11institutetext:  Department of Astrophysics/IMAPP, Radboud University, PO Box
9010, 6500 GL Nijmegen, the Netherlands 22institutetext: Max-Planck-Institut
für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

# Polarimetric signatures of hot spots in black hole accretion flows

J. Vos Polarimetric signatures of hot spots in black hole accretion
flowsPolarimetric signatures of hot spots in black hole accretion flows
<EMAIL_ADDRESS>M. Mościbrodzka Polarimetric signatures of hot spots in
black hole accretion flowsPolarimetric signatures of hot spots in black hole
accretion flows M. Wielgus Polarimetric signatures of hot spots in black hole
accretion flowsPolarimetric signatures of hot spots in black hole accretion
flows

(Received ¡date¿ / Accepted ¡date¿)

###### Abstract

Context. The flaring events observed in the Sagittarius A* supermassive black
hole system can be attributed to the non-homogeneous nature of the near-
horizon accretion flow. Bright regions in this flow may be associated with
density or temperature anisotropies, so-called “bright spot” or “hot spots”.
Such orbiting features may explain observations at infrared wavelengths as
well as recent findings at millimeter wavelengths.

Aims. In this work, we study the emission from an orbiting equatorial bright
spot, imposed on a radiatively inefficient accretion flow background, to find
polarimetric features indicative of the underlying magnetic field structure
and other system variables including inclination angle, spot size, black hole
spin, and more. Specifically, we investigate the impact of these parameters on
the Stokes $\mathcal{Q}-\mathcal{U}$ signatures that commonly exhibit a
typical double loop (pretzel-like) structure.

Methods. Our semi-analytical model, describing the underlying plasma
conditions and the orbiting spot, is built within the framework of the
numerical radiative transfer code ipole, which calculates synchroton emission
at 230 GHz.

Results. We showcase the wide variety of $\mathcal{Q}-\mathcal{U}$ loop
signatures and the relation between inner and outer loops. For the _vertical_
magnetic field topology, the inner $\mathcal{Q}-\mathcal{U}$ loop is explained
by the suppression of the synchrotron emission as seen by the distant
observer. For the radial and toroidal magnetic field topologies, the inner
$\mathcal{Q}-\mathcal{U}$ loop corresponds to the part of the orbit where the
spot it is receding with respect to the observer.

Conclusions. Based on our models we conclude that it is possible to constrain
the underlying magnetic field topology with an analysis of the
$\mathcal{Q}-\mathcal{U}$ loop geometry, particularly in combination with a
circular polarization measurements.

###### Key Words.:

black hole physics – relativistic processes – radiative transfer – methods:
numerical

##  1 Introduction

The supermassive black hole (SMBH/BH) at the center of the Milky Way,
Sagittarius A* (hereafter Sgr A*), is surrounded by a radiatevely inefficient
accretion flow (RIAF) with bolometric luminosity at a strongly sub-Eddington
level ($L_{\rm bol}\approx 10^{-10}L_{\rm Edd}$, where $L_{\rm Edd}$ is the
Eddington luminosity; for reviews of Sgr A* observational properties see,
e.g., Genzel et al. 2010; Morris et al. 2012). The environment surrounding
SMBH is emitting photons across the entire electromagnetic spectrum from radio
to $\gamma-$rays, but the source spectral energy distribution peaks in the
millimeter band. The millimeter band emission is dominated by the synchrotron
process occurring in the immediate vicinity of the SMBH (e.g., Narayan et al.,
1995; EHTC et al., 2022a, d). Sgr A* is known to display flaring behaviour on
the top of the base ”quiescent” stochastic variability. During these flaring
events, observed on a daily basis, the signal can increase by up to $\sim 100$
times the mean quiescent flux density in the near-infrared (NIR) (e.g., Dodds-
Eden et al., 2009; Witzel et al., 2018, 2021) and X-ray (e.g., Porquet et al.,
2008; Nowak et al., 2012; Do et al., 2019). The flux density is much larger at
millimeter to radio wavelengths than the mean NIR and X-ray values and
displays only marginal flaring behavior (e.g., EHTC et al., 2022b; Wielgus et
al., 2022a). This apparent discrepancy is attributed to effects associated
with particle heating/acceleration occurring near the SMBH but the details of
such flaring sites such as their location, geometry, magnetization, and
evolution are currently not well-understood (Witzel et al., 2018, and
references therein).

One of the proposed models for explaining the flaring activity of Sgr A* is
the formation of plasmoids111A number of other non-plasmoid explanations (see,
e.g., Witzel et al., 2021) have also been proposed to explain the NIR
variability. near the BH event horizon. The term ”plasmoid” is broadly used,
but we employ the term to describe a ”localized energized accretion flow
feature”. These bright blob-like features are also referred to as ”hot spots”
or ”bright spots”. Such a scenario has been adopted to explain NIR flares
observed using the GRAVITY instrument (Gravity Collaboration et al., 2018,
2020) and polarimetric variability at millimeter wavelengths (Wielgus et al.,
2022b) observed with the Atacama Large Millimeter/submillimeter Array (ALMA).

It is still largely unknown how a plasmoid would observationally manifest
itself at millimeter wavelengths. With the Event Horizon Telescope (EHT; EHTC
et al., 2019a, b) it has recently become possible to reach a sufficient
angular resolution to image event horizon scale features of Sgr A*, both in
total intensity (EHTC et al., 2022a, c) and in linear polarization (so far
only published for the M87* SMBH; EHTC et al., 2021a, b). While there is no
unambiguous trace of orbiting hot spots in the EHT images of Sgr A* published
so far, we turn to investigating their impact on the millimeter light curves.
In particular, Sgr A*’s linear polarization observations (during flares) in
both the NIR and millimeter bands display similar looping behavior in the
Stokes ${\mathcal{Q}}-{\mathcal{U}}$ diagrams (Marrone, 2006; Gravity
Collaboration et al., 2020; Wielgus et al., 2022b), suggesting that the
flaring events should manifest themselves in the millimeter band light curves
of the unresolved compact source. Additionally, in the millimeter band we
observe persistent net circular polarization (Stokes $\mathcal{V}$) of the Sgr
A* light curves at the level of $\sim-1\%$ (Muñoz et al., 2012; Goddi et al.,
2021; Wielgus et al., 2022b). Circular polarization measurements obtained
during a ${\mathcal{Q}}-{\mathcal{U}}$ loop may help us to further constrain
the physics of the flares.

On the theoretical front there has been a significant effort to model Sgr A*
flares with plasmoids. Plasmoids emission or their formation are currently
modeled using a few distinct approaches: (i) semi-analytic geometric models
(Gelles et al., 2021), (ii) semi-analytic accretion flow models with a
Gaussian fluctuation in it (Broderick & Loeb, 2005, 2006; Gravity
Collaboration et al., 2020), (iii) general relativistic magnetohydrodynamic
(MHD/GRMHD) simulations (see, e.g., Murphy et al., 2013; Ripperda et al.,
2022) and (iv) general relativistic particle-in-cell (PIC/GRPIC) simulations
(see, e.g., Cerutti et al., 2014; Sironi et al., 2016; Crinquand et al.,
2021). The approaches (ii), (iii) and (iv) include all general relativistic
effects. Approaches (i) and (ii) focus on predicting radiative properties of
plasmoids; approaches (iii) and (iv) mostly aim at following plasmoid
formation and dynamics. In approaches (iii) and (iv) plasmoids are associated
with magnetic reconnection events that heat the plasma (e.g., Rowan et al.,
2017). Under certain conditions the reconnection current sheets break into a
chain of plasmoids which then grow by merging with other plasmoids and evolve
further (Loureiro et al., 2012). Plasmoids may therefore observationally
appear as hot spots.

The main advantage of semi-analytic models of plasmoids (approach (i) and
(ii)) is that their observational appearance can be accurately predicted using
first-principle radiative transfer calculations. The first semi-analytical
radiative models of plasmoid were developed in Broderick & Loeb (2005), where
the authors investigated the total intensity and polarized light curves and
images of an optically thick sphere that orbits the BH in the equatorial
plane. In Broderick & Loeb (2006), the previous models are expanded by adding
a RIAF background emission and a different underlying magnetic field. Other
notable papers that discuss radiation from (semi-analytical) Gaussian
structures include: Hamaus et al. (2009); Zamaninasab et al. (2010); Broderick
et al. (2011a, b); Pu et al. (2016); Tiede et al. (2020). Another class of
these models discusses the hot spot’s formation and evolution along the jet of
the BH (Younsi & Wu, 2015; Ball et al., 2021).

The main focus of this work is to investigate the Stokes $\mathcal{I}$,
$\mathcal{Q}$, $\mathcal{U}$, and $\mathcal{V}$ properties of the semi-
analytic Kerr black hole RIAF + hot spot model, and their dependence on the
model parameters, particularly the magnetic field topology. We argue that the
polarimetric features introduced by an equatorially orbiting hot spot, in
addition to a RIAF background, are indicative of the underlying magnetic field
geometry and spacetime. More specifically, we demonstrate the utility of
Stokes $\mathcal{Q}-\mathcal{U}$ diagrams (first shown in Marrone 2006; also
discussed in Gravity Collaboration et al. 2020; Gelles et al. 2021) for
differentiating between a wide variety of simulated scenarios that can help
constrain black hole spin, inclination, and orbital velocity of the entire
system. Additionally, we discuss how circular polarization from orbiting
plasmoids can help differentiate between dominant magnetic field geometries.
While the NIR emission is well explained by an orbiting hot spot alone, the
millimeter emission is expected to be dominated by the accretion disk
background. The interplay between background and orbiting hot spot is another
problem of interest that is investigated in this work.

Our work is motivated by the new full Stokes, high-sensitivity millimeter
light curve datasets available for Sgr A* (Wielgus et al., 2022a, b), and in
particular by the identification of orbiting hot spot signatures in ALMA 230
GHz observations (Wielgus et al., 2022b). The paper is structured as follows.
We describe our model and its parameters in Sect. 2. In Sect. 3 we report the
polarimetric properties of hot spots as a function of the model free
parameters. We focus on discussing polarimetric light curves,
$\mathcal{Q}-\mathcal{U}$ loops, and circular polarization. We discuss our
results and compare them to previous works in Sect. 4. Finally, we summarize
our main findings and conclude in Sect. 5.

