#  Black Hole Magnetic Fields and Their Imprint on Circular Polarization
Images

Angelo Ricarte1,2, Richard Qiu3,4, and Ramesh Narayan1,2
1 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
2 Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA
02138, USA
3 Department of Physics, Harvard University, 17 Oxford Street Cambridge, MA
02138, USA
4 John A. Paulson School of Engineering and Applied Sciences, Harvard
University, 29 Oxford Street, Cambridge, MA 02138, USA

###### Abstract

The circular polarization of black hole accretion flows can encode properties
of the underlying magnetic field structure. Using general relativistic
magnetohydrodynamics (GRMHD) simulations, we study the imprint of magnetic
field geometry on circular polarization images potentially observable by the
Event Horizon Telescope (EHT). We decompose images into the different
mechanisms that generate circular polarization in these models, which are
sensitive to both the line of sight direction and twist of the magnetic field.
In these models, a stable sign of the circular polarization over time, as
observed for several sources, can be attributed to a stability of these
properties. We illustrate how different aspects of a generic helical magnetic
field geometry become imprinted on a circular polarization image. We also
identify novel effects of light bending that affect the circular polarization
image on event horizon scales. One consequence is the sign flipping of
successive photon rings in face-on systems, which if observable and
uncorrupted by Faraday rotation, can directly encode the handedness of the
approaching magnetic field.

###### keywords:

accretion, accretion discs — black hole physics — galaxies: individual (M87) —
magnetohydrodynamics (MHD) — polarization — techniques: polarimetric

††pagerange: LABEL:firstpage–LABEL:lastpage††pubyear: 2020

##  1 Introduction

Supermassive black holes (SMBHs) are thought to reside at the centres of all
massive galaxies, where they sometimes shine as active galactic nuclei (AGN)
and help regulate star formation via AGN feedback (e.g., Kormendy & Ho, 2013).
Detailed studies of their central engines are important to better understand
their accretion and feedback processes. By performing very long baseline
interferometry (VLBI) at 1.3 millimetres, the Event Horizon Telescope (EHT)
collaboration produced the first resolved images of the SMBH at the centre of
M87, enabling novel studies of accretion physics on event horizon scales
(Event Horizon Telescope Collaboration et al., 2019a, b, c, d, e, f, 2021a,
2021b).

The SMBH of M87, henceforth M87*, is believed to be fed by an advection
dominated accretion flow (ADAF) of hot, tenuous plasma (Ichimaru, 1977; Rees
et al., 1982; Narayan & Yi, 1994, 1995; Abramowicz et al., 1995; Reynolds et
al., 1996b; Yuan & Narayan, 2014). Its image is sensitive to properties
intrinsic to the SMBH, including spin and accretion rate, as well as those of
the plasma in the surrounding environment, such as the magnetic field
configuration, temperature, and underlying electron distribution function.
Combining multi-wavelength constraints with the first resolved image of M87*
enabled Event Horizon Telescope Collaboration et al. (2019e) to narrow the
allowed parameter space of M87*, ultimately ruling out all non-spinning SMBH
models.

Recently, the EHT Collaboration completed an analysis of the linear
polarization of M87*, providing novel insights into its magnetic field
structure in particular (Goddi et al., 2021; Event Horizon Telescope
Collaboration et al., 2021a, b). Synchrotron emission, which dominates the
millimetre image, is initially generated perpendicular to the local magnetic
field, such that its polarization carries with it an imprint of the field
geometry (Palumbo et al., 2020). Then, as this polarization propagates, it is
further modified by Faraday effects, important for depolarizing accretion
flows down to observed levels and generating the observed rotation measure
(Ballantyne et al., 2007; Mościbrodzka et al., 2017; Jiménez-Rosales & Dexter,
2018; Ricarte et al., 2020). Fully polarized radiative transport simulations
on EHT scales have been developed in the past few decades, allowing us to link
polarized images to the detailed physics of the underlying plasma and the
space-time producing them (Bromley et al., 2001; Broderick & Loeb, 2006;
Broderick & McKinney, 2010; Porth et al., 2011; Shcherbakov et al., 2012;
Dexter, 2016; Mościbrodzka & Gammie, 2018). This enabled (Event Horizon
Telescope Collaboration et al., 2021b) to discriminate between two major
classes of accretion disk: a “Magnetically Arrested Disk” (MAD) and “Standard
and Normal Evolution” (SANE). MAD accretion disks have magnetic fields strong
enough to affect the disk dynamics and exhibit stronger poloidal (or non-
toroidal) magnetic field components (Bisnovatyi-Kogan & Ruzmaikin, 1974;
Igumenshchev et al., 2003; Narayan et al., 2003; Chael et al., 2019).
Meanwhile, the weaker magnetic fields of a SANE disk are sheared out by the
motion of the plasma into a mostly toroidal configuration (Narayan et al.,
2012; Sądowski et al., 2013; Ryan et al., 2018). The fractional linear
polarization of M87*, the upper limit on its circular polarization, and most
importantly the “twisty pattern” of its spatially resolved linear