$ | $g_{r}=0.5$ | $g_{r}$ | $g_{r}$ | 0.00 | 0.25 | 0.75 | $g_{r}$ | $g_{r}$ | $g_{r}$ | $g_{r}$ | $g_{r}$ | $g_{r}$ | $g_{r}$ | $g_{r}$
| $p_{\rm max}$ | $p_{r}=0.5$ | $p_{r}$ | $p_{r}$ | $p_{r}$ | $p_{r}$ | $p_{r}$ | 0.20 | 0.80 | $p_{r}$ | $p_{r}$ | $p_{r}$ | $p_{r}$ | $p_{r}$ | $p_{r}$
| $\chi$ |  $\chi_{r}$=32.5∘ | $\chi_{r}$ | $\chi_{r}$ | $\chi_{r}$ | $\chi_{r}$ | $\chi_{r}$ | $\chi_{r}$ | $\chi_{r}$ | $12.5^{\circ}$ | $57.5^{\circ}$ | $82.5^{\circ}$ | $\chi_{r}$ | $\chi_{r}$ | $\chi_{r}$
| $\alpha$ | $\alpha_{r}=10^{\circ}$ | $\alpha_{r}$ | $\alpha_{r}$ | $\alpha_{r}$ | $\alpha_{r}$ | $\alpha_{r}$ | $\alpha_{r}$ | $\alpha_{r}$ | $\alpha_{r}$ | $\alpha_{r}$ | $\alpha_{r}$ | $5^{\circ}$ | $15^{\circ}$ | $20^{\circ}$
para | $i$ |  |  |  |  |  |  |  |  |  |  |  |  |  |
$\overline{I}/I_{\star}[\%]$ | 7.5 | 2.89 | 0.82 | 7.87 | 4.03 | 3.81 | 1.50 | 2.92 | 2.85 | 3.30 | 1.82 | 0.41 | 1.36 | 4.62 | 6.65
| 32.5 | 2.89 | 0.84 | 7.48 | 3.69 | 3.52 | 1.68 | 2.90 | 2.87 | 3.79 | 1.25 | 0.73 | 1.35 | 4.66 | 6.71
| 57.5 | 1.66 | 0.48 | 4.36 | 2.58 | 2.18 | 0.93 | 1.66 | 1.66 | 4.62 | 0.98 | 0.89 | 0.78 | 2.66 | 3.82
| 77.5 | 0.90 | 0.24 | 2.66 | 2.25 | 1.51 | 0.38 | 0.89 | 0.91 | 2.06 | 0.86 | 0.80 | 0.46 | 1.08 | 0.87
$\overline{Q}_{\varphi}/I_{\star}[\%]$ | 7.5 | 0.90 | 0.34 | 1.52 | 1.36 | 1.27 | 0.44 | 0.36 | 1.45 | 1.22 | 0.53 | 0.13 | 0.42 | 1.43 | 2.02
| 32.5 | 0.69 | 0.26 | 1.17 | 1.00 | 0.92 | 0.36 | 0.28 | 1.12 | 1.04 | 0.31 | 0.10 | 0.33 | 1.10 | 1.56
| 57.5 | 0.33 | 0.12 | 0.58 | 0.56 | 0.47 | 0.17 | 0.13 | 0.54 | 0.75 | 0.13 | 0.06 | 0.15 | 0.54 | 0.78
| 77.5 | 0.11 | 0.04 | 0.21 | 0.26 | 0.18 | 0.05 | 0.04 | 0.18 | 0.38 | 0.05 | 0.04 | 0.06 | 0.14 | 0.13
$-Q_{000}/\overline{Q}_{\varphi}$ | 7.5 | 0.147 | 0.145 | 0.148 | 0.170 | 0.155 | 0.143 | 0.144 | 0.147 | 0.140 | 0.160 | 0.272 | 0.145 | 0.150 | 0.152
| 32.5 | 0.086 | 0.081 | 0.092 | 0.151 | 0.110 | 0.076 | 0.085 | 0.087 | 0.067 | 0.147 | 0.431 | 0.079 | 0.094 | 0.104
| 57.5 | 0.035 | 0.036 | 0.034 | 0.070 | 0.046 | 0.034 | 0.034 | 0.036 | 0.015 | 0.101 | 0.190 | 0.028 | 0.043 | 0.051
| 77.5 | -0.005 | 0.005 | 0.025 | 0.001 | -0.008 | 0.004 | -0.005 | -0.004 | -0.003 | 0.016 | 0.030 | -0.011 | 0 | 0.018
$-U_{045}/\overline{Q}_{\varphi}$ | 7.5 | 0.152 | 0.153 | 0.154 | 0.170 | 0.160 | 0.148 | 0.154 | 0.152 | 0.146 | 0.162 | 0.252 | 0.149 | 0.154 | 0.156
| 32.5 | 0.135 | 0.128 | 0.142 | 0.201 | 0.163 | 0.119 | 0.133 | 0.135 | 0.111 | 0.211 | 0.234 | 0.126 | 0.144 | 0.152
| 57.5 | 0.153 | 0.141 | 0.171 | 0.222 | 0.182 | 0.136 | 0.154 | 0.154 | 0.083 | 0.273 | 0.161 | 0.134 | 0.173 | 0.195
| 77.5 | 0.265 | 0.245 | 0.295 | 0.290 | 0.271 | 0.263 | 0.265 | 0.267 | 0.089 | 0.351 | 0.163 | 0.223 | 0.306 | 0.373
$Q_{090}/\overline{Q}_{\varphi}$ | 7.5 | 0.161 | 0.160 | 0.162 | 0.162 | 0.163 | 0.162 | 0.160 | 0.162 | 0.161 | 0.164 | 0.175 | 0.162 | 0.162 | 0.162
| 32.5 | 0.213 | 0.212 | 0.217 | 0.219 | 0.223 | 0.204 | 0.215 | 0.213 | 0.205 | 0.248 | 0.193 | 0.212 | 0.214 | 0.214
| 57.5 | 0.375 | 0.375 | 0.371 | 0.341 | 0.368 | 0.367 | 0.375 | 0.373 | 0.329 | 0.298 | 0.322 | 0.379 | 0.366 | 0.356
| 77.5 | 0.422 | 0.424 | 0.420 | 0.404 | 0.421 | 0.414 | 0.422 | 0.421 | 0.478 | 0.348 | 0.402 | 0.441 | 0.382 | 0.303
$U_{135}/\overline{Q}_{\phi}$ | 7.5 | 0.166 | 0.167 | 0.165 | 0.149 | 0.159 | 0.170 | 0.167 | 0.165 | 0.170 | 0.154 | 0.063 | 0.167 | 0.164 | 0.162
| 32.5 | 0.177 | 0.183 | 0.168 | 0.109 | 0.147 | 0.193 | 0.177 | 0.177 | 0.201 | 0.095 | 0.001 | 0.187 | 0.168 | 0.159
| 57.5 | 0.079 | 0.090 | 0.066 | 0.033 | 0.055 | 0.102 | 0.082 | 0.080 | 0.176 | 0.003 | 0.002 | 0.096 | 0.065 | 0.052
| 77.5 | -0.002 | -0.002 | -0.003 | -0.003 | -0.002 | -0.002 | -0.002 | -0.002 | 0.020 | -0.002 | 0.004 | -0.003 | 0 | 0
$-Q_{180}/\overline{Q}_{\varphi}$ | 7.5 | 0.168 | 0.168 | 0.163 | 0.141 | 0.156 | 0.170 | 0.170 | 0.165 | 0.173 | 0.149 | 0.020 | 0.169 | 0.163 | 0.160
| 32.5 | 0.135 | 0.144 | 0.125 | 0.062 | 0.096 | 0.166 | 0.137 | 0.136 | 0.167 | 0.017 | 0 | 0.144 | 0.126 | 0.117
| 57.5 | 0.007 | 0.008 | 0.006 | 0.001 | 0.002 | 0.013 | 0.006 | 0.007 | 0.069 | 0 | 0 | 0.009 | 0.005 | 0.004
| 77.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\langle p_{\varphi}\rangle$ | 7.5 | 0.31 | 0.41 | 0.19 | 0.34 | 0.33 | 0.29 | 0.12 | 0.51 | 0.37 | 0.29 | 0.33 | 0.31 | 0.31 | 0.31
| 32.5 | 0.24 | 0.31 | 0.16 | 0.27 | 0.26 | 0.21 | 0.10 | 0.39 | 0.27 | 0.25 | 0.14 | 0.24 | 0.24 | 0.23
| 57.5 | 0.20 | 0.26 | 0.13 | 0.22 | 0.22 | 0.18 | 0.08 | 0.32 | 0.16 | 0.14 | 0.07 | 0.20 | 0.20 | 0.21
| 77.5 | 0.12 | 0.16 | 0.08 | 0.12 | 0.12 | 0.12 | 0.05 | 0.20 | 0.18 | 0.06 | 0.05 | 0.12 | 0.13 | 0.15
$\langle p_{Q}\rangle$ | 7.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.01 | 0 | 0.01 | 0.02 | 0 | 0 | 0
| 32.5 | 0.05 | 0.06 | 0.03 | 0.06 | 0.06 | 0.04 | 0.02 | 0.08 | 0.05 | 0.08 | -0.01 | 0.05 | 0.05 | 0.05
| 57.5 | 0.14 | 0.18 | 0.09 | 0.13 | 0.15 | 0.12 | 0.06 | 0.23 | 0.09 | 0.07 | 0.03 | 0.14 | 0.14 | 0.13
| 77.5 | 0.10 | 0.14 | 0.07 | 0.09 | 0.10 | 0.10 | 0.04 | 0.17 | 0.18 | 0.04 | 0.04 | 0.11 | 0.10 | 0.09
${\rm max}(p_{\varphi})$ | 7.5 | 0.35 | 0.49 | 0.22 | 0.37 | 0.37 | 0.35 | 0.16 | 0.55 | 0.49 | 0.34 | 0.43 | 0.34 | 0.37 | 0.38
| 32.5 | 0.32 | 0.44 | 0.21 | 0.36 | 0.35 | 0.32 | 0.14 | 0.53 | 0.42 | 0.30 | 0.35 | 0.32 | 0.34 | 0.36
| 57.5 | 0.32 | 0.43 | 0.22 | 0.37 | 0.35 | 0.30 | 0.13 | 0.52 | 0.40 | 0.31 | 0.27 | 0.32 | 0.33 | 0.35
| 77.5 | 0.33 | 0.42 | 0.23 | 0.38 | 0.37 | 0.29 | 0.13 | 0.53 | 0.40 | 0.26 | 0.21 | 0.33 | 0.34 | 0.36
$I_{180}/I_{000}$ | 7.5 | 1.23 | 1.29 | 1.14 | 0.96 | 1.09 | 1.39 | 1.23 | 1.23 | 1.37 | 0.98 | 0.08 | 1.22 | 1.24 | 1.26
| 32.5 | 1.93 | 2.37 | 1.37 | 0.65 | 1.13 | 3.20 | 1.93 | 1.95 | 3.00 | 0.16 | 0 | 1.91 | 1.98 | 2.04
| 57.5 | 0.29 | 0.44 | 0.16 | 0.04 | 0.11 | 0.75 | 0.29 | 0.29 | 6.11 | 0 | 0 | 0.32 | 0.27 | 0.26
| 77.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.97 | 0 | 0 | 0 | 0 | 0