##  2 Model

###  2.1 Numerical setup

Our semi-analytic model of the accretion flow with an orbiting bright spot
around a Kerr black hole is built within ipole \- a ray-tracing code for
covariant general relativistic transport of polarized light in curved
spacetimes. The detailed description of the numerical scheme is presented by
Mościbrodzka & Gammie (2018). The radiative transfer model requires defining
the underlying plasma density, electron temperature, and magnetic field
strength and geometry to calculate thermal synchrotron emission. Given a
plasma configuration, ipole generates maps of Stokes $\mathcal{I}$,
$\mathcal{Q}$, $\mathcal{U}$, $\mathcal{V}$ at chosen observing frequencies
including emission, self-absorption and Faraday effects (rotation and
conversion). In this work, we will limit ourselves to model thermal
synchrotron emission at 230 GHz ($\lambda=1.3$ mm, i.e. millimeter band) with
a standard image resolution of $256\times 256$ (pixels). The images are pixel
integrated to generate light curves. The radiative transfer model takes into
account the finite speed of light which indicates a ”slow light” approach.

All calculations are carried out using the Kerr metric described by the Kerr-
Schild coordinate system ($t,r,\theta,\phi$) (which is well-described in,
e.g., McKinney & Gammie, 2004) with the standard metric signature (- + + +).
In the following sections, we measure distances in units of gravitational
radii ($r_{g}=GM/c^{2}$, where $G$ and $c$ are the gravitational constant and
the speed of light, respectively). Time is measured in units of $r_{g}/c$.

###  2.2 Background accretion flow

Our background accretion flow model is inspired by a radiatively inefficient
accretion disk presented in Broderick & Loeb (2006) (see also Fraga-Encinas et
al. 2016). At the equator, we assume that the accretion flow around the Kerr
black hole is Keplerian. The dimensionless black hole spin parameter is
defined as $a_{*}\equiv Jc/GM^{2}$, where $J$ is the black hole angular
momentum. In this work, we consider $a_{*}=0,\pm 0.5,\pm 0.9375$. The various
quantities describing the disk properties, with the subscript $b$ for
background, are denoted by:

$\displaystyle
n_{e,b}=n_{0}\,\left(\frac{r}{r_{g}}\right)^{-3/2}\,\exp\left(-\tfrac{\cos^{2}(\theta)}{2\sigma^{2}}\right),$
(1)
$\displaystyle\Theta_{e,b}=\Theta_{0}\left(\frac{r}{r_{g}}\right)^{-0.84}\,,$
(2) $\displaystyle B=B_{0}\left(\frac{r}{r_{g}}\right)^{-1}.$ (3)

Here, $n_{e}$ is the electron density, $\Theta_{e}\equiv
k_{B}T_{e}/m_{e}c^{2}$ is the dimensionless electron temperature, $B$ is the
magnetic field strength, and $\sigma$ parameter characterises the accretion
disk vertical thickness.

All user-defined scaling factors and variables (including $n_{0}$ and
$\Theta_{0}$) are listed in Table 1. Note that we do not fit these parameters
to any existing observations of Sgr A* in this work, because the source is
variable and our static background model is not. Nevertheless, we gauge the
model parameters so that the radial, $i=30^{\circ}$ case approximately matches
the observed flux density of Sgr A* at 230 GHz, which is $\sim$2.5 Jy (Johnson
et al., 2015; Bower et al., 2019; Wielgus et al., 2022a).

The electron density $n_{e}$ is set to a small value for radii smaller than
the innermost stable circular orbit (ISCO, see Bardeen et al., 1972).
Therefore, emission from within this region is minimal except from the photon
ring in certain configurations. Additionally, as the ISCO is an spin-dependent
quantity (see Table 2) that becomes smaller for increasing $a_{*}$, we find
that the overall $n_{e,b}$ is higher for the positive spin cases. Therefore,
as the spot’s $n_{e}$ (see next subsection) is unchanged for all cases, the
background emission will be more pronounced in the positive spin ($a_{*}>0$)
than in the negative spin ($a_{*}<0$) cases.

What is the model’s magnetic field geometry? We investigate three basic
magnetic field topologies denoted simply as _radial_ , _toroidal_ ,
_vertical_. In purely poloidal cases (radial and vertical topology), the
magnetic fields, in a given coordinate frame, are described with the azimuthal
component of the vector potential. For the radial (monopolar) magnetic field
this is;

$A_{\phi}=1-\cos(\theta),$ (4)

and for the vertical magnetic field this is;

$A_{\phi}=r\sin(\theta).$ (5)

While the magnetic fields of the former cases are defined through the vector
potential, the toroidal magnetic field is straightforwardly defined by

$B_{\phi}=1.$ (6)

###  2.3 Orbiting spot

In addition to the background, we introduce a spot in a circular orbit in the
equatorial plane at a fixed distance to the SMBH. The spot’s location is
denoted, in Kerr-Schild coordinates ($t,r,\theta,\phi$), by:

$\displaystyle x_{s}^{0}=t\,,$ (7) $\displaystyle
x_{s}^{1}=r_{\textrm{spot}}=11\,r_{\text{g}}\,,$ (8) $\displaystyle
x_{s}^{2}=\theta_{\textrm{spot}}=\pi/2\,,$ (9) $\displaystyle
x_{s}^{3}=\phi_{\textrm{spot}}=2\pi t/P.$ (10)

Here, $t$ is the time coordinate along the null-geodesic, which implies that
the spot’s position is not fixed as a light ray moves through it. The radius
at which the spot orbits, $r_{\textrm{spot}}$, is fixed for the majority of
this work. Only the $\phi_{\textrm{spot}}$ coordinate changes over time and is
determined by the orbital period $P$ (see Table 2). We typically assume
Keplerian rotation, that is

$P=2\pi\left(r^{3/2}+a_{*}\right)\,[r_{\mathrm{g}}/c],$ (11)

and that the spot co-rotates with the background RIAF. For observers with
viewing angles $i<90^{\circ}$ ($i>90^{\circ}$) the spot orbits the SMBH
counterclockwise (clockwise).

Our implementation of a bright spot can be thought of as an over-density with
a higher temperature, which effectively makes the spot brighter than the
background emission. The spot’s electron density $n_{e,s}$ and dimensionless
electron temperature $\Theta_{e,s}$ are defined as:

$\displaystyle n_{e,s}=\tfrac{n_{0}}{3}\exp\left(-\frac{\Delta x^{k}\Delta
x_{k}}{2R_{s}^{2}}\right)\,,$ (12)
$\displaystyle\Theta_{e,s}=\Theta_{0}\left(\frac{r}{r_{g}}\right)^{-0.84}\exp\left(-\frac{\Delta
x^{k}\Delta x_{k}}{2R_{s}^{2}}\right),$ (13)

where $\Delta x^{k}\equiv x^{k}-x_{s}^{k}$, with $k\in\\{1,2,3\\}$, is the
Euclidean distance from the spot center ($x_{s}^{k}$) to the photon (on the
null-geodesic) which eventually falls on the camera. Note that in our model
the spot continues its orbit while the radiative transfer equations are
evolved (a.k.a. “slow light”). The entire, disk + spot, system is then simply
described by;

$\displaystyle n_{e}=n_{e,b}+n_{e,s}\,,$ (14)
$\displaystyle\Theta_{e}=\Theta_{e,b}+\Theta_{e,s}.$ (15)

The characteristic radius of the Gaussian hot spot is fixed to $R_{s}=1.5$
$r_{\rm g}$, which makes it consistent with the assumptions of Broderick &
Loeb (2005, 2006). Additionally, we enforce a sharp spot boundary by only
including matter if $n_{e,s}>10^{5}$ cm-3. Compared with plasmoids formed in
recent GRMHD studies (Ripperda et al., 2020; Bransgrove et al., 2021), our
spot’s size is large but still possible. On the other hand, simulations of
Ripperda et al. (2022) indicate formation of flux tubes that could manifest as
observable orbiting hot spots, and are generally larger than the size that we
assume.

Figure 1:  An edge-on view ($i=90^{\circ}$) of the disk background in the radial (_top row_), toroidal (_middle row_) and vertical (_bottom row_) magnetic field configurations for a field of view of $40\,r_{g}\times 40\,r_{g}$. The _left panel_ displays the zero-spin cases ($a_{*}=0$), while the _right panel_ gives an overview of the other spin-cases. The white ticks, which are plotted over the Stokes $\mathcal{I}$ maps, denote the EVPA scaled according to $\widetilde{\text{LP}}$.  Figure 2:  Stokes $\mathcal{I}$ and polarimetric images of the background accretion flow as a function of the observer’s viewing angle $i$ and magnetic field topology. The _left panel_ corresponds to $i=0^{\circ}$ while $i=20^{\circ},40^{\circ},60^{\circ},80^{\circ}$ are shown in the _right panel_. Total intensity and polarimetric maps are displayed in the same fashion as in Fig. 1.  Table 1: Fiducial background accretion flow model parameters Parameter | Value
---|---
$B_{0}$ | 100 G
$n_{0}$ |  $6\times 10^{6}$ cm-3
$\Theta_{0}$ | 200
$\sigma$ | $0.3$
Table 2: List of Keplerian orbital periods ($P$) and ISCOs

$a_{*}$ | P [$r_{g}/c$] | ISCO [$r_{g}$]
---|---|---
0.9375 | 235.1 | 2.04
0.5 | 232.4 | 4.23
0 | 229.2 | 6.00
-0.5 | 226.1 | 7.55
-0.9375 | 223.3 | 8.82
222All periods are calculated for a spot orbital radius of
$r_{\text{spot}}=11$ $r_{\rm g}$.

##  3 Results

###  3.1 Definitions

In the following subsections we present results in terms of net fractional
linear (LP) and net fractional circular polarization fractions (CP) which are
defined as:

$\displaystyle\text{LP}=\frac{\sqrt{(\sum_{i}\mathcal{Q})^{2}+(\sum_{i}\mathcal{U})^{2}}}{\sum_{i}\mathcal{I}},$
(16) $\displaystyle\text{CP}=\frac{\sum_{i}\mathcal{V}}{\sum_{i}\mathcal{I}},$
(17)

where $\mathcal{I},\mathcal{Q},\mathcal{U},\mathcal{V}$ are the Stokes
parameters and the sums ($\sum_{i}$) are over image pixels. We also define:

$\displaystyle\widetilde{\text{LP}}=\frac{\sqrt{\mathcal{Q}^{2}+\mathcal{U}^{2}}}{\mathcal{I}},$
(18) $\displaystyle\widetilde{\text{CP}}=\frac{\mathcal{V}}{\mathcal{I}},$
(19)

where $\widetilde{\text{LP}}$ and $\widetilde{\text{CP}}$ denote the linear
and circular polarization per pixel. Additionally, we often discuss the net
Stokes parameters’ behavior, which are denoted as:
$F_{\nu}=\sum_{i}\mathcal{I}$, $\text{U}=\sum_{i}\mathcal{U}$,
$\text{Q}=\sum_{i}\mathcal{Q}$, and $\text{V}=\sum_{i}\mathcal{V}$. The net
electric vector position angle (EVPA) is defined as:

$\text{EVPA}=\frac{1}{2}\mathrm{arg}\left(\sum_{i}\mathcal{Q}+j\sum_{i}\mathcal{U}\right),$
(20)

where $j$ is the imaginary unit and $\text{EVPA}\in(-90^{\circ},90^{\circ}]$.
EVPA is measured East of North on the sky. Therefore, increasing EVPA
indicates counter-clockwise rotation of the EVPA. $\text{EVPA}=0^{\circ}$
translates into the polarization tick being aligned with the North-South and
$\text{EVPA}=\pm 90^{\circ}$ with East-West direction in the sky (or image).

In the following subsections, we will outline distinguishing features for the
suite of models. To characterize the spot’s position in the image (over time),
and thereby the behaviour of $\mathcal{I,Q,U,V}$, we introduce the concept of
a location angle $\zeta$. This angle is a simple aiding quantity to specify
the spot’s location in the image (and does therefore not outline the
geometrical location of the spot in Kerr-Schild coordinates). For a face-on
view of the system ($i=0^{\circ}$), however, $\zeta$ will be equal to
$\phi_{\textrm{spot}}$, but this is not the case for other inclination angles.
From this viewing angle, the spot’s (counterclockwise) orbit will describe a
circle. Then, $\zeta=0^{\circ}$ corresponds to the spot being in the topmost
and $\zeta=180^{\circ}$ to the spot being in the bottom-most positions.
$\zeta=90^{\circ}$ and $\zeta=270^{\circ}$ correspond to the spot being in
leftmost and rightmost positions, respectively.

###  3.2 Static background emission: Stokes ${\mathcal{I}}$ and polarization

The left panels of Figure 1 display images of the background emission of the
zero-spin ($a_{*}=0$) model for the three different magnetic field
configurations as seen from an edge-on viewing angle ($i=90^{\circ}$). For
this viewing angle, the radial and vertical flux density $F_{\nu}$ structure
is similar – a bright (relativistically beamed) left side of the disk, a faint
(relativistically debeamed) right side of the disk, and a faint lensed ring
surrounding the black hole shadow. Notice, that the photon ring around the
central black hole shadow is only weakly visible in the total intensity maps.
Total intensity images are similar for different magnetic field geometries
although the radial and vertical cases look slightly different than the
toroidal case.

Linear polarization carries information about the magnetic field topology.
Generally, for all magnetic field topologies, $\widetilde{\text{LP}}$ is the
lowest for the brightest (beamed) region and the highest for the diffuse halo.
The toroidal case differs substantially from the others as it has two lower
$\widetilde{\text{LP}}$ (of $\sim 50\%$) ”lobes” above and below the
equatorial plane. This behavior is specific to this particular case and is not
recovered for other configurations based on inclination or otherwise. For the
radial and vertical cases, the higher flux density regions are less polarized
($60\%$ – $65\%$). Polarization ticks in the total intensity maps show the
direction of the linear polarization vector $(\mathcal{Q},\mathcal{U})$. These
are different for different magnetic field geometries.

For $\widetilde{\text{CP}}$, the structural differences based on magnetic
field configuration are even more substantial and they reflect the underlying
magnetic field polarity (as we have checked that in our models most of the
$\widetilde{\text{CP}}$ is produced intrinsically rather then via the Faraday
conversion process). The radial case $\widetilde{\text{CP}}$ image is
predominantly negatively circularly polarized with positive polarization in
the Doppler beamed part of the image where the sign (handedness) of the
$\widetilde{\text{CP}}$ is due to the arbitrarily assumed magnetic field
polarity (the inversion of B sign will produce opposite polarity of
$\widetilde{\text{CP}}$ maps). In the toroidal case, the
$\widetilde{\text{CP}}$ map is positive on the Doppler beamed side and
negative elsewhere. The vertical case displays a positive
$\widetilde{\text{CP}}$ for the top half and negative $\widetilde{\text{CP}}$
for the lower half resulting in null total CP, as expected. Notice that in all
cases the magnitude of the fractional circular polarization is much lower when
compared to the fractional linear polarization.

The photon ring (and emission from its direct vicinity) is another feature
that differs per magnetic field case. Overall, the photon ring is expected to
be relatively less linearly polarized than other regions due to a combinations
of plasma (i.e., absorption and Faraday rotation) and gravitational effects,
which are further elaborated in Jiménez-Rosales et al. (2021). Although, since
photon rings are not the main topic of this paper, we discuss the linear and
circular polarization of the photon rings for different magnetic field
geometries in Appendix B.

The right panels of Figure 1 display the model background emission for
rotating black holes for an edge-on view with $a_{*}=\pm 0.5,\pm 0.9375$.
There, we find that both the flux density and linear polarization change
significantly when the black hole spin becomes negative. The negative spin
($a_{*}=-0.5,-0.9375$) cases have a higher $\widetilde{\text{LP}}$ (and LP)
than the positive spin ($a_{*}=0.5,0.9375$) cases for all magnetic field
configurations. This is explained by the increased contribution of
relativistic beaming (to the flux) and the higher electron densities,
especially in the inner regions, for the positive spin cases. As we set the
electron density to a floor value for radii smaller than the ISCO, the
emission from this region will be small. Additionally, the emission from these
inner regions will experience more Doppler boosting as the (Keplerian) orbital
velocity is greater close to the BH. This explains why the Stokes
$\mathcal{I}$ maps of these ($a_{*}=0.5,0.9375$) cases are dominated by a
relatively small emission region. Why the negative spin ($a_{*}=-0.5,-0.9375$)
cases have a bigger emission structure is explained with the same arguments,
namely the relatively lower densities and lower orbital velocities at these
radii result in an emission structure that is more diffuse and therefore not
dominated by a singular feature. For the fixed magnetic field configuration
changing the black hole spin from positive to negative produces significantly
different $\widetilde{\text{LP}}$ map. Importantly, two observables remain
weakly affected by the BH spin value and sign: the linear polarization
direction (the EVPA maps) and the $\widetilde{\text{CP}}$ maps.

Figure 2 shows images of the zero-spin ($a_{*}=0$) models as a function of the
viewing angle. Starting with $i=0^{\circ}$, the total intensity maps show
background disk emission where the inner edge corresponds to the ISCO and the
photon ring is located around the black hole shadow. As the viewing angle
increases, a characteristic brightness depression appears in the upper left of
the flux density maps for the vertical case. This ”hollow” region is present
for viewing angles that are considered, broadly speaking, to be face-on
($0^{\circ}<i\lesssim 45^{\circ}$) and is due to the Doppler effect which
facilities alignment of the wavevector ${\bf k}$ with the magnetic field
vector ${\bf B}$ and effectively results in the suppression of the synchrotron
emissivity (see Eq. 12 in Narayan et al. 2021 or notice that the synchrotron
emissivity is $\propto{\bf k}\times{\bf B}$ as measured in the comoving fluid
frame). This effect will be critical when describing the variability of the
models (presented in Sect. 3.3).

In Figure 2 the fractional linear polarization at $i=0^{\circ}$ varies per
magnetic field configurations. The EVPAs form distinct patterns: for radial
fields the polarization ticks are purely azimuthal, for toroidal fields they
are purely radial and for vertical field the ticks are nearly azimuthal. This
is consistent with previous predictions (EHTC et al., 2021c). Such EVPA
patterns are determined by the relative orientation of the electromagnetic
wavevector ${\bf k}$ and magnetic field vector ${\bf B}$ and the effect of
relativistic beaming as explained by and in agreement with Narayan et al.
(2021). For the face-on viewing angle(s) the circular polarization of the
radial field configuration is similar to the vertical one, with positively
polarized disk emission and photon ring emission with opposite polarization.
The radial and vertical $\widetilde{\text{CP}}$ maps are similar because both
of these fields have a poloidal component which is visible at this viewing
angle. The toroidal field model, on the other hand, has
$\widetilde{\text{CP}}$ that is entirely negative. For small viewing angles
(up to about $i\leq 30^{\circ}$) the polarimetric properties of the background
emission are similar to the properties of models viewed face-on. For
$i>30^{\circ}$, the linear and circular polarization maps become more complex.
Note that when $\widetilde{\text{CP}}$ is produced intrinsically, there is a
degeneracy between inverting the viewing angle (from $i=30^{\circ}$ to
$i=150^{\circ}$, for example) and reversal of the polarity of the field. We
extensively comment on this in App. B.

We conclude that while the background emission EVPA and
$\widetilde{\text{CP}}$ depend more on the observer’s viewing angle rather
than the black hole spin, the background emission $\widetilde{\text{LP}}$ is
strongly dependent on both the viewing angle and the black hole spin. This
gives us a hint as to what polarimetric properties of the bright spot emission
may be more geometry (viewing angle and magnetic field) dependent and less
dependent on the model details.

###  3.3 Variable emission: Stokes ${\mathcal{I}}$

Figure 3:  Left panels: An edge-on views ($i=90^{\circ}$) of the non-spinning
($a_{*}=0$) full (disk + spot) model in radial (_top_), toroidal (_middle_),
and vertical (_bottom_) magnetic configurations for a field of view of
$40\,r_{g}\times 40\,r_{g}$. The rows, within each panel, display Stokes
$\mathcal{I}$, linear polarization $\widetilde{\text{LP}}$, and circular
polarization $\widetilde{\text{CP}}$ at times
$t=25,50,75,100\,r_{\mathrm{g}}/c$. The EVPA orientation is represented with
white ticks on top of the Stokes $\mathcal{I}$ panels. Their length is
determined by the $\widetilde{\text{LP}}$ strength. Right panels: Comparison
of the evolution of the flux density (top), the net linear polarization
(middle) and the net circular polarization (bottom) over time for a radial
(solid), toroidal (dashed), and vertical (dotted) magnetic field configuration
shown in the left panels. Here, we additionally plot light curves for black
holes with $a_{*}\neq 0$ (different black hole spins are color coded).