111Results are given for the reference model for
$i=7.5^{\circ},\,32.5^{\circ},\,57.5^{\circ}$, and $77.5^{\circ}$, and models
that deviate for one of the following parameters: scattering albedo $\omega$,
asymmetry $g$, maximum scattering polarization $p_{\max}$, wall slope $\chi,$
and angular wall height $\alpha$. Numerical uncertainties for the calculated
model results are at a level of about one or two units for the last digit of
the given values.

###  4.3 Quantitative disk model results

The radiation parameters introduced for the reference model are also used for
the investigation of the disk parameter dependencies for disks with different
parameters. Table 1 shows numerical values for a quantitative assessment.
Results for the inclinations $i=7.5^{\circ},\,32.5^{\circ},\,57.5^{\circ},$
and $77.5^{\circ}$ for 14 models are given including the reference model and
models where one of the parameters $\omega$, $g$, $p_{\rm max}$, $\chi,$ or
$\alpha$ deviates from the reference model.

The quadrant polarizations are given as values relative to the integrated
azimuthal polarization $(\pm)\,Q_{xxx}/\overline{Q}_{\varphi}$ or
$(\pm)\,U_{xxx}/\overline{Q}_{\varphi}$. Because our disk models are
rotationally symmetric, the absolute values for all relative quadrant
parameters are for a pole-on view $i=0^{\circ}$ equal to $1/(2\pi)=0.159$ (see
Schmid 2021). The disk-averaged fractional polarization values $\langle
p_{\varphi}\rangle$ and $\langle p_{Q}\rangle$ can be deduced from $\langle
p_{\varphi}\rangle=\overline{Q}_{\varphi}/\overline{I}$ and $\langle
p_{Q}\rangle=\langle
p_{\varphi}\rangle(Q_{000}/Q_{\varphi}+2\,Q_{090}/Q_{\varphi}+Q_{180}/Q_{\varphi})$.
However, these are important observational parameters and are therefore also
listed. In addition, Table 1 includes the values for the maximum fractional
polarization ${\rm max}(p_{\varphi})$, and the front-to-back intensity ratio
$I_{180}/I_{000}$ for the comparison with the front-to-back polarization ratio
$Q_{180}/Q_{000}$.

Numerical uncertainties in Table 1 are at the level of less than one to a few
units for the last digit of the given values. Uncertainties are larger for
small inclination $i$, because the solid angles per bin for the collection of
escaping photons are smaller $(\propto\sin i)$ for pole-on viewing directions.
The uncertainties can also be estimated from the jitter in the expectedly
smooth curves in Fig. 7.

The maximum fractional polarization ${\rm max}(p_{\varphi}(x,y,i))$ occurs in
the two-dimensional image at a disk location $(x,y)$ where the scattering
angle is close to $\theta_{S}\approx 90^{\circ}$ and produces a high
fractional polarization close to the maximum value for given scattering
parameters (Fig. 6). For a large range of disk inclinations $i,$ there exists
a surface region where $\theta_{S}\approx 90^{\circ}$ and therefore ${\rm
max}(p_{\varphi}(x,y,i))$ does not depend strongly on the disk inclination
$i$, unlike the disk-averaged value $\langle p_{\varphi}(i)\rangle$ as
illustrated in Fig. 7. This is confirmed by the values in Table 1 which also
show that the dependencies of ${\rm max}(p_{\varphi})$ on $\alpha$, $\chi,$
and $g$ are small. Therefore, the ${\rm max}(p_{\varphi}(x,y))$ parameter is
very useful for constraining dust scattering parameters, in particular
$\omega$ and $p_{\rm max}$.

For most of the models in Table 1, Appendix B provides curves of the radiation
parameters as a function of disk inclination, as in the reference model in
Fig. 7. These are meant to help with assessment of the dependencies of the
reflected radiation on the disk parameters.

##  5 Diagnostic parameters for dust scattering

The calculated radiation parameters from the transition disk models can be
compared with observational data in order to constrain the properties of real
disks, in particular the dust scattering parameters $\omega$, $g,$ and $p_{\rm
max}$. However, the model results show quite complex dependencies on the disk
geometry and the dust scattering properties. Therefore, we investigate in this
section the diagnostic potential of ”observational” radiation parameters for
constraining and deriving scattering parameters of the dust and refine
geometric properties of the disk.

###  5.1 Typical radiation parameters for transition disks

The model results for the radiation parameters $\overline{I}/I_{\star}$,
$\overline{Q}_{\varphi}/I_{\star}$, and $\langle p_{\varphi}\rangle$ for
$i=12.5^{\circ}$, $32.5^{\circ}$, $57.5^{\circ}$, and $77.5^{\circ}$ for the
reference disk and 13 other models where one intrinsic disk parameter differs
from the reference case are listed in Table 1. Figure 9 shows their
distribution in the $\overline{I}/I_{\star}$–$\langle p_{\varphi}\rangle$
diagram and the points cover ranges of about a factor of 30 for
$\overline{I}/I_{\star}$ and $\overline{Q}_{\varphi}/I_{\star}$ and about a
factor of 10 for the fractional polarization $\langle p_{\varphi}\rangle$.
Typical values for this selection of models are about
$\overline{I}/I_{\star}\approx 2\leavevmode\nobreak\ \%$,
$\overline{Q}_{\varphi}/I_{\star}\approx 0.4\leavevmode\nobreak\ \%$ and a
fractional polarization of $\langle p_{\varphi}\rangle\approx
20\leavevmode\nobreak\ \%$, which are in rough agreement with measurements
obtained for transition disks; for example, with the
$\overline{Q}_{\varphi}/I_{\star}$ values for transition disks compiled by
Garufi et al. (2017), or the fractional polarization measurements of $\langle
p_{\varphi}\rangle$ provided by Perrin et al. (2009), Monnier et al. (2019),
Hunziker et al. (2021), and Tschudi & Schmid (2021).

The highlighted reference model is located in the middle of the point cloud
and clearly shows enhanced values for disk intensities, polarized intensities,
and fractional polarization for low inclination $i$ and small values for high
$i$. This pattern is shared by most disk models and therefore the parameters
are systematically displaced towards the upper right for low $i$ and towards
the lower left for high $i$, which are marked with different colors. The
points for the reference models are surrounded by values from the other models
with one parameter changed and this yields offsets in the
$\overline{I}/I_{\star}$–$\langle p_{\varphi}\rangle$ plane by some amount,
but typically by less than a factor of a few for the parameter range
considered by our models.

Figure 9: Distribution of the scattered intensity
$\overline{I}/\overline{I}_{\star}$ and the fractional polarization $\langle
p_{\varphi}\rangle$ (Y-axis) for all disk models given in Table 1. The dotted
lines indicate the constant values for the azimuthal polarization
$\overline{Q}_{\varphi}/\overline{I}$. The colors indicate the inclination $i$
of the disks and the red line marks the results for the reference model.