Figure 4: The same description as in Fig. 3 except for the viewing angle which
is $i=20^{\circ}$ and we show maps corresponding to times
$t=25,75,125,175\,r_{\mathrm{g}}/c$.

Before we discuss the polarimetric findings for an orbiting spot, we will
outline a number of characteristic features that appear in Stokes
$\mathcal{I}$. We discuss in more detail two cases with an edge-on viewing
angle and a nearly face-on viewing angle.

In Fig. 3, in left panels, we show model images with $a_{*}=0$ and all three
magnetic field configurations for an edge-on viewing angle ($i=90^{\circ}$).
These images are naturally paired with the light curves shown in the right
panels in Fig. 3. Our full (background plus spot) model describes a periodic
system, hence the light curves can be arbitrarily shifted. Therefore, we align
the light curves in such a way that the spot’s position is similar between
different magnetic field cases. The maximum flux density, in the light curve
shown in Fig. 3, always corresponds to the formation of an ”Einstein
ring”-like image at edge-on viewing angles. Then, as the Einstein ring
disappears, we perceive a decrease in flux density. This decrease is slight
for the radial case, more substantial for the vertical case, and significant
for the toroidal case. After this decrease, the flux density will increase
again as the spot moves in view and is relativistically beamed (or Doppler
boosted) towards the observer. The recovery of the flux density is also highly
dependent on the magnetic field geometry. The second peak of the radial case
almost reaches the level of the first maximum. For the vertical case, we see
that the second maxima recovers to about half of the first maximum amplitude.
The toroidal case only increases to a third of the first maximum. These
‘recoveries’ are all quantified with respect to the corresponding curves’
minimum.

Figure 4 shows the same models as in Fig. 3 but for the nearly face-on viewing
angle ($i=20^{\circ}$). Here the observer has an unobstructed view of the spot
during the entire orbit. As the emission of the spot is not heavily distorted
by strong gravitational lensing, we recover a simpler, single-peaked,
structure in the light curve. The flux density maximum occurs when the spot’s
emission is beamed towards the observer (at $t\approx 100\,r_{g}/c$ or
$\zeta\approx 45^{\circ}$). However, contrary to the edge-on case, the
vertical ($i=20^{\circ}$) case has a minimum at this point in the spot’s
orbit, where the wave vector ($\bm{k}$) and the magnetic field vector
($\bm{B}$) are parallel to each other and the synchrotron emissivity emission
becomes negligible.

How sensitive is the variable component of the total intensity to the black
hole spin? For the edge-on cases (in Fig. 3), we find that the contribution of
the spot is largest for the $a_{*}=-0.9375$ case, with (up to) $50\%$ being
determined by the variable component, and smallest for the $a_{*}=0.9375$
case, with only (up to) $25\%$ being determined by the variable component. As
the orbital velocity increases as $a_{*}$ decreases, the contribution of the
spot increases as a result of relativistic beaming. Additionally, this
difference in contribution to the total flux density is also partly explained
by the $r<r_{\textrm{ISCO}}$ cut we apply to the disk background. For the
nearly face-on cases (in Fig. 4), the spot’s contribution to the total flux
density is less dependent on spin and seems to be roughly $10\%-20\%$. As the
observer inclination no longer lines up with spot’s orbital plane,
relativistic beaming will be less strong than for the edge-on case. It also no
longer has the first (greater) flux density peak which corresponds to the
occurrence of the Einstein ring.

###  3.4 Variable emission: polarization

####  3.4.1 Polarimetric signatures of bright spots: $\mathcal{Q}-\mathcal{U}$
loops

Figure 5: Normalized $\mathcal{Q}$-$\mathcal{U}$ diagrams for the three basic
magnetic field configurations and a black hole spin of $a_{*}=0$. Both the
spot-only (dotted lines) and full models’ (solid lines)
$\mathcal{Q}-\mathcal{U}$ loops are shown for four viewing angles
$i=20^{\circ},40^{\circ},60^{\circ},80^{\circ}$.

The evolution of EVPA is best described using $\mathcal{Q}-\mathcal{U}$
diagrams. Looping structure in ${\mathcal{U}}$ versus ${\mathcal{Q}}$ diagram
(or put simply as ”$\mathcal{Q}-\mathcal{U}$ diagram”) is a hallmark signature
of an orbiting hot spot (Marrone, 2006). However, one could argue that the
(disk) background emission is equally important in shaping the loop structure
and position in the $\mathcal{Q}-\mathcal{U}$ diagram. This is illustrated in
Fig. 5, where we show $\mathcal{Q}-\mathcal{U}$ diagrams for both the spot-
only and the full (disk+spot) configurations for _radial_ (left panel),
_toroidal_ (middle panel), and _vertical_ (right panel) magnetic fields.
Without commenting yet on the loops’ finer substructure, we find that the
spot-only loops are greater in size and broadly describe a circle for the
radial and toroidal configurations which are largely independent of viewing
angle. The vertical configuration is more dependent on the system’s
inclination angle - only the $i=20^{\circ}$ case shows looping structure. The
smaller size of the full configuration’s $\mathcal{Q}-\mathcal{U}$ loops is
largely due to the compensation with the flux density $F_{\nu}$. As was
commented in Sect. 3.2, the spot-only configuration has much smaller flux
densities than the full model. Surprisingly, there are clear structural
differences between the spot-only and full model descriptions for the
${\mathcal{Q}}/{\mathcal{I}}$,${\mathcal{U}}/{\mathcal{I}}$ loops, whereas
this is not the case for the $\mathcal{Q}-\mathcal{U}$ loops (unnormalized
diagrams where ${\mathcal{Q}}$ and ${\mathcal{U}}$ are shown in units of
Jansky). For the latter diagrams, which are currently not shown, we find an
identical $\mathcal{Q}-\mathcal{U}$ structure for both the spot-only and full
configurations but they are displaced from each other by an offset as a result
of the disk background. Therefore, the clear differences in the
${\mathcal{Q}}/{\mathcal{I}}$,${\mathcal{U}}/{\mathcal{I}}$ diagrams (between
the spot-only and full models) originate from the disk component that
dominates the flux density (or Stokes $\mathcal{I}$), especially when the spot
is receding from the observer. The (near) circular structure in the
${\mathcal{Q}}/{\mathcal{I}}$,${\mathcal{U}}/{\mathcal{I}}$ for the spot-only
cases is an indication that the variation in ${\mathcal{I}}$, ${\mathcal{Q}}$,
and ${\mathcal{U}}$ are largely (but not perfectly) correlated. This circular
symmetry for the radial and toroidal cases is, mainly, broken by relativistic
beaming and, partially, by Faraday rotation – see Appendix A.

Figure 6: Schematic diagrams of normalized $\mathcal{Q}-\mathcal{U}$ loops of
the full models (for $i=20^{\circ}$, $a_{*}=0$) with the radial, toroidal, and
vertical magnetic field configurations in the _left_ , _middle_ , and _right_
panel, respectively. The loop color encodes the model LP. The inner loop
surface $S_{1}$ (in _pink_) and rest of surface (excluding $S_{1}$) enclosed
by the entire $\mathcal{Q}-\mathcal{U}$ loop structure is denoted by $S_{2}$
(in _yellow_). The total surface is $S=S_{1}+S_{2}$, surface fraction is
$\bar{S}=S/S_{1}$ and time spent in inner loop with respect to the total
period is $\bar{R}=P/(T_{4}^{c}-T_{2}^{c})$ (similar to Wielgus et al.,
2022b). The ‘+’ symbol marks the normalized $\mathcal{Q}-\mathcal{U}$ value of
the static background emission. All shown loops are traversed in the
counterclockwise direction.

The properties of the $\mathcal{Q}-\mathcal{U}$ loops depend on a number of
model parameters. Although, before we describe how the
$\mathcal{Q}-\mathcal{U}$ loops depend on them, we will introduce metrics to
quantitatively describe the loops. Figure 6 displays a schematic overview of
the typical $\mathcal{Q}-\mathcal{U}$ loop structure that appears in our
models. Similar to Wielgus et al. (2022b), we describe the loops (i.e., a
pretzel-like shape) using two dimensionless metrics. Another, additional,
metric we consider (which Wielgus et al. (2022b) does not) is the total area
of the loop structure in the $\mathcal{Q}-\mathcal{U}$ space, $S=S_{1}+S_{2}$.
The first of the dimensionless metrics is the ratio of total loop area ($S$)
to small loop area ($S_{1}$), which is denoted as $\bar{S}=S/S_{1}$. The last
metric is the inner loop timing ratio
$\bar{R}=P/(T^{c}_{\text{exit}}-T^{c}_{\text{enter}})$. Here, the
$T^{c}_{\text{enter}}$ is the time at which the $\mathcal{Q}-\mathcal{U}$
signature enters the inner loop and $T^{c}_{\text{exit}}$ is the time at which
it exits the inner loop. Then, $\bar{R}$ gives the period $P$ divided by the
time spent within the inner loop.

By inspecting Fig. 6 one finds that for radial and toroidal magnetic fields
the $\mathcal{Q}-\mathcal{U}$ signature will be following the inner loop when
the spot is receding from the observer ($\zeta=180^{\circ}$ to
$\zeta=360^{\circ}=0^{\circ}$). The $\mathcal{Q}-\mathcal{U}$ signature will
follow the outer loop as the spot approaches the observer ($\zeta=0^{\circ}$
to $\zeta=180^{\circ}$). For the vertical magnetic field case, this behaviour
is quite the opposite. Roughly speaking, the vertical magnetic field’s
$\mathcal{Q}-\mathcal{U}$ signature will traverse the left half of the
$\mathcal{Q}-\mathcal{U}$ loop, roughly the outer loop, for $\zeta\approx
160^{\circ}$ to $320^{\circ}$. There, generally speaking, the spot is receding
with respect to the observer. Therefore, the right half of the
$\mathcal{Q}-\mathcal{U}$ diagram, including the inner loop, corresponds to
the approaching spot. The $\mathcal{Q}-\mathcal{U}$ signature follows the
inner loop when the contribution of the spot (almost completely) disappears in
the region where synchrotron emission is suppressed (as at $T_{3}$ we only can
see a lensed image of the spot). The vertical cases that do not possess an
inner $\mathcal{Q}-\mathcal{U}$ loop (for $i\gtrsim 25^{\circ}$) are still
affected by the emission suppression region. Then, the indentation (where the
inner loop would have been) corresponds to the part of the spot’s orbit where
the emission is suppressed (for $i=25^{\circ}$). When we look at Fig. 6, we
find that for the vertical cases only a relatively small part of the spot’s
orbital period is spent inside the inner $\mathcal{Q}-\mathcal{U}$ loop when
compared to toroidal/radial cases.

Next, we investigate the $\mathcal{Q}-\mathcal{U}$ loop characteristics,
(using the metrics defined above) as a function of a few model parameters such
as magnetic field geometry, observers’ viewing angle, radius of the spot,
black hole spin, and plasma parameters (we consider only small deviations from
our fiducial values of density, temperature and magnetic field strength). We
summarize our main findings in the paragraphs below.

Figure 7: Normalized $\mathcal{Q}-\mathcal{U}$ diagrams of the full model
colored according to their LP fraction as a function of increasing inclination
$i$ (panels from top to bottom). Each column corresponds to a different
magnetic field configuration. All models have $a_{*}=0$. While for vertical
magnetic field the $\mathcal{Q}-\mathcal{U}$ loops disappear for $i\gtrsim
25^{\circ}$, for other field geometries they are present for all viewing
angles except exact edge-on cases (see also Figure 5). The bottom row shows
the behavior of $\bar{S}$ (in blue) and $\bar{R}$ (in orange) as a function of
inclination $i$. The observables $S$, $\bar{S}$, and $\bar{R}$ are explained
in Fig. 6.

##### Viewing angle:

Figure 7 shows that the dominant decider for $\mathcal{Q}-\mathcal{U}$ loops
shape is the inclination angle of the system. For a face-on viewing angle
($i=0^{\circ}$), we will only recover circle-shaped loops, regardless of the
field configuration. Notice that for a single orbital period this circle is
traversed two times, which indicates that the inner and outer loops are still
present but have the same size (corresponding to $\bar{S}=1$ and $\bar{R}=2$).
Notice also that all our loops are traversed in the counter-clockwise
direction because the spot is orbiting counter-clockwise on the sky (we only
show results for viewing angles $i>90^{\circ}$). As the inclination $i$
increases, the properties of the $\mathcal{Q}-\mathcal{U}$ loops strongly
depend on the underlying magnetic field geometry. For the radial and toroidal
cases, for increasing $i$, the inner loop starts to shrink while the outer
loop remains the same size. For the vertical cases, we find that the inner
loop disappears already for $i\gtrsim 25^{\circ}$ (with the last inner loop
structure being present at $i\approx 24^{\circ}$ to be more exact). For radial
and toroidal cases the loop persist for all viewing angles except the exact
edge-on case. For all three magnetic field topologies, $\bar{S}$ increases
with increasing viewing angle, but for vertical field the dependency is
stronger. $\bar{R}$ can be used to discriminate between radial/toroidal and
vertical cases: $\bar{R}$ slowly decreases with $i$ for radial/toroidal
fields, while $\bar{R}$ increases with $i$ for vertical field. The sharp
increase of $\bar{R}$ as a function of $i$ is related to the duration of the
${\bf k}\times{\bf B}$ suppression as a result of relativistic beaming which
is a strong function of viewing angle.

##### Spot orbital radius:

Figure 8 displays the $\mathcal{Q}-\mathcal{U}$ loop structure as a function
of the spot’s orbiting radius $r_{\text{spot}}$. The viewing angle and spin
are fixed to $20^{\circ}$ and $a_{*}=0$, respectively. For the vertical case
with $r_{\text{spot}}\leqslant 11$ $r_{\rm g}$, we still find an inner loop,
but the inner loop decreases in size for increasing $r_{\text{spot}}$ until it
is no longer present which becomes clear for $r_{\text{spot}}=15$ $r_{\rm g}$.
For the radial and toroidal cases, we recover the double
$\mathcal{Q}-\mathcal{U}$ loop structure for all $r_{\text{spot}}$ values.
Their total surface $S$ increases till $r_{\text{spot}}=9$ $r_{\rm g}$ after
which it decreases. $\bar{R}$ seems insensitive to different $r_{\text{spot}}$
values, while $\bar{S}$ is almost constant except for $r_{\text{spot}}=6$
$r_{\rm g}$. This slight increase in $\bar{S}$ is likely related to cut-off at
ISCO radius. The vertical magnetic field cases display a largely constant
$\bar{R}$ for increasing $r_{\text{spot}}$, but $\bar{S}$ does increase with
$r_{\text{spot}}$. This behavior has the same explanation as was stated in the
previous paragraph, i.e. the orbital speed increases for decreasing
$r_{\text{spot}}$.

Figure 8: Normalized $\mathcal{Q}-\mathcal{U}$ diagrams colored according to
their LP fraction. Each row corresponds to a different orbital radius
$r_{spot}$. The inclination and black hole spin are set to $i=20^{\circ}$ and
$a_{*}=0$, respectively. The bottom row shows the behavior of $\bar{S}$ (in
blue) and $\bar{R}$ (in orange) as a function of orbital radius
$r_{\text{spot}}$. The observables $S$, $\bar{S}$, and $\bar{R}$ are explained
in Fig. 6.  Figure 9: Normalized $\mathcal{Q}-\mathcal{U}$ diagrams colored
according to their LP fraction. The inclination is set to $i=20^{\circ}$. Each
row corresponds to a different black hole spin parameter $a_{*}$. The bottom
row shows the behavior of $\bar{S}$ (in blue) and $\bar{R}$ (in orange) as a
function of $a_{*}$. The observables $S$, $\bar{S}$, and $\bar{R}$ are
explained in Fig. 6.  Figure 10: Normalized $\mathcal{Q}-\mathcal{U}$ diagrams
colored according to their LP fraction for $i=20^{\circ}$ and $a_{*}=0$ cases.
Each row corresponds to different spot sizes ($R_{s}$). The default in the
rest of this work is $R_{s}=1.5$ $r_{\rm g}$. The observables $S$, $\bar{S}$,
and $\bar{R}$ are explained in Fig. 6.  Figure 11: Normalized
$\mathcal{Q}-\mathcal{U}$ diagrams colored according to their LP fraction for
$i=20^{\circ}$ and $a_{*}=0$ cases. Each row corresponds to different orbital
velocities denoted as a fraction of the standard velocity in this work which
is Keplerian ($v_{kep}$). The observables $S$, $\bar{S}$, and $\bar{R}$ are
explained in Fig. 6.  Figure 12: Normalized $\mathcal{Q}-\mathcal{U}$ diagrams
colored according to their LP fraction. The _top row_ corresponds to the base
model with $r_{\text{spot}}=11$ $r_{\rm g}$, $a_{*}=0$, and $i=20^{\circ}$ and
the _bottow row_ shows some exploratory models with doubling of $\Theta_{e}$
and $B$ and halved $n_{e}$. The observables $S$, $\bar{S}$, and $\bar{R}$ are
explained in Fig. 6.

##### Black hole spin:

Figure 9 displays $\mathcal{Q}-\mathcal{U}$ loops and their properties as a
function of black hole spin. We find that the size of the
$\mathcal{Q}-\mathcal{U}$ loop is the greatest for the lowest spin
($a_{*}=-0.9375$). This behavior is a direct result of our model assumptions.
We assume a low floor electron density $n_{e}$ at radii below the ISCO. The
contribution of the disk background will therefore vary between spin-cases,
while the contribution of spot will remain unaffected. Hence, the
$a_{*}=0.9375$ case has the largest disk background contribution, whereas the
$a_{*}=-0.9375$ case has the lowest disk background contribution. This is an
explanation for the increase in $\mathcal{Q}-\mathcal{U}$ loop size ($S$) with
decreasing spin parameter. The toroidal cases’ $\mathcal{Q}-\mathcal{U}$ loops
are similar between the five spin cases. To a lesser extent this can also be
stated about the vertical cases. However, for the most positive
($a_{*}=0.9375$) spin-case, we observe that the upper part of the loop is
relatively larger with respect to the lower part. For the most negative
($a_{*}=-0.9375$) spin, this is the other way around but less pronounced. The
other spin-cases ($a_{*}=0,\pm 0.5$) display a clearer symmetry between upper
and lower half of the loop.

We find that the total surface $S$ increases when the spin decreases for all
cases. For the radial and toroidal cases, we recover an almost constant
$\bar{S}$ and $\bar{R}$ similar to what we saw in the previous paragraph (for
$r_{\text{spot}}$). For the vertical case, we find that $\bar{R}$ is only a
weak function of spin, but $\bar{S}$ is a exceptionally strong function of
spin. Given the immutable nature of the $\mathcal{Q}-\mathcal{U}$ loop
geometry and the strong dependence of $\bar{S}$ (and $S$) on spin, it would be
possible to assess the black hole spin $a_{*}$ of the system for the vertical
case (if the inclination is well-constrained). For the radial and toroidal
cases, this is less promising. However, the total surface $S$ displays a
stronger dependence on spin than it has shown for the other investigated
quantities. We do recover near identical $S$ between the radial and toroidal
fields (as we have seen previously for other $\mathcal{Q}-\mathcal{U}$
figures).

Lastly, we discuss the radial cases in the left column of Fig. 9 in more
detail. Each $\mathcal{Q}-\mathcal{U}$ loop has a point where one crosses from
the inner to the outer loop. This point will henceforth be referred to as the
crossing-point. For the radial cases, we find that the crossing-point lies in
the upper half of the $\mathcal{Q}-\mathcal{U}$ loop for the positive spin-
cases. However, as the spin decreases, the crossing-point gradually moves
towards the lower half of the $\mathcal{Q}-\mathcal{U}$ loop. This behavior is
still present if we remove the background, which indicates that it is likely
related to the underlying spacetime.

##### Spot size:

Figure 10 displays the dependency of $\mathcal{Q}-\mathcal{U}$ loops on the
spot size parameter $R_{s}$. Variations in $R_{s}$ bring about significant
changes in, especially, $\mathcal{Q}-\mathcal{U}$ loop size but not so much
the structure, except for the $R_{s}=3.0$ $r_{\rm g}$. Note that the
($\mathcal{Q}/\mathcal{I},\mathcal{U}/\mathcal{I}$) domain shown in this
figure is significantly greater than for the other $\mathcal{Q}-\mathcal{U}$
figures. The spot’s contribution to the overall flux density increases with
increasing spot size. This is reflected in the total loop size $S$. $\bar{R}$
remains independent of the spot size. $\bar{S}$, on the other hand, tends to
decrease for increasing $R_{s}$ in the radial and toroidal magnetic field
configurations. As was established in the previous paragraphs (and shown
schematically in Fig. 6), the inner $\mathcal{Q}-\mathcal{U}$ loop is followed
when the spot is receding from the observer. For the vertical, the inner loop
has a different origin, namely a region where $\bm{k}\times\bm{B}\sim 0$. This
explains why the $R_{s}=3.0$ $r_{\rm g}$ case breaks the trend of steadily
decreasing $\bar{S}$ for increasing $R_{s}$ as the spot has become
sufficiently large to no longer be fully suppressed in the
$\bm{k}\times\bm{B}\sim 0$ region. Note that the relative growth of the inner
loop size indicates that the spot’s emission starts eclipsing the contribution
of the disk. If this were to continue, then we find that we end up with
$\mathcal{Q}-\mathcal{U}$ loop behavior that is similar to what we have seen
in Fig. 5 for the spot-only case(s).