###  5.2 Dependencies on model parameters

Table 2: Dependence of the disk scattering intensity and polarization on model
parameters

| $X_{r}$ |  $X_{r-1}$, $X_{r+1}$ | $\overline{I}/I_{\star}$ | $\overline{Q}_{\varphi}/I_{\star}$ | $\langle p_{\varphi}\rangle$
---|---|---|---|---|---
$\omega$ | 0.5 | 0.2,0.8 |  $1.92\pm 0.52$ bb$b$Here the linear relation does not fit well, so we use a power law instead: $\overline{I}/\overline{I}_{r}=\exp{[A^{\prime}_{\omega}(\omega-\omega_{r})/\omega_{r}]}$, $A^{\prime}_{\omega}=1.59\pm 0.04$ | $1.11\pm 0.02$ | $-0.54\pm 0.03$
$g$ | 0.5 | 0.25,0.75 | $-0.64\pm 0.14$ | $-0.81\pm 0.10$ | $-0.20\pm 0.01$
$p_{\rm max}$ | 0.5 | 0.2,0.8 | $-0.01\pm 0.01$ | $1.01\pm 0.01$ | $1.02\pm 0.01$
$\alpha$ | 10 | 5, 20 | $1.16\pm 0.06$ | $1.12\pm 0.04$ | $-0.02\pm 0.01$
$\chi$ | 32.5 | 12.5, 57.5 | $-0.65\pm 0.08$ | $-0.76\pm 0.03$ | $-0.07\pm 0.09$
$\cos i$ | $0.84$ |  $0.99,\,0.61$ aa$a$The linear relation for $\overline{I}/I_{\star}$ is valid to $\cos i=0.61$ corresponding to $i=42.5^{\circ}$. | $0.12\pm 0.06$ | $1.62\pm 0.03$ | $1.51\pm 0.08$
222 The gradients of the radiation parameters integrated intensity
$\overline{I}/I_{\star}$, azimuthal polarization
$\overline{Q}_{\varphi}/I_{\star}$, and averaged fractional polarization
$\langle p_{\varphi}\rangle$ for the scattered light from transition disk with
respect to the geometric parameters inclination $i$, angular wall height
$\alpha$, wall slope $\chi,$ and the dust parameters scattering albedo
$\omega$, asymmetry $g,$ and maximum scattering polarization $p_{\rm max}$. .

The radiation parameters for the transition disks depend in various ways on
the model parameters and we roughly summarize these dependencies with linear
relations $Y=aX+b$. We express the dependencies with a relative gradient
$A_{X}$ at a reference point $X_{r},\,Y_{r}$:

$\frac{Y(X)-Y_{r}}{Y_{r}}=A_{X}\,\frac{X-X_{r}}{X_{r}}\,,$ (13)

where $Y$ represents radiation parameters like $\overline{I}/I_{\star}$,
$\overline{Q}_{\varphi}/I_{\star}$, or $\langle p_{\varphi}\rangle$ and $X$
represents one of the model parameters for the dust scattering $\omega$, $g$,
$p_{\rm max}$, or $\alpha$, $\cos i$, $\chi$ for the disk geometry. Also,
$Y_{r}$ and $X_{r}$ are the corresponding values for the reference model and
$A_{X}=a\,(X_{r}/Y_{r})$ describes the relative gradient for the parameters of
the reference model $Y_{r}$, $X_{r}$. Derived values for $A_{X}$ are
summarized in Table 2. For example, if $\omega$ is enhanced by
$20\leavevmode\nobreak\ \%$ from $\omega_{r}=0.5$ to $\omega=0.6$, $\langle
p_{\varphi}\rangle$ decreases by $-0.54\cdot 20\leavevmode\nobreak\
\%=-10.8\leavevmode\nobreak\ \%$ with respect to the reference model, taking
the gradient $A_{\omega}[\langle p_{\varphi}\rangle]=-0.54$.

The gradients $A_{X}$ listed in Table 2 are derived from the gradient between
the two bracketing values $Y(X_{r-1})$ and $Y(X_{r+1})$ and the uncertainties
represent the mean gradient difference for the points $Y(X_{r-1})$, $Y_{r}$
and $Y_{r}$, $Y(X_{r+1})$ with respect to $A_{X}$. The simple linear relation
in Eq. 13 accurately describes the dependencies $Y(X)$ for many combinations
of radiation parameters $Y$ and model parameters $X,$ as can be inferred from
the small uncertainties indicated in Table 2.

For $A_{\omega}[I/I_{\star}],$ a large uncertainty of $\pm 0.52$ or a strong
deviation from a linear relation is obtained, because the reflected intensity
has a dependence on the scattering albedo which is roughly exponential
$\overline{I}/I_{\star}=c\,\exp(a\,\omega)$ in the range $\omega=0.2-0.8,$
which is similar to the results of the plane parallel case (e.g., Fig. 5(a)).
This exponential function or the corresponding Taylor expansion can also be
expressed with the relative gradient at the reference point $A^{\prime}_{X}$:

$\frac{Y(X)-Y_{r}}{Y_{r}}=\exp\Bigl{(}A^{\prime}_{X}\frac{X-X_{r}}{X_{r}}\Bigr{)}-1=Z+\frac{1}{2}Z^{2}+\frac{1}{6}Z^{3}+\cdots\,,$
(14)

where $Z=A^{\prime}_{X}(X-X_{r})/X_{r}$. For the gradient
$A^{\prime}_{\omega}=1.59\pm 0.04$, a good fit is obtained for
$\overline{I}/\overline{I}_{r}=\exp{[A^{\prime}_{\omega}(\omega-\omega_{r})/\omega_{r}}]$.

The inclination dependence of the polarized intensity
$\overline{Q}(i)/I_{\star}$ and the fractional polarization $\langle
p_{\varphi}(i)\rangle$ show a broad maxima for pole-on disks or $i\approx
0^{\circ}$ and a decrease similar to a $\cos i$-function for larger $i$. This
can be well fitted with the linear function as in Eq. 13, if $\cos i$ is used
as model parameter $X$. However, for $X=\cos i,$ the obtained gradients
$A_{\cos i}$ must be interpreted accordingly. First, a positive $A_{\cos i}$
means that the $Y(i)$ decreases with $i$ similar to $\cos i$. For the
scattered intensity $\overline{I}/I_{\star}$, the gradient is close to zero
and this is equivalent to a very flat dependence. Second, the large gradients
of $A_{\cos i}\approx 1.5$ to $1.6$ for $\overline{Q}(i)/I_{\star}$ and
$\langle p_{\varphi}(i)\rangle$ mean that there is still a relatively small
deviation of $<|\pm 15\leavevmode\nobreak\ \%|$ from the exact $\cos
i$-function, because only a very small $\cos i$-range around the reference
value is sampled for the covered inclination range from $7.5^{\circ}$ to
$42.5^{\circ}$. Third, the $i$-range for the fitting had to be restricted to
$42.5^{\circ}$, because for larger $i$ the radiation parameters show bumps and
strong changes due to the disappearance of the inner wall on the front side.

The dependencies given in Table 2 apply for the reference case $Y_{r}(X_{r})$,
but should represent the behavior of the disk radiation parameters described
by the range $[X_{r-1},X_{r+1}]$ quite well; this range covers the region
where the radiation functions $Y(X)$ ($\overline{I}/I_{\star}$,
$\overline{Q}_{\varphi}/I_{\star}$, $\overline{Q}/I_{\star}$, $\langle
p_{\varphi}\rangle$, and $\langle p_{Q}\rangle$) show smooth dependencies in
Fig. 19. This range excludes highly inclined disks $i>57.5^{\circ}$ or disks
with steep inner walls $\chi>57.5^{\circ}$ where disk visibility and
obscuration effects play an important role and produce strong and ”bumpy”
parameter dependencies.

The value for radiation parameters with two or more model parameters different
from the reference value $Y(X_{1},X_{2})$ can be estimated from

$\frac{Y(X_{1},X_{2})}{Y_{r}}\approx\Bigl{(}A_{1}\frac{X_{1}-X_{1,r}}{X_{1,r}}+1\Bigr{)}\,\Bigl{(}A_{2}\frac{X_{2}-X_{2,r}}{X_{2,r}}+1\Bigr{)}\,.$
(15)

This assumes that the relative gradient $A_{1}$ for the $Y(X_{1})$ remains
roughly constant for small offsets along model parameter $X_{2}$ and vice
versa. This approximation is certainly good for moderate offsets
$X_{1}-X_{1,r}$ and $X_{2}-X_{2,r}$ from the reference model.

It is clear from Table 2 that the integrated intensity
$\overline{I}/I_{\star}$, azimuthal polarization
$\overline{Q}_{\varphi}/I_{\star}$, and fractional polarization $\langle
p_{\varphi}\rangle$ depend on several model parameters. This creates
ambiguities between the six model parameters involved and it is not
straightforward to derive or constrain individual disk or dust parameters from
the scattered radiation.

The following subsections discuss the different radiation parameters that are
particularly useful for constraining the dust scattering parameters $\omega$,
$g,$ and $p_{\rm max}$. Powerful diagnostic information can be obtained from
radiation parameters that do not depend on disk wall height $\alpha$ or slope
$\chi,$ because these are often poorly known geometric parameters of the disk.

###  5.3 Wavelength dependencies for the scattering parameters

The most basic observational parameters for the scattered radiation from
circumstellar disk are the disk-integrated intensity $\overline{I}/I_{\star}$
and the azimuthal polarization $\overline{Q}_{\varphi}/I_{\star}$. However,
both parameters depend strongly on the disk geometry and the dust scattering
parameters and it is difficult to disentangle the complex dependencies.

An important approach to constrain dust scattering parameters is therefore the
determination of the wavelength dependence for parameters of the reflected
radiation, like $(\overline{I}/I_{\star})_{\lambda}$,
$(\overline{Q}_{\varphi}/I_{\star})_{\lambda}$, or $Q_{180}/Q_{\varphi}$,
because it seems reasonable to assume that the disk scattering geometry is
identical or close to identical for different wavelengths. This means that
color effects in the reflected light of circumstellar disks are predominantly
produced by the wavelength dependence of the dust scattering parameters
$\omega(\lambda)$, $g(\lambda),$ and $p_{\rm max}(\lambda)$. Colors for a
radiation parameter $Y$ can be quantified as differences $\Delta
Y/\Delta\lambda=(Y_{\lambda 2}-Y_{\lambda 2})/(\lambda_{2}-\lambda_{1})$ or
ratios $\Lambda Y/\Lambda\lambda=(Y_{\lambda 2}/Y_{\lambda
2})/(\lambda_{2}/\lambda_{1})$. This requires high-quality measurements of the
same source for different wavelengths and such measurements or estimates have
been obtained for a few transition disks, such as TW Hya (Debes et al. 2013),
HD 100546 (Mulders et al. 2013), HD 135344B (Stolker et al. 2016), HD 142527
(Hunziker et al. 2021), or HD 169142 (Tschudi & Schmid 2021).

Figure 10: Relative change in disk integrated intensity
$\overline{I}/\overline{I}_{r}$, azimuthal polarization
$\overline{Q}/\overline{Q}_{r}$, and fractional polarization $\langle
p_{\varphi}\rangle/\langle p_{\varphi}\rangle_{r}$ with respect to the
reference model as a function of the dust scattering parameters $\omega$, $g,$
and $p_{\rm max}$. The dashed gray line is the fit from Table 2. In most
panels the fitted lines coincide with the calculated results.

The model results and the gradients $A_{X}(Y)$ given in Tables 1 and 2 provide
the relative changes ($\Lambda(Y/Y_{r})-1$) for the disk intensity
$\overline{I}$, azimuthal polarization $\overline{Q}_{\varphi}$, and
fractional polarization $\overline{p}_{\varphi}$ for parameters ”near” the
reference model as plotted in Fig. 10. For disks with properties comparable to
those of the reference model, an observed color like
$\Lambda(\overline{Q}_{\varphi}/I_{\star})$ for $\lambda_{1}$ and
$\lambda_{2}$ can then be compared with these results to constrain the
wavelength dependencies of the dust scattering parameters $\omega(\lambda)$,
$g(\lambda)$, and $p_{\rm max}(\lambda)$ compatible with the measured color.

In several observational studies the transition disk was found to reflect
radiation at longer wavelengths $\lambda>1\leavevmode\nobreak\ \mu$m more
efficiently than that at shorter wavelengths $\lambda<1\leavevmode\nobreak\
\mu$m (e.g., Mulders et al. 2013; Stolker et al. 2016; Hunziker et al. 2021).
The diagrams in Fig. 10 indicate that this requires dust with either a larger
single scattering albedo $\omega$ for longer wavelengths, a lower scattering
asymmetry $g$, or a solution where the combined effect of both parameters can
explain the observed color change.

###  5.4 Fractional polarization

The parameters for the fractional polarization $\langle{p}_{\varphi}\rangle$,
$\langle{p}_{Q}\rangle$, or ${\rm max}(p_{\varphi})$ are relative quantities
between the polarization and intensity of the scattered radiation and
therefore they are independent of the size of the scattering region; for our
disk model they are almost independent of the wall height $\alpha$. Figure 11
shows the dependencies for the azimuthal polarization
$\langle{p}_{\varphi}\rangle$ according to the results in Table 1 with
dependencies as described by the gradients given in Table 2.

The fractional polarization $\langle p_{\varphi}\rangle$ shows strong
dependencies on $\omega$ and $p_{\rm max}$, but is independent of $\alpha$,
and is therefore ideal for constraining these two dust scattering parameters.
Important for the fractional polarization is the scattering angle $\theta_{S}$
and therefore $\langle p_{\varphi}\rangle$ depends on the inclination $i$,
which is fortunately often well known for disks. The wall slope $\chi$ has
little but still some influence on $\langle p_{\varphi}\rangle$ because it
influences the relative contributions of scatterings with different
$\theta_{S}$ to the flux-weighted average $\langle p_{\varphi}\rangle$.

The dependence of the fractional polarization on $i$ and $\chi$ can be further
reduced if only the maximum fractional polarization of a disk
max($p_{\varphi}$) is considered as shown in the lower panel row of Fig. 11.
This parameter selects the disk surface regions where the scattering angle is
close to $\theta_{S}\approx 90^{\circ}$, which produces the highest possible
fractional polarization for a surface with the given scattering parameters.
Such regions exist for disks with high and low inclination, or steep and flat
inner walls.

An important aspect is that the determination of ${\rm max}(p_{\varphi})$
requires only a measurement of $p_{\varphi}(x,y)\approx
Q_{\varphi}(x,y)/I(x,y)$ for disk locations where the fractional polarization
is close to the maximal fractional polarization. No full disk integrations for
$Q_{\varphi}$ and $I$ are required. Therefore, the ${\rm max}(p_{\varphi})$
may be obtained for disks where reliable $\langle p_{\varphi}\rangle$
measurements for an estimate of scattering parameters are not possible,
because the disk deviates strongly from axisymmetry, because some disk regions
are shadowed by an inner disk, or because it is observationally difficult to
measure the disk intensity $I$ or polarized intensity $Q_{\varphi}$ well for
all disk regions for the determination of integrated quantities $\overline{I}$
and $\overline{Q}_{\phi}$. For inclined disks, ${\rm max}(p_{\varphi})$ is
predicted for locations near the major axis (see Fig. 6) which are well
separated from the central star and therefore favorable for accurate
measurements.

Figure 11: Dependencies of the disk-averaged fractional polarization
$\langle{p}_{\varphi}\rangle$ and maximum fractional polarization ${\rm
max}(p_{\varphi})$ on the scattering parameters $\omega$, $g,$ and $p_{\rm
max}$ and the geometric parameters $i$ and $\chi$ near the reference case.

Figure 12: Maximum fractional polarization $p_{\varphi}$ as a function of
scattering parameter $p_{max}$ and $\omega$. The gray dashed lines represent
$\omega=0.2,0.5,$ and $0.8$ for the reference parameters
$X_{r}:\,g=0.5,\,i=32.5^{\circ},\,\chi=32.5^{\circ}$. The points near the
reference case illustrate $p_{max}$ values for $X_{r-1}$ and $X_{r+1}$.

The parameter dependencies for ${\rm max}(p_{\varphi})$ can be quite
accurately approximated by two linear dependencies on $p_{\rm max}$ and
$\omega$ while ignoring minor effects on $g$, $i$ , $\chi,$ and $\alpha$:

$\frac{{\rm max}(p_{\varphi})}{{\rm
max}(p_{\varphi})_{r}}\approx\Big{(}1+A_{\rm pmax}\dfrac{p_{\rm max}-p_{\rm
max,r}}{p_{\rm
max,r}}\Big{)}\Big{(}1+A_{\omega}\dfrac{\omega-\omega_{r}}{\omega_{r}}\Big{)}$
(16)

with the gradients $A_{p{\rm max}}=1.00\pm 0.03$ and $A_{\omega}=-0.59\pm
0.01$ based on the values from Table 1. Because of the strict proportionality
${\rm max}(p_{\varphi})\propto p_{\rm max}$, and using the reference values
for ${\rm max}(p_{\varphi})_{r}$, $p_{\rm max,r}$, and $\omega_{r}$, we obtain
an even more simple dependence:

${\rm max}(p_{\varphi})\approx p_{\rm max}(1.02-0.76\cdot\omega)\,.$ (17)

This relation is compared in Fig. 12 with model results and the agreement is
excellent for a large range of $p_{\rm max}$ and $\omega$ parameters. Thus,
the fractional polarization $\langle p_{\varphi}\rangle$ or ${\rm
max}(p_{\varphi})$ of the scattered radiation from disks provides strong
constraints on the possible value combination for the single scattering albedo
$\omega$ and the maximum scattering polarization $p_{\rm max}$. If one of
these parameters can be constrained with additional observations or
theoretical considerations, then the other parameter is also well defined.

###  5.5 Azimuthal distribution of the scattered polarization

Figure 13: Quadrant polarization ratios $Q_{180}/Q_{000}$ and
$I_{180}/I_{000}$ as a function of $\omega$, $g$, $\chi,$ and $\alpha$. The
largest inclination $i$ is restricted to $42.5^{\circ}$ to avoid effects
related to the disappearance of the inner wall on the front side.

Figure 14: Quadrant polarization ratios $Q_{180}/Q_{000}$ as a function of
asymmetry $g$ and disk wall slope $\chi$. The gray dashed lines represent
$g=0,0.25,0.5,$ and $0.75$ for the reference parameters
$X_{r}:\omega=0.5,p_{max}=0.5,i=32.5^{\circ}$.

The quadrant polarization parameters given in Table 1 describe the azimuthal
dependence of the polarized radiation. A combination of different quadrant
parameters could be investigated (Schmid 2021), but we restrict our discussion
to the front-to-back ratio $Q_{180}/Q_{000}$ and the corresponding intensity
ratio $I_{180}/I_{000}$ shown in Fig. 13 for different $i$ and as functions of
$\omega$, $g$, $\chi$, and $\alpha$. The ratios are almost independent of
$p_{\rm max}$ and therefore this is not shown. Nearly face-on disks
($i=7.5^{\circ}$) are close to axisymmetric and the ratios are close to one
for all model parameters. The front–backside asymmetries become obvious for
moderately inclined disks ($i=32.5^{\circ}$ and $42.5^{\circ}$) and they are
rather similar for the polarization ratio $Q_{180}/Q_{000}$ and the intensity
ratio $I_{180}/I_{000}$. The values for larger inclinations are not shown,
because the front side disappears and the $Q_{180}$ and $I_{180}$ values go to
zero. For the wall slope $\chi=57.5^{\circ}$ this happens already for
$i=32.5^{\circ}$ and $42.5^{\circ}$.

For inclined disks, the dependence of the front-to-back ratios is particularly
strong with respect to the scattering asymmetry parameter $g$ and the disk
slope $\chi$ and the ratio $Q_{180}/Q_{000}$ can be used as diagnostics. For
example, for $i=32^{\circ}$, $\omega=0.5,$ and $\alpha=10^{\circ}$, the
dependence can be fitted with the linear relation:

$\dfrac{Q_{180}/Q_{000}}{(Q_{180}/Q_{000})_{r}}\approx\Big{(}1+A_{g}\dfrac{g-g_{r}}{g_{r}}\Big{)}\Big{(}1+A_{\chi}\dfrac{\chi-\chi_{r}}{\chi_{r}}\Big{)}$
(18)

where $A_{g}=0.83\pm 0.04$ and $A_{\chi}=-1.11\pm 0.08$ and the relation is
plotted in Fig. 14. Much brighter front sides, as seen in many moderately
inclined disks, indicate that the dust favors forward scattering $g>0.25$ and
the illuminated inner disk wall on the front side is flat enough to be well
visible. If the back side is brighter, then the illuminated wall on the front
side becomes hardly visible. Other quadrant polarization ratios, such as
$Q_{090}/Q_{000}$, can also be used to determine $g$ and $\chi$ using the
numerical results presented in Table 1. Results can be verified with the
investigation of different quadrant ratios and additional effects, such as
deviations of the disk geometry from rotational symmetry, the presence of
shadows from hot dust near the star, or significant contributions from diffuse
dust located above the disk surface, and other effects may be identified.

###  5.6 Comparison of the scattered and thermal disk emission

The spectral energy distributions of transition disk systems show a well-
defined far-IR emission excess which is explained by the thermal emission from
the dust in the disk, which is heated mainly by the absorption of stellar
radiation. It was further shown by Garufi et al. (2017) that for transition
disks (Meeus group I disks) there exists a strong correlation between this
far-IR excess and the relative polarized intensity $F({\rm
fIR})/F_{\star}\approx 35\times Q_{\varphi}/I_{\star}$ which are both
quantities normalized to the emission of the stars. Therefore, the comparison
between the scattered intensities $I/I_{\star}$ or polarized intensity
$Q_{\varphi}/I_{\star}$ and the IR excess $F_{\rm IR}/F_{\star}$ can be used
to obtain rough constraints on the disk surface reflectivity and therefore
also the single scattering albedo $\omega$.

For our scattering models, we can assume that the reflecting dust of the inner
disk wall is also strongly absorbing and as discussed for a flat surface, the
absorbed radiation ${\cal{K}}$ is directly related to the integrated surface
reflectivity or surface albedo ${\cal{R}}$ according to
${\cal{K}}=1-{\cal{R}}$. The absorbed radiation integrated over all
wavelengths heats the dust which produces thermal emission measurable in the
spectral energy distribution of the system as IR excess $F_{\rm
IR}/F_{\star}$.

We use the double ratios

$\Lambda_{I}=\frac{\langle\overline{I}(i)/I_{\star}\rangle}{F_{\rm
IR}(i)/F_{\star}}\quad{\rm
and}\quad\Lambda_{p}=\frac{\langle\overline{Q}_{\varphi}(i)/I_{\star}\rangle}{F_{\rm
IR}(i)/F_{\star}}\,,$ (19)

which can be considered as an apparent disk scattering albedo and polarized
albedo (according to Tschudi & Schmid 2021), and one can expect that these
quantities depend strongly on the single scattering albedo $\omega$ of the
dust.

In this approach, the relationship between measurable radiation parameters and
the dust scattering parameters is less close. A major complication is that the
IR excess $F_{\rm IR}(i)/F_{\star}$ provides a ”bolometric” measure of the
absorbed radiation energy integrated over all wavelengths, while the scattered
intensity $\overline{I}(i)/I_{\star}$ is typically only measured for one or a
few wavelengths. Thus, one needs to derive or predict the wavelength
dependence of the scattered radiation
$\langle\overline{I}(i)/I_{\star}\rangle$ or
$\langle\overline{Q}_{\varphi}(i)/I_{\star}\rangle$ and estimate the entire
energy input contributing to the far-IR excess. This includes the stellar
radiation, energetic radiation from the disk accretion, near-IR radiation from
hot dust located near the star, the galactic radiation field, and cosmic rays.
In most cases, the absorption of stellar radiation will be the dominant source
of energy for the far-IR emission of transition disks.

In this work, as a first approximation, we neglect the wavelength dependencies
for the scattering and the absorption by the disk. We assume that results from
such simple models will probably introduce uncertainties of a factor of a few
for an estimate on $\omega$ from the apparent disk albedo $\Lambda_{I}$ or
polarized albedo $\Lambda_{p}$ determinations. First estimates by Garufi et
al. (2017) for a large sample or the detailed determination by Tschudi &
Schmid (2021) yield $\Lambda_{p}\approx 0.03$, which is compatible with the
$\omega=0.5$ of the reference model. Most likely, the apparent disk albedo
determination and the corresponding constraints on $\omega$ could be improved
with detailed investigations of wavelength-dependent effects and the
derivation of ”bolometric” correction factors, but this is beyond the scope of
this paper. In this section, we provide only a simplified assessment of how
the apparent albedo is expected to depend on the parameters of our disk
models.

Figure 15: Reflected emission $\overline{I}_{\rm Lam}(i,\chi)/I_{\star}[\%]$
for a scattering disk with a perfect Lambert surface ${\cal{R}}=1$ and a wall
height of $\alpha=10^{\circ}$. Table 3: Reflected emission $\overline{I}_{\rm
Lam}(i,\chi)/I_{\star}[\%]$ for a perfect Lambert disk surface

$i$ | $\chi=$ | $12.5^{\circ}$ | $32.5^{\circ}$ | $57.5^{\circ}$ | $82.5^{\circ}$
---|---|---|---|---|---
$7.5^{\circ}$ |  | 33.6 | 29.1 | 18.5 | 4.6
$32.5^{\circ}$ |  | 28.6 | 24.7 | 15.8 | 11.9
$57.5^{\circ}$ |  | 18.2 | 15.8 | 17.0 | 18.2
$77.5^{\circ}$ |  | 7.4 | 11.9 | 17.1 | 18.8
333Numerical results of the reflected emission at different inclinations $i$
for disks with different wall slopes $\chi$ and fixed wall height
$\alpha=10^{\circ}$, by a Lambert surface with ${\cal{R}}=1$, as in Fig. 15.

####  5.6.1 Derivation of the IR excess

There exist very sophisticated models for the thermal emission of the dust in
circumstellar disks which calculate the detailed spectral energy distribution
based on hydrodynamical disk geometries and considering thermo-chemical
processes of the gas and the dust (e.g., Woitke et al. 2016). These models are
characterized by ten or more parameters and they are not appropriate for our
simple models. We use a rough approximation for the integrated IR excess
$\overline{F}_{\rm IR}/F_{\star}$ which is compatible with our six-parameter
transition disk models.

Therefore, for the calculation of the IR excess of the disk $\overline{F}_{\rm
IR}/F_{\star}$ , we adopt the same disk geometry as for the scattered
radiation, assuming that the disk is also optically thick for infrared light
similar to the model of D’Alessio et al. (2005). The emitted infrared energy
from point $S(r,h)$ is assumed to be equal to the energy of absorbed photons
at $S(r,h)$ as calculated by the scattering models and each point on the disk
$S(r,h)$ emits in the same way as a Lambert surface $I_{\rm
Lam}(\theta)\propto\cos\theta$ (see Fig 4). We only consider the total energy
of the emitted thermal radiation from the transition disk and disregard the
spectral energy distribution.

The thermal emission for this kind of model can be calculated with a
scattering model which assumes that all stellar light is absorbed and
converted into IR thermal emission. In this case, the thermal emission is
equal to the results of the scattering model for a disk with a perfectly
reflecting Lambert surface,

$\frac{F_{\rm IR}(\omega=0)}{F_{\star}}=\frac{\overline{I}_{\rm
Lam}({\cal{R}}=1)}{I_{\star}}\,.$ (20)

The reflected intensity of such a disk $\overline{I}_{\rm
Lam}(i,\chi,\alpha)/I_{\star}$ depends on the disk geometry $i$, $\alpha$, and
$\chi$ only. Figure 15 shows $\overline{I}_{\rm Lam}(i)/I_{\star}$ for
$\alpha=10^{\circ}$ as a function of inclination for different wall slopes
$\chi$ and assuming perfect reflection ${\cal{R}}=1,$ and Table 3 gives
numerical values for four inclinations. The Lambertian disk $\overline{I}_{\rm
Lam}(i)/I_{\star}$ shows strong differences in the $i$-dependence compared to
the $g=0.5$ multiple-scattering model $\overline{I}(i,\chi)/I_{\star}$ in Fig.
19 because the latter produces a strong forward scattering peak for inclined
disks with small $\chi=12.5^{\circ}$ and $32.5^{\circ}$.

The disk with $\alpha=10^{\circ}$ intercepts 17.6 % of the stellar light and
the perfect Lambert surface produces for small $\chi$ an intensity of more
than $\overline{I}_{\rm Lam}/I_{\star}>25\leavevmode\nobreak\ \%$ for polar
directions. It should be noted that the results in Fig. 15 or Table 3 consider
only the radiation scattered at one location on the disk. Not considered is
for example the reflected light from the back side which would contribute
significantly for high ${\cal{R}}$ to the illumination of the inner wall on
the front side for a high-albedo disk surface as can be seen by the drop in
$\overline{I}_{\rm Lam}(i)/I_{\star}$ for $i>90^{\circ}-\alpha=80^{\circ}$.
This neglected second-order effect would add some light to the reflected light
$\overline{I}_{\rm Lam}/I_{\star}$ for smaller $i$. This effect is very small
for low-albedo dust. The dependence of the thermal radiation on the disk
height $\alpha$ can be described approximately by combining the results in
Fig. 15 and Table 3 with the scaling factor $\sin\alpha/\sin\alpha_{r}$ (where
$\alpha_{r}=10^{\circ}$), which accounts for the amount of stellar radiation
intercepted by the disk.

For the calculation of the thermal emission, we adopt the description for the
scattered light for a disk with a perfect Lambert surface $\overline{I}_{\rm
Lam}(i,\chi,\alpha)/I_{\star}$ and simply scale with ${\cal{K}}=1-{\cal{R}}$
for the absorbed radiation:

$\frac{F_{\rm
IR}(i,\chi,\alpha,\omega,g)}{F_{\star}}\approx\frac{\overline{I}_{\rm
Lam}(i,\chi,\alpha_{r})}{I_{\star}}\,\frac{\sin\alpha}{\sin\alpha_{r}}\,(1-{\cal{R}}(\chi,\omega,g))\,.$
(21)

The disk-integrated reflectivity ${\cal{R}}(\chi,\omega,g)$ is a function of
the scattering parameters $\omega$ and $g$, and also depends on the wall slope
$\chi$ as given in Table 4 for 36 parameter combinations. The
${\cal{R}}$-values vary by up to a factor of 100 between ${\cal{R}}\approx
0.43$ for high-scattering albedo $\omega=0.8$, close to isotropic scattering
$g=0.25$, and flat wall slope $\chi=12.5^{\circ}$ and ${\cal{R}}\approx 0.04$
for low albedo $\omega=0.8$, strong forward scattering $g=0.75$, and steep
wall slopes $\chi=82.5^{\circ}$. The impact of the model parameters is much
less dramatic for the absorption ${\cal{K}}=1-{\cal{R}}$ and the thermal
emission, because the extreme values are ${\cal{K}}\approx 0.57$ and 1.0 which
differ by less than a factor of two.

It is obvious that most of the radiation interacting with the disk is absorbed
for the parameters selected for our model grid and the values for the
reference model are about ${\cal{R}}\approx 0.10$ for the reflectivity or
${\cal{K}}\approx 0.9$ for the absorption.This yields together with
$\overline{I}_{\rm
Lam}(i=32.5^{\circ},\chi=32.5^{\circ})/I_{\star}=24.7\leavevmode\nobreak\ \%$
for the reference model an IR excess which is consistent with the typical
results $(F_{mid-IR}+F_{far-IR})/F_{\star}\approx 20\%\pm 10\%$ derived for a
larger disk sample studied within the DIANA project (Woitke et al. 2019).

Table 4: Integrated surface reflectivities ${\cal{R}}(\chi,\omega,g)$

$\omega$ | $g$ | $\chi=12.5^{\circ}$ | $32.5^{\circ}$ | $57.5^{\circ}$ | $82.5^{\circ}$
---|---|---|---|---|---
0.8 | 0.25 | 0.4287 | 0.3095 | 0.2312 | 0.1951
0.8 | 0.5 | 0.4011 | 0.2575 | 0.1741 | 0.1393
0.8 | 0.75 | 0.3408 | 0.1706 | 0.0969 | 0.0712
0.5 | 0.25 | 0.2058 | 0.1277 | 0.0859 | 0.0690
0.5 | 0.5 | 0.1885 | 0.0968 | 0.0558 | 0.0415
0.5 | 0.75 | 0.1497 | 0.0537 | 0.0252 | 0.0172
0.2 | 0.25 | 0.0696 | 0.0391 | 0.0245 | 0.0190
0.2 | 0.5 | 0.0630 | 0.0278 | 0.0145 | 0.0103
0.2 | 0.75 | 0.0476 | 0.0139 | 0.006 | 0.004
444 The values are derived for $p_{\rm max}=0.5$ and $\alpha=10^{\circ}$, but
they are essentially independent of $p_{\rm max}$ and can be scaled with
$\alpha$. The 0.5-model is given with italic fonts.

####  5.6.2 Apparent disk scattering albedos

Figure 16: Dependencies of the apparent disk scattering albedo $\Lambda_{I}$
and polarized albedo $\Lambda_{P}$ on the scattering parameters $\omega,g,$
and $p_{max}$ and geometric parameters $i$ and $\chi$. For parameters not
indicated in the panel, the reference parameters $\omega,\,g,p_{\rm max}=0.5,$
and $\chi=32.5^{\circ}$ are used. The values of the reference model for
$i=7.5^{\circ}$, $32.5^{\circ}$, $57.5^{\circ}$ , and $77.5^{\circ}$ are
plotted with open red symbols.

The approximation for the IR excess $F_{\rm IR}/I_{\star}$, together with the
disk integrated scattered intensity $\overline{I}/I_{\star}$ or polarized
intensity $\overline{Q}_{\varphi}/I_{\star}$ , provides the apparent disk
scattering albedo $\Lambda_{I}$ or polarized albedo $\Lambda_{P}$ defined in
Eq. (19). The albedo is a relative quantity which does not depend on the
fraction of stellar photons interacting with the disk. Therefore, the
dependence of the albedo on $\alpha$ is very small for our models. On the
other side, the apparent disk albedo is a wavelength-averaged quantity and
this introduces issues with the $\lambda$-dependence of the scattering
parameters which should be known for the range covered by the stellar
radiation.

The dependencies of the apparent disk albedos $\Lambda_{I}(i)$ and
$\Lambda_{p}(i)$ on the scattering parameters $\omega$, $g$, $p_{\rm max}$,
and the wall slope $\chi$ are shown in Fig. 16 for different inclination $i$.
This shows that $\Lambda_{I}(i)$ correlates very strongly with the single
scattering albedo $\omega$ and anti-correlates with the asymmetry parameter
$g$ as expected from the relative dependencies
$\overline{I}(\omega)/\overline{I}_{r}$ and $\overline{I}(g)/\overline{I}_{r}$
given in Fig. 10. However, for the albedos, the effect of higher reflectivies
${\cal{R}}$ for large $\omega$ and low $g$ is further enhanced by the reduced
absorption and the overall disk albedo dependence is roughly
$\Lambda\propto{\cal{R}}/(1-{\cal{R}})$.

For small disk inclinations $i=7.5^{\circ}$ or $i=32.5^{\circ}$, the
dependence of $\Lambda_{I}$ on $i$ and the wall slope $\chi$ is very small and
$\Lambda_{I}$ is therefore mainly a function of the two scattering parameters
$\omega$ and $g$ and is very useful for constraining the dust properties for
disks seen close to pole-on. This is of particular value because disk slopes
and scale heights are difficult to derive for disks seen close to pole-on
(e.g., Rich et al. 2021).

The polarized scattered intensity $\overline{Q}_{\varphi}/I_{\star}$ is often
easier to measure than $\overline{I}/I_{\star}$ with high-contrast disk
observations (Schmid 2021) and this provides the polarized scattering albedo
$\Lambda_{P}$. The dependencies for $\Lambda_{P}$ are more complex because the
polarization introduces additional strong effects related to $i$, $p_{\rm
max}$. Also, the dependence on the wall slope is larger for $\Lambda_{P}$ than
for $\Lambda_{I}$, but still quite small for $i$ close to pole-on
($i=7.5^{\circ},\,32.5^{\circ}$).

The typical $\Lambda_{P}$-values in Fig. 16 are in surprisingly good agreement
with the study of Garufi et al. (2017), who, for a sample of transition disks,
find a strong correlation between IR excess and polarized light contrast of
$F_{\rm IR}/F_{\star}\approx 35\times\overline{Q}_{\varphi}/I_{\star}$ or
$\Lambda_{P}\approx 0.029$ while the tabulated values (Tables 1,3,4) for the
reference model with $i=32.5^{\circ}$ yield $\Lambda_{P}\approx
0.0069/(0.247\,(1-0.0968))=0.031$ using Eqs. (19) and (21). An agreement
within a factor of a few indicates that the model calculations in this work
accurately represent the overall budget between scattering and absorption of
real transition disks. The almost perfect match between the reference model
and the relation of Garufi et al. (2017) is a coincidence when considering the
model parameter dependencies of $\Lambda_{P}$ and the large dispersion and
uncertainties in the compiled observational values. The intensity albedo of
the reference model is $\Lambda_{I}\approx\Lambda_{p}/\langle
p_{\varphi}\rangle=0.13$ and this value also matches well with the measured
value for the inner wall (inner ring) of HD 169142 by Tschudi & Schmid (2021).

##  6 Discussion and conclusions

This paper presents model simulations of the intensity and polarization of the
scattered stellar light from the inner wall of transition disks. Results for
many model parameter combinations are calculated in order to quantify the
dependencies of the ”observable” radiation parameters on the disk geometry and
the dust scattering properties. The results can be readily compared with
measurements and used to constrain the properties of transition disks; in
particular, the light scattering parameters of the dust.

The presented transition disk models are very simple and fully characterized
by only six parameters. This confers the important advantage that the
dependencies on each parameter can be easily understood and investigated in
detail and that the very basic relations describing the disk properties can be
easily quantified.

Indeed, real transition disks are much more complex than the selected
description of an axisymmetric inner wall described by the parameters
inclination $i$, wall height $\alpha,$ and slope $\chi,$ and the dust
scattering parameters albedo $\omega$, asymmetry $g$, and polarization $p_{\rm
max}$. In addition, the dust scattering in the disk is treated in the same way
as multiple scattering on a semi-infinite (optically thick) plane parallel
surface, assuming that the dust properties are the same everywhere in the
disk.

When using the presented results as tools for the interpretation of
observational data, it is important to be aware of these simplifications and
to therefore compare model results only to measurements that do not depend
strongly on system properties not considered in the simple model. In the
following, we review our main simulation results and critically discuss their
application for the interpretation of observations.

###  6.1 Dust scattering parameters

The description of the dust scattering presented in Sect. 3 explains the most
important properties of the radiation reflected from a multi-scattering dusty
surface. We also find a strong correlation between the single scattering
albedo $\omega$ and the total surface reflectivity ${\cal{R}}$, an anti-
correlation between ${\cal{R}}$ and the forward scattering parameter $g$, a
higher ${\cal{R}}$ for grazing incidence, and the highest fractional
polarization for scattering angles around $\theta_{S}\approx 90^{\circ}$ , as
described in the literature (e.g., Chandrasekhar 1960; Abhyankar & Fymat 1970;
Dlugach & Yanovitskij 1974; van de Hulst 1980). In addition, we add many
results for the polarization of the reflected light for forward scattering
dust which are specifically useful for the investigation of the reflection
from optically thick circumstellar disks. A particularly interesting result is
the obtained maximum fractional polarization parameter $max(p(\theta_{pp}))$
for the reflected polarization, which shows a very strong dependence on the
scattering albedo $\omega$ and polarization $p_{\rm max}$ but very little
dependence on the scattering asymmetry $g$ and very moderate dependence on the
incidence angle $\theta_{0}$. This is very useful for tightly constraining
$\omega$ and $p_{\rm max}$ from fractional polarization measurements of disks.

Describing the scattering in the disk surface using only three dust parameters
is certainly a simplification. However, which extension to the dust scattering
model is required to achieve a significant improvement for the interpretation
of disk observations seems to be unclear. The adoption of a Rayleigh-
scattering-like polarization phase function $p_{\rm sca}(\theta_{S})$, which
is symmetric with respect to $\theta_{S}=90^{\circ}$, could introduce a
significant bias effect for the interpretation of the azimuthal distribution
of the polarization signal in very well observed disks. The dust in debris
disks seems to have a different $p_{\rm sca}(\theta_{S})$ function (e.g.,
Milli et al. 2019; Arriaga et al. 2020) and this possibility should also be
investigated for transition disks.

For a given model, we adopt the same dust scattering parameters everywhere in
the disk. This could be a strong over-simplification as dust particles in
disks are known to have different size distributions at different locations
and this is probably also the case for vertical or radial directions within
the scattering surface of disks. For example, there could exist a significant
vertical stratification with small particles with Rayleigh scattering-like
properties in the uppermost layers and larger, more forward-scattering
particles in deeper layers. This could produce reflected radiation
$I_{\lambda}(\theta,\phi)$, $p_{\lambda}(\theta,\phi)$ from a surface with
wavelength dependencies which are not compatible with one homogeneously
distributed dust-particle population. An investigation of such stratification
effects would be very interesting, because they could produce signatures that
might be clearly recognizable in high-quality data.

###  6.2 Scattering in a plane parallel surface layer

Our disk models adopt a plane-parallel surface geometry everywhere on the
disk, where the scattered photon escapes from the same location as the
incident photon. This is certainly a reasonable assumption for surface areas
$S(r,h)$ in the ”middle” of an extended flat wall (at $r_{0}<r<r_{w}$) where
diffuse dust above the surface behaves like the uppermost layers of a plane-
parallel surface. However, the plane-parallel surface assumption breaks down
for optically thin, diffuse dust inside of $r_{0}$, for diffuse dust above the
rim of the wall, and for illuminated dust above the outer flat surface of the
disk, because scattered photons may escape for directions $\theta>90^{\circ}$
for which escape is not possible in the plane-parallel geometry adopted for
our transition disk models. For example, diffuse dust above the disk at
$h(r)/r>\tan\alpha$ will produce optically thin forward scattering and this
could significantly enhance the intensity brightness of the front side for
highly inclined disks. Therefore, the calculated diagnostic parameters for the
intensity and polarized intensity ratios between the front and back side
$I_{180}/I_{000}$ and $Q_{180}/Q_{000}$, respectively, should only be used
very cautiously for the interpretation of strongly inclined disks
$i>45^{\circ}$.

The model results for highly inclined disks are still useful because they
provide an estimate or upper limit for the expected scattered light produced
by the optically thick disk wall. Any additional signal above this level could
then be attributed to scattering occurring above the disk surface (Glauser et
al. 2008; Villenave et al. 2020) and this could be used to quantify the
diffuse dust above the disk surface.

For small disk inclinations $i,$ or for the reflected radiation from the back
wall, the forward scattering effect by the diffuse dust is much less important
and the signal from the quasi plane-parallel disk wall surface is more visible
and therefore dominant. For all these cases, the results from our simple model
should provide a good description of the scattered intensity and polarization.

###  6.3 Disk geometry

For the geometry of the transition disk, the simulations adopt a rotationally
symmetric inner wall with a fixed wall slope $\chi$ and an angular wall height
$\alpha$ and no variations in the azimuthal direction. Real disks are often
much more complicated, showing a large diversity of structures for transition
disks including spirals (e.g., Garufi et al. 2013; Benisty et al. 2015),
eccentric rings (Rameau et al. 2012), circular gaps (Quanz et al. 2013),
irregular small-scale structures (Ginski et al. 2021), outer low-surface-
brightness regions for extended flaring disks (Ginski et al. 2016), and
shadows from hot dust near the star (Marino et al. 2015; Pinilla et al. 2018).
All these features are not captured by our simple models and the comparison of
calculated results with such observations must consider the impact of the more
complex geometry. One can study only disks that are close to centro-symmetric,
but this would strongly restrict the available sample for the detailed
investigation of the dust in circumstellar disks. For investigations of a
larger sample, it is useful to study the uncertainties introduced by the not-
so-well-defined disk geometry on the observational parameters and the derived
dust properties.

Very significant discrepancies between observations and the results of the
axisymmetric models can occur for disks with large-scale azimuthal structures
which have a significant impact on the disk-integrated
$\overline{I},\,\overline{Q}_{\varphi}$ or disk-averaged $\langle
p_{\varphi}\rangle$ radiation parameters. Less critical are small-scale
structures, such as small spiral features, localized shadows, or bumps in the
disk surface because their effect will at least partly average out, meaning
that comparisons of measured values $\overline{I}$ and
$\overline{Q}_{\varphi}$ with axisymmetric models are still useful. Also, more
complicated radial disk structures, such as a bright inner disk wall or faint
but extended contributions from the outer regions of a flaring disk could at
first approximation be compared with a model with one wall slope $\chi_{\rm
model}$, intermediate between the steep slope of the inner wall $\chi_{\rm
inner}>\chi_{\rm model}$ and the flat slope of the outer flaring disk surface
$\chi_{\rm outer}<\chi_{\rm model}$.

Azimuthal inhomogeneities of disks can be measured as brightness differences
between the left and right quadrant polarization parameters with respect to
the projected minor disk axis as

$\Delta=\frac{|Q_{090}|-|Q_{270}|}{|Q_{090}|+|Q_{270}|}\,,$ (22)

and the equivalent for $|U_{045}|$ and $|U_{315}|$ or $|U_{135}|$ and
$|U_{225}|$ (Schmid 2021). This gives a rough measure of the strength of the
azimuthal feature from which one can estimate its impact on the disk-
integrated or disk-averaged radiation parameters when comparing observations
of non-symmetric disks to the results of the symmetric disk models.

###  6.4 Reflected radiation and diagnostic parameters

The presented model calculations provide two-dimensional images for the
scattered intensity $I(x,y)$, azimuthal polarization $Q_{\varphi}(x,y)$, and
other radiation parameters. It is useful for the comparison with observations
to deduce disk-integrated radiation parameters from these model images that
are scaled to the stellar intensity, as in $\overline{I}(i)/I_{\star}$,
$\overline{Q}_{\phi}(i)/I_{\star}$, the disk-averaged fractional polarization
$\langle p_{\varphi}\rangle$, or quadrant polarization values
$Q_{xxx}(i)/I_{\star}$ or $U_{xxx}(i)/I_{\star}$ which can also be deduced
from observations without introducing significant ambiguities by the diversity
of disk morphologies. We note that the measured values must be corrected for
instrumental effects, in particular for the signal convolution with the
instrumental PSF, which can introduce significant cancellation of the disk
signal (Schmid et al. 2006; Avenhaus et al. 2014, 2017; Heikamp & Keller 2019;
Tschudi & Schmid 2021). Unfortunately, available observational data with high
accuracy and quantified uncertainties are still very limited and often only
some of the radiation parameters can be determined for a given disk.
Therefore, the derivation of disk parameters from the comparison with model
results can be rather ambiguous. The model results presented in this work
allow us to carry out a detailed investigation of the important parameter
ambiguities involved in the interpretation of observations of transition disks
and provide diagnostic relations that can be used to constrain key model
parameters for different types of observational data. For example, Fig. 19
shows that, for a given inclination $i$, $\overline{I}/I_{\star}$,
$\overline{Q}_{\varphi}/I_{\star}$, and $\overline{Q}/I_{\star}$ all strongly
correlate with the angular wall height $\alpha$ and single scattering albedo
$\omega$, and anti-correlate with wall slope $\chi$ and scattering asymmetry
parameter $g$. In addition, the polarization parameters
$\overline{Q}_{\varphi}/I_{\star}$ and $\overline{Q}/I_{\star}$ also
correlated with $p_{\rm max}$. If only the scattered intensity
$\overline{I}/I_{\star}$ or only the polarized intensity is measured then it
is almost impossible to constrain individual disk parameters without
additional information. Therefore, it is important to select and obtain more
observational information and better diagnostic parameters with more
diagnostic power.

Strong ambiguities can be partly broken if both $\overline{I}/I_{\star}$ and
$\overline{Q}_{\varphi}/I_{\star}$ or $\langle
p_{\varphi}\rangle=\overline{Q}_{\varphi}/\overline{I}$ are known. The
fractional polarization correlates with $p_{\rm max}$, but anti-correlates
with $\omega$, while the dependencies are small for the parameter $g$, the
wall slope, if $\chi\leq 57.5^{\circ}$, and zero for the wall height $\alpha$,
while $i$ and the relative dependence $\langle p_{\varphi}(i)\rangle$ are
typically well known. Essentially, the impact of uncertainties in the disk
geometry on the resulting disk-averaged fractional polarization $\langle
p_{\varphi}(i)\rangle$ or the maximum fractional polarization
max($p_{\varphi}$) are strongly reduced, these parameters are mainly used to
probe the dust-scattering parameters $\omega$, $p_{\rm max}$, and $g$.

Similarly, the measurement of the wavelength dependence of the radiation
parameters $(\overline{I}/I_{\star})_{\lambda}$ or
$(\overline{Q}_{\varphi}/I_{\star})_{\lambda}$ does not depend strongly on the
relatively poorly understood disk geometry because one can assume that the
scattering geometry for the radiation with different wavelengths is very
similar. This provides well-defined constraints on the wavelength dependence
of the scattering parameters $\omega(\lambda)$, $g(\lambda),$ and $p_{\rm
max}(\lambda),$ which can be compared with the prediction of different dust
models.

More constraints on the dust-scattering parameters, in particular on the
scattering asymmetry parameter, can be gained from derivations of the
azimuthal dependence of the scattered radiation in inclined disks using for
example ratios for the front-to-back intensity ratio $I_{180}/I_{000}$ or
ratios of quadrant polarization parameters $Q_{180}/Q_{000}$. In addition, one
may constrain the single scattering albedo $\omega$ from derivations of the
effective disk albedos $\Lambda_{I}$ or $\Lambda_{p}$. These are less direct
methods which might be strongly affected by the not-well-known scattering
geometry. However, observations seem to confirm general trends, such as
$g>0.25$ from the often-observed brighter front side in inclined disks, and
$\omega\leq 0.75$ from the correlation between scattered and thermal emission
from resolved transition disks $\Lambda_{p}\approx 0.03$ as derived by Garufi
et al. (2017). These findings can at least exclude dust with a close-to-
isotropic scattering phase function $g\nless 0.25$ and very high scattering
albedos of $\omega\ngtr 0.9$.

###  6.5 Dust properties in transition disks

Ultimately, we are interested in deriving dust properties for transition
disks, such as the particle size distribution $n(a)$ where $a$ is the particle
radius, the particle composition, and structure. These parameters are strongly
linked to the used dust scattering parameters $\omega$, $g$, and $p_{\rm max}$
but there exist no well-established relations between dust properties and dust
scattering parameters for propto-planetary disks. Detailed dust models are
required for a translation between these two types of dust description.

Mie theory describes scattering for homogeneous spherical particles and is
useful for a first rough characterization. For wavelengths much longer than
the size of the particles $\lambda>2\pi a$, this theory predicts a fast
decrease in the absorption cross-section with wavelengths, roughly as
$\kappa(\lambda)\propto\lambda^{-2}$ and an even faster decrease for the
scattering $\sigma(\lambda)\propto\lambda^{-4}$. Therefore, the extinction
cross-section is very small and mostly due to absorption. Therefore, according
to Eq. (4), $\omega$ is also small. Thus, for long wavelengths, the expected
scattering parameters are $\omega\leq 0.5$, $g<0.5$, and $p_{\rm max}>0.5$ as
expected for scattering of small particles in the Rayleigh regime such as
interstellar dust at $\lambda>2\leavevmode\nobreak\ \mu$m (Weingartner &
Draine 2001). For short wavelengths $\lambda\leq 2\pi\alpha$, the cross-
sections $\sigma$ and $\kappa$ are roughly independent of $\lambda$ and are of
the same order as the particle size $\approx\pi\,a^{2}$. The scattering
parameters are therefore $\omega>0.5$, $g>0.5,$ and $p_{\rm max}<0.5,$ as
expected from scattering and Fraunhofer diffraction by large particles similar
to the scattering of cometary dust at $\lambda<1\leavevmode\nobreak\ \mu$m
(e.g., Lasue et al. 2009).

Derived scattering parameters, for example for the disk around HD 142527, but
also for other transition disks, seem to indicate a higher reflectivity and
therefore higher $\omega$ for longer wavelengths, a roughly constant asymmetry
parameter from $0.7\leavevmode\nobreak\ \mu$m to $2.2\leavevmode\nobreak\
\mu$m, and an increasing max$(p_{\varphi})$ with wavelength (Hunziker et al.
2021). This is not easily compatible with a simple dust model based on Mie
theory. Usually, dust models based on Mie theory seem to be in qualitative
agreement with observations of the thermal and scattered emission if only the
reflected light at one wavelength is considered. However, we are not aware of
a detailed study using Mie theory that compares models and observations of the
thermal emission, the wavelength dependence of the reflected radiation, and
the fractional polarization of a protoplanetary disk in detail. This would
provide the dust scattering parameters $\sigma(\lambda,a)$,
$\omega(\lambda,a)$, $g(\lambda,a)$, and $p_{\rm max}(\lambda,a)$ as functions
of wavelength. Alternative dust models based on large porous grains or
aggregates $a>\lambda$ with a lot of small-scale structures could quite
naturally explain $\omega>0.5$ at $2\leavevmode\nobreak\ \mu$m, a $g>0.5$ over
a broad wavelength range, and still a quite high scattering polarization
$p_{\rm max}\approx 0.3$ at $0.7\leavevmode\nobreak\ \mu$m. These produce a
high scattering polarization by the interaction of the light with small
substructures, and strong forward scattering and high scattering albedo