##### Orbital velocity:

Figure 11 displays the $\mathcal{Q}-\mathcal{U}$ loop behavior for deviations
from the base model’s Keplerian orbital velocity (shown in the middle row). We
modify the orbital velocity of the entire system (disk + spot) with a factor
that ranges from 0.6 (sub-Keplerian) to 1.4 (super-Keplerian). For the radial
and toroidal cases, where the inner and outer $\mathcal{Q}-\mathcal{U}$ loop
correspond to the receding and approaching spot, we find that the inner and
outer loop become more similar in size (lower $\bar{S}$ but higher $\bar{R}$)
for decreasing orbital speed. This behavior is explained by the increased
(decreased) contribution of relativistic beaming for increasing (decreasing)
orbital velocity, which makes the inner and outer loop diverge more (less)
from eachother. For the vertical cases, we find an opposite relation (in the
metrics) as the inner loop has a different physical cause, i.e. suppression of
the emission in the $\bm{k}\times\bm{B}\sim 0$ region. This is also explained
by the varying strength of relativistic beaming for the different orbit
velocities, but it has the opposite effect with respect to the other magnetic
field configurations, i.e. decreasing (increasing) inner loop size for
decreasing (increasing) orbital speed. This indicates that $\bm{k}$, and
subsequently $\bm{k}\times\bm{B}$, are, unsurprisingly, quite sensitive to the
orbital velocity. The increase in $\bar{R}$ indicates that the
$\bm{k}\times\bm{B}\sim 0$ region grows in size for increasing orbital
velocity.

##### Plasma parameters:

Figure 12 displays an exploratory model where we half our base electron
density $n_{e}$ and double both the electron temperature $\Theta_{e}$ and
magnetic field strength $B$. Unsurprisingly, the (Stokes) quantities have
changed with respect to the base model (shown in the top row of the figure).
The most notable difference are in LP and total $\mathcal{Q}-\mathcal{U}$
surface $S$ of the system, which have both decreased substantially. The
overall shape of the $\mathcal{Q}-\mathcal{U}$ structure remains relatively
unaffected. Variations in $\bar{S}$ and $\bar{R}$ are consistent with what was
found in earlier figures. Stokes $\mathcal{I}$ has almost quadrupled but
Stokes $\mathcal{Q}$ and $\mathcal{U}$ have only doubled/tripled which
explains both the decrease in LP and $\mathcal{Q}-\mathcal{U}$ loop metrics.
Additionally, the intensity-averaged Faraday depth $\tau_{V}$ is three times
smaller, but this is largely inconsequential as both cases are still very
Faraday thin at $\tau_{V}\sim 10^{-2}\,\textrm{--}\,10^{-3}$. In the
exploratory model, we have entered a relatively less polarized regime (at 230
GHz). This example is illustrative of the fact that (polarized) synchrotron
emission is highly sensitive to the underlying plasma quantities and is
affected differently per magnetic field configuration. This illustrates that
fitting these models to observations would therefore be a non-trivial multi-
parameter endeavour.

####  3.4.2 Linear and circular polarization of bright spots

Bright spots orbiting in the disk are periodically highlighting (and adding
to) emission regions with different linear and circular polarization. Can we
use LP and CP light curves to infer the magnetic field topology or remove
degeneracy between e.g. radial/toroidal cases (since they look similar in the
EVPA evolution)? The interpretation of LP and CP evolution is challenging
because the polarization of light curves is modified by the beam
depolarization. Also LP and CP magnitudes depend strongly on the underlying
plasma parameters. However, the models we discuss in this work are all
Faraday-thin which implies that almost all CP emission is intrinsic and
Faraday conversion/rotation plays a negligible role in shaping
$\widetilde{\text{CP}}$ maps. This helps to understand the behaviour of the CP
light curves.

In Figure 3 Stokes $\mathcal{I}$ exhibits variations that are expected for
edge-on viewing angle. During the spot’s passage the double peaked light curve
is produced by the Einstein ring and followed by emission from Doppler beaming
of the spot along its orbit, as was already explained. The LP is mostly
constant in all cases, except during the Einstein ring phase where LP
decreases and the Doppler beamed phase where LP increases – clearly seen in
radial case, but is somewhat spin-dependent in toroidal case. The decrease in
LP is caused by cancellation effect (beam depolarization) because places with
opposite polarization direction cancel out. The LP increase is caused by
amplification of the spot emission which is small in size and therefore
amplifies only one polarization direction. Due to symmetry, for the edge-on
inclination with a vertical magnetic field geometry, CP becomes zero
independently of the black hole spin. For toroidal case CP decreases during
Einstein ring moment, due to a cancellation effect, and then increases during
the Doppler beaming time. For radial case we see that CP flips sign during the
spot’s passage. When the Einstein ring forms, CP is most negative and then
gains positive sign during Doppler beaming which is a direct result of the
underlying field topology.

In Figure 4 Stokes $\mathcal{I}$ exhibits small variations that are expected
for low viewing angles. For the radial/toroidal cases we find a slightly
enhanced flux when the spot is approaching the observer, but for vertical case
the total intensity decreases due to suppression of synchrotron emissivity at
$\zeta\approx 45^{\circ}$ (or inner $\mathcal{Q}-\mathcal{U}$ loop passage).
For radial fields the increase in Stokes $\mathcal{I}$, when spot is
approaching the observer, is accompanied with a single peak in LP. For
toroidal fields, increase in Stokes $\mathcal{I}$ is associated with two peaks
in LP. Finally, for vertical fields, the decrease in Stokes I in the middle of
spot’s approach, is associated with a single peak in LP. When the spot is
receding from the observer, LP increases to the maximum for the vertical case.
For vertical field case, the background flow polarity $\widetilde{\text{CP}}$
is uniform across the image and the bright spot added to the model simply
amplifies the CP, except during $\zeta\approx 45^{\circ}$ time moment (the
inner $\mathcal{Q}-\mathcal{U}$ loop time moment) when the CP light curves
tend to show a small decrease. For toroidal fields the left and right parts of
the $\widetilde{\text{CP}}$ map have opposite polarities which results in a
decrease of the CP amplitude when the spot is Doppler boosted. Finally, the
radial field case is more complex because the top part of the
$\widetilde{\text{CP}}$ map has opposite polarity compared to the bottom one.
Here one observes the decrease in CP light curves just before the Doppler
boost moment marked with a hump in Stokes $\mathcal{I}$.

Let us conclude with a number of statements about CP and how one could utilize
it to differentiate between the underlying magnetic field topologies during a
$\mathcal{Q}-\mathcal{U}$ loop event (with utilization of Figs. 4 and 6).
Notice that our models assume that the spot contributes significantly to both
LP and CP which, in reality, may not be always the case. Starting with the
vertical topology, we immediately find that the $\mathcal{Q}-\mathcal{U}$ loop
structure has a significant offset in $\mathcal{Q}-\mathcal{U}$ space with
respect to the other magnetic field cases. The $\mathcal{Q}-\mathcal{U}$ loop
signature alone would therefore be sufficient to identify the vertical
magnetic field (at this inclination and orbital velocity, see Fig. 11).
Additionally, during the inner $\mathcal{Q}-\mathcal{U}$ loop passage, a
slight drop in CP is perceivable. Although the $\mathcal{Q}-\mathcal{U}$ loops
of the radial and toroidal cases span a similar $\mathcal{Q}-\mathcal{U}$
range, they have different CP behavior (and $\mathcal{Q}-\mathcal{U}$ loop
structure). Due to the underlying $\widetilde{\text{CP}}$ morphology we only
see an increase in CP when the spot is approaching the observer (and the outer
$\mathcal{Q}-\mathcal{U}$ loop is traversed) for the toroidal case. For
radial, we find the most complex $\widetilde{\text{CP}}$ (image) morphology
with an overall positive $\widetilde{\text{CP}}$ structure with a section at
$\zeta=0^{\circ}\pm 25^{\circ}$ where $\widetilde{\text{CP}}$ is negative.
This implies that when the inner $\mathcal{Q}-\mathcal{U}$ loop is traversed
(as the spot recedes from the observer) the CP keeps decreasing over time.
Note that the CP minimum is not reached when inner $\mathcal{Q}-\mathcal{U}$
loop is still begin traversed, but rather when it has just entered the outer
$\mathcal{Q}-\mathcal{U}$ loop after which CP quickly increases.

##  4 Discussion

In this work we have studied polarimetric signatures of bright spots produced
in accretion flow close to the black hole event horizon. Semi-analytic
modelling enables the exploration of a parameter space without needing to run
computationally expensive plasma simulations (GRMHD or GRPIC) where plasma
simulation’s resolution may impact the results. Though not as complex as a
full plasma simulation, our simplified approach allows for a relatively
straight-forward decomposition of the various physical effects that are
important in the system.

Even in the semi-analytic models the parameter space is very large so we can
still only present a limited number of models. All models studied here have
small optical depths $\tau_{\nu}\ll 1$ and small Faraday depths $\tau_{F}\ll
1$ (see Appendix C for detailed definitions of optical and Faraday depths and
their values for all models). We also fixed the magnetic field polarities and
the orbital direction of the spot. In Appendix C we discuss the impact of the
magnetic fields polarity, Faraday effects and spot rotation direction onto all
polarimetric observables of a bright spot.

Several works have presented polarized emission from hot spots around Kerr
black hole. Broderick & Loeb (2005) is one of the pioneering works on the
semi-analytical modeling of spots polarization. They use an ‘optically thick
sphere’ to investigate the effect of a curved spacetime on polarized emission
in a single vertically-aligned magnetic field configuration. Especially the
gravitational magnifying effect and the image centroid motion are topics they
comment on. In Broderick & Loeb (2006), they include a disk background and
change the magnetic field to toroidal. Additionally, they improve their
(optically thin) hot spot description by including a localized population of
non-thermal electrons. Much of our work is inspired by Broderick & Loeb (2005,
2006), but there are numerous differences - mostly with respect to our focus
which lies predominantly on the polarimetric signatures. While the
aforementioned works mainly display net polarization fractions, we focus on
the entire structure of the Stokes parameters (especially in
$\mathcal{Q}-\mathcal{U}$ diagrams). Overall, when directly comparing the net
polarization, there is little similarity between our model and the
aforementioned ones by Broderick & Loeb (2005, 2006).

A more recent study by Gelles et al. (2021) outlines a toy model for
synchrotron point source emission by analytically computing Kerr spacetime
null-geodesics (Gralla & Lupsasca, 2020) and enforcing conservation of the
Penrose-Walker constant, without any more sophisticated radiative transfer
considerations. This approach is fundamentally different from the numerical
radiative transfer approach used in ipole (while it is also a Penrose-Walker
constant conserving scheme). The authors comment extensively on the
$\mathcal{Q}-\mathcal{U}$ loop structure they find. However, they only model a
hot spot without a disk background, which makes their result better comparable
with NIR emission observed during flares in Sgr A* than with observations at
millimeter wavelengths. Additionally, they effectively adopt a ‘fast light’
approach where the spot is stationary when the geodesics are evaluated. In our
work, the spot continues its orbit while the radiative transfer equations are
being integrated along the null geodesics. Lastly, note that they display
$\mathcal{Q}-\mathcal{U}$ loops and not the normalized
$\mathcal{Q}-\mathcal{U}$ loops we show in this work, so they are not
intuitively comparable. Their methodology does not produce images, which makes
tracking down underlying effects difficult. Even though our magnetic field
descriptions are not identical (and the disk background is missing), we find
very consistent $\mathcal{Q}-\mathcal{U}$ loops structures in the low Faraday
depth regime. The direct comparison of our model with
$\mathcal{Q}-\mathcal{U}$ loops from Gelles et al. (2021) is shown in Wielgus
et al. 2022b.

Another aspect investigated in this work, which was not addressed in detail
previously, is the potential of using the circular polarization (or Stokes
$\mathcal{V}$) to further constrain the underlying magnetic field structure.
If it is difficult to identify the model unambiguously based on the
$\mathcal{Q}-\mathcal{U}$ loop behavior alone, then it is possible to get
additional information from the CP signature as we discussed in Sect. 3.4.2. A
simple measurement of the CP sign can already help break this ambiguity,
although imaging circular polarization structure (as shown in, e.g., Figure 4)
would be even more effective. Note that polarity of the magnetic field is
arbitrarily chosen and can possibly be reversed in reality, but the behavior
of CP, especially while traversing an inner $\mathcal{Q}-\mathcal{U}$ loop,
will remain diverging (even when inverted) for the magnetic field cases
evaluated in this work.

In a parallel work we demonstrated a remarkable consistency between
predictions of our model and the linear polarization light curves of Sgr A*
observed with ALMA at 230 GHz shortly after an X-ray flare (Wielgus et al.,
2022b). Our findings strongly suggest low inclination of the Sgr A* system and
dominance of the vertical magnetic field component. In the future, one can
expand the semi-analytic hot spot model in a number of ways which include, but
are not limited to, elliptical and/or non-equatorial orbits, separate magnetic
field description for the spot, mixed (global) magnetic field descriptions,
cooling of the hot spot over time, and presence of a non-thermal electron
population. These more sophisticated models may provide a better fit to the
existing data, and a deeper understanding of the physical processes.

Our current base model consists of a RIAF disk and a hot spot, but one would
expect an additional contribution of the jet (Blandford et al., 2019). If one
was to consider a disk-jet model (e.g., Vincent et al., 2019) instead of a
pure disk model, the emission from the disk would need to be reduced for the
full model to end up at the same flux density. As this will require
rebalancing the plasma parameters (mainly through the electron temperature and
density), it is possible that the disk will enter a more optically and/or
Faraday thick regime. Then, the background flow can become an internal Faraday
screen, which may severely alter the polarimetric characteristics of the
models (Mościbrodzka et al., 2017; Jiménez-Rosales & Dexter, 2018).
Additionally, it has been shown that strong variations in rotation measure as
a result of the internal Faraday screen are expected from GRMHD simulations
(Ricarte et al., 2020).

The physical origin and lifetime of hot spots are also uncertain.
Alternatively to the plasmoid interpretation, the orbiting hot spots reported
in Sgr A* by Gravity Collaboration et al. (2018) and Wielgus et al. (2022b)
may correspond to flux tubes powered by reconnection and filled with hot
electrons confined by the vertical magnetic field. Such features have been
observed in numerical GRMHD simulations of magnetically arrested accretion
disks (Ripperda et al., 2022). This origin would also support the notion that
the hot spot emission should be (at least partially) described by non-thermal
electron population, which is especially needed to explain the NIR
observations (Chael et al., 2017; Davelaar et al., 2018, 2019; Fromm et al.,
2022). The persistence and integrity of an orbiting hot spot also remain
largely unknown. We expect hot spots to be disrupted by effects such as
shearing in a differentially rotating flow (Tiede et al., 2020) or by the
Rayleigh-Taylor instability (Ripperda et al., 2022) on dynamical timescales.
Accounting for these effects would further modify the outcome of our
calculations.

These topics are best investigated in a dedicated GRMHD and/or GRPIC
simulations. A number of studies have been conducted to better understand the
behaviour and evolution of plasmoids, but these typically do not include a
study of the radiation. These simulation can be run within a shearing box
(see, e.g., Murphy et al., 2013; Cerutti et al., 2014; Sironi et al., 2016) or
global GRMHD and GRPIC simulations of magnetized accretion disks around a
SMBH. There, one generally finds more erratic plasmoid chains due to the less
confining nature of the setup (Crinquand et al., 2021; Bransgrove et al.,
2021; Ripperda et al., 2020, 2022).

##  5 Summary

This study has outlined the various distinguishing polarimetric signatures of
a circularly orbiting bright spot on top of a disk background at a frequency
of 230 GHz. For this model we have showcased the utility of
$\mathcal{Q}-\mathcal{U}$ loops and CP for establishing underlying magnetic
field structure. We have shown what signatures are indicative of the
underlying _radial_ , _toroidal_ , and _vertical_ magnetic fields. In the
following paragraphs, we summarise the main findings of this work.

  1. (i)

The shape and size of a $\mathcal{Q}-\mathcal{U}$ loop is indicative for, at
least, three parameters we investigate, namely; inclination $i$, magnetic
field, and BH spin ($a_{*}$). The inner $\mathcal{Q}-\mathcal{U}$ loop
structure is particularly informative. The point where the
$\mathcal{Q}-\mathcal{U}$ path over time moves from the outer to inner loop
was previously referred to as the crossing-point. This point lies on opposite
sides for the _toroidal_ /_vertical_ magnetic field cases and the _radial_
case. This does not change with inclination (including $i>90^{\circ}$), but
the exact position of the crossing point is sensitive to rotation of the
image. Additionally, as becomes clear from Figure 6, the timing (of
entering/exiting of the inner loop) is strikingly different between the
_radial_ /_toroidal_ and _vertical_ cases. While for the former case the
division between inner and outer $\mathcal{Q}-\mathcal{U}$ loop lies in the
approaching versus receding part of the orbit, for the _vertical_ case, it is
connected to the part of the orbit where the direct emission from the spot
disappears (as outlined in Sect. 3.4).

  2. (ii)

Moreover, inclination ($i$) is especially influential on the
$\mathcal{Q}-\mathcal{U}$ loop structure for the _vertical_ magnetic field
case, as demonstrated in Figs. 5 and 7. In this configuration the inner loop
structure disappears for $i\gtrsim 25^{\circ}$ and for even greater
inclination angles one only recovers thin crescents which have little
resemblance to the typical $\mathcal{Q}-\mathcal{U}$ loop structure. Another
influential parameter is the orbital velocity of the system, particularly for
the _vertical_ case (as highlighted in Fig. 11). To a lesser degree, this
behavior is also present for _radial_ and _toroidal_ cases as the inner loop
relative size increases as function of inclination. However, for those cases,
we find that even for near edge-on viewing angles there is still some inner
loop structure present as shown in Fig. 5.

  3. (iii)

The _vertical_ magnetic field case behaves substantially different from the
two others – both in terms of $\mathcal{Q}-\mathcal{U}$ loop structure and LP
strength. Overall, the _vertical_ cases are more ${\mathcal{Q}}$-negative and
variation of certain parameters results in significant shift in Stokes
${\mathcal{Q}}$ (as shown in Figs. 7, 9, and 11 for $i$, $a_{\ast}$, and
$v_{\textrm{kep}}$, respectively). The _radial_ and _toroidal_ cases generally
span an almost identical $\mathcal{Q}-\mathcal{U}$ range and display only
minor variations in Stokes ${\mathcal{Q}}$. Interestingly, Stokes
${\mathcal{U}}$ is (almost) insensitive to any of the parameter variations
with an average value of $\mathcal{U}/\mathcal{I}\approx 0$ for the _radial_
/_toroidal_ cases. Some variation in $\mathcal{U}$ is found for the vertical
case (where it becomes more positive), but this remains minimal when compared
to the variations in $\mathcal{Q}$.

  4. (iv)

When $\mathcal{Q}-\mathcal{U}$ structure is similar or unclear, CP could break
the degeneracy. _Radial_ and _toroidal_ cases for low inclinations ($i\lesssim
30\degr$) would observationally be hard to differentiate. However, they do
differ structurally in CP. The region close to the photon ring has negative CP
(for all cases) while the surrounding emission structure (from radii beyond
ISCO) has a predominantly positive CP for the _radial_ case and negative CP
for the _toroidal_ case for $40\degr\gtrsim i\gtrsim 150\degr$. However, for
edge-on viewing angles ($70\degr\lesssim i\lesssim 110\degr$), CP structure
becomes more similar, but then one can differentiate based on LP. Therefore,
we argue that all Stokes parameters, $\mathcal{I}$, $\mathcal{Q}$,
$\mathcal{U}$, and $\mathcal{V}$, are required to robustly determine dominant
underlying magnetic field structure.

## Acknowledgements

We thank Alejandra Jiménez-Rosales, Hector Olivares, Aristomenis Yfantis, and
Zachary Gelles for the insightful discussions and comments that have improved
the quality of the manuscript. Additionally, we thank the internal EHT
reviewer Michi Baubock for his comments. MM acknowledges support by the Dutch
Research Council (NWO) grant No. OCENW.KLEIN.113. MW thanks Alexandra Elbakyan
for her contributions to the open science initiative.

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##  Appendix A Linear and circular polarization

Figure 13 displays the relation between LP and CP for all spins and
inclinations of $i=20^{\circ},\,40^{\circ},\,60^{\circ},\,80^{\circ}$. Each
magnetic field configuration displays a unique relation between LP and CP. For
increasing inclination, LP rises for all cases. However, for increasing
inclination, we find that CP decreases radial and vertical cases and increases
for the toroidal case. Another diverging feature is the size of path followed
in the LP and CP. The radial case is more variable in both LP and, especially,
CP than the other cases. Additionally, it has a clear crescent-like shape,
which is only deformed for the higher inclination cases (due to boosting). The
toroidal case has a vaguely similar shape, but it has almost no difference
between the part of the orbit when the spot is receding (lower part of CPLP
structure) and when the spot is approaching (upper part of CPLP structure).
Lastly, for the vertical case, we observe that there is little CP and LP
variation during the spot’s orbit for any inclination. Overall, for all cases,
we find that CP and LP variability decreases with increasing spin, which is
due to the increased contribution of the spot to the overall flux density for
lower spin cases.

Figure 14 continues to outline the relation between LP and CP, but here we
vary the underlying plasma parameters modestly to investigate slight
deviations from our base configuration. The base model has $a_{*}=0$ and
$i=20^{\circ}$ and corresponds to the modification factor of 1 in the figure.
In the top row, we vary only the electron (number) density $n_{e}$. There, we
find that LP is almost invariant for variations in $n_{e}$. The LP-CP
structure decreases slightly and CP becomes more negative as the $n_{e}$
modification factor increases. This effect is largest for the radial case, but
is also present for the toroidal case. For the vertical case, we find that the
relation is reversed - CP increases minimally with increasing multiplication
factor. In the second row, we vary electron temperature $\Theta_{e}$. There,
we find a larger response both in LP-CP structure alterations and
corresponding CP values. For the radial case, we observe that both the LP and
CP variation decreases if the temperature increases. For the toroidal case, CP
is most negative and largest in size for smallest $\Theta_{e}$ value but
increases when the multiplication factor increases. A similar relation is
observed for the vertical case, but again in reverse - CP becomes smaller for
increasing multiplication factor. In the third row, we observe that variations
in magnetic field strength $B$ once again produce unique alterations to the
model’s LP and CP. For the radial case, we find that LP decreases and CP
increases for increasing $B$. For the toroidal case, both LP and CP increase
as $B$ increase. For the vertical case, LP remain constant but we observe a
significant increase in CP. For the bottom row, all quantities ($n_{e}$,
$\Theta_{e}$, $B$) are scaled simultaneous. The behaviour here clearly shows
that trying to predict the plasma behaviour based on slight deviations to the
base model is already a non-trivial endeavour. Each model once again describes
a relation between LP and CP that’s different from the behaviour in the three
top rows. Interestingly, both the radial and toroidal cases react similarly,
but the vertical case is most similar to the $B$ variation.

Figure 13: LP versus CP diagrams for models with different magnetic fields,
black hole spins and viewing angles. The crosses denote the disk-only values.
The dotted lines are shown to highlight a trend. Figure 14: LP versus CP
diagrams for models with different magnetic fields, black hole spin of
$a_{*}=0$, and viewing angle of $i=20^{\circ}$. In each row we show the base
plasma property (i.e., $n_{e}$, $\Theta_{e}$, $B$) which are color-coded
according to the multiplication factor with which they are scaled. The bottom
row shows the behavior where all quantities ($n_{e}$, $\Theta_{e}$, $B$) are
scaled simultaneously.

##  Appendix B Circular polarization from regions close to the photon ring

A feature that further magnifies the differences between the magnetic field
cases is the $\widetilde{\text{CP}}$ signature of the lensed emission directly
outside the photon ring. The photon ring is a infinitesimal feature, with a
well-defined radius (see, e.g., Bardeen et al. 1972), that is not well-
resolved in our images but generally has a similar polarization (if slightly
reduced) with respect to the lensed emission directly outside it (Jiménez-
Rosales et al. 2021). For the rest of the section, we will be discussing the
emission that lies directly outside the photon ring that we will simply refer
to as ”emission” from now on. For the radial case, we find a strictly negative
$\widetilde{\text{CP}}$, which is most pronounced for face-on viewing angles
($110^{\circ}\lesssim i\lesssim 70^{\circ}$) as demonstrated in Figs. 2 and
15. For the toroidal case, we find the vertical division as outlined in Sect.
3.4.2 persists in the photon ring’s $\widetilde{\text{CP}}$ signature (as
shown in Figs. 1 and 2) - positive $\widetilde{\text{CP}}$ on the left and
negative $\widetilde{\text{CP}}$ on the right. For the face-on viewing angle
(from top or bottow), we find no apparent difference in the sign of
$\widetilde{\text{CP}}$ between photon ring and disk as shown in Fig. 16.

The vertical case is deviating in the sense that the $\widetilde{\text{CP}}$
signature of the emission directly outside the photon ring is strongly
determined by viewing angle. For $i<90^{\circ}$, we find negative
$\widetilde{\text{CP}}$ for the photon ring and an overall positive
$\widetilde{\text{CP}}$ from disk and spot emission. For $i>90^{\circ}$, the
situation is reversed. This is clearly shown in Fig. 17. As can be seen in
Fig. 1, the edge-on view reveals two modes - a top half which has negative
$\widetilde{\text{CP}}$ and a bottom half with positive
$\widetilde{\text{CP}}$ for the photon ring. This overall
$\widetilde{\text{CP}}$ structure is oppositely polarized with respect to the
photon ring. The dividing line, which lies along the equatorial plane, is
unpolarized and splits these regions.

If we were to look at $\widetilde{\text{LP}}$ alone, then the radial and
toroidal case describe a similar structure which would be hard to
differentiate. For $\widetilde{\text{CP}}$, however, the structure is
completely different, which allows for a straight-forward identification.

Figure 15:  Stokes $\mathcal{I}$ (_left_ panel) and linear (_middle_ panel)
and circular (_right_ panel) polarization images of the background accretion
flow for observer viewing angles of $i=10^{\circ}$ and $i=170^{\circ}$ and a
_radial_ magnetic field topology.  Figure 16: Same description as for Fig. 15
only for the _toroidal_ magnetic field topology.  Figure 17: Same description
as for Fig. 15 only for the _vertical_ magnetic field topology.

##  Appendix C Optical/Faraday Depth and other effects

Figure 18: Impact of Faraday effects on the Stokes $\mathcal{IQUV}$, LP and CP
evolution in model with vertical magnetic field viewed at $i=20\deg$. In the
left panel we show light curves of a model where we disabled the Faraday
rotation and conversion effects by setting the Faraday rotativities to zero
(radiative transfer model with $\rho_{QV}=0$). In the right panels we show the
same model but with the Faraday effects taken into account. Solid/dashed lines
in each sub-panel indicate model with accretion disk and the bright spot on
clockwise (C) / counterclockwise (CC) orbit. Arrows in the QU panels indicate
the direction of time evolution.

Models considered in this work are mostly optically thin, with optical depth
of $\tau_{a}\sim 10^{-3}-10^{-2}$. Models are also Faraday thin, with Faraday
depths $\tau_{V}\sim 10^{-3}-10^{-2}$. The Faraday thickness defined as
$\tau_{V}=\int\rho_{V}dl$ (where $\rho_{V}$ is the transfer coefficient for
Faraday rotation) gives us an estimate of how many rotations (in radians) the
EVPA is undergoing when a ray of light is transported from the emission origin
to the observer. This means that Faraday rotation has small impact on linear
polarization of models.

Nevertheless, we discuss in more detail how Faraday effects impact the linear
and circular polarization of our models. This is done for two reasons: i)
Faraday conversion can alter circular polarization, ii) Faraday effects may
break degeneracies in observables such as fractional linear and circular
polarization of hot spot models rotating in different directions (on clockwise
and counter-clockwise orbits) or in models with opposite magnetic field
polarity. Even for completely Faraday thin models changing the magnetic field
polarity results in Stokes ${\mathcal{U}}$ with opposite sign.

The discussion is based on analysis of a model of accretion disk with a spot
around non-rotating black hole threaded by vertical magnetic field seen at a
viewing angle of 20 degrees. In this model a QU loop is formed and circular
polarization experiences significant variation.

First we consider a case when both Faraday rotation and Faraday conversion are
absent ($\rho_{Q,V}=0$). Changing the direction of system rotation from
clockwise to counterclockwise reverses the sign of Stokes ${\mathcal{U}}$ and
hence the direction in which the QU loop is traversed (the
clockwise/counterclockwise rotating systems with display
clockwise/counterclockwise QU loops). The fractional linear polarization and
circular polarization remain intact. In the model without Faraday effects the
circular polarization is produced intrinsically, hence the change of the
system rotation changes neither the sign nor the magnitude of the circular
polarization. In Fig. 18 (left panel) we demonstrate that this is true for our
chosen model, as expected. Notice also that, for optically thin emission, when
Faraday conversion is absent, the sign of the Stokes ${\mathcal{V}}$ depends
only on the polarity of the magnetic fields. Changing the magnetic field
polarity to the opposite one will simply reverse the sign of ${\mathcal{V}}$.

Secondly, we consider the same model but with the Faraday effects included
(shown in the right panels in Fig. 18). Stokes ${\mathcal{Q}U}$ are slightly
rotated by Faraday rotation, hence clockwise and counter-clockwise rotating
models will have slightly different linear polarizations and the clockwise
system QU loop will not be identical reflection of the counterclockwise
rotating system. The LP will be changed. It is interesting to note that
reversing the sign of the magnetic fields in this case will result in Faraday
rotation in the opposite direction (since $\rho_{V}\sim B_{||}$ along the line
of sight), so the QU loop of clockwise model with $+B$ should be a reflection
of the QU loop of the counterclockwise model with $-B$ (this should be true
only for spin zero). Notice also that when Faraday effects are included, the
magnitude of Stokes ${\mathcal{V}}$ is changed meaning that in our models
significant portion of the observed circular polarization is produced in the
process of Faraday conversion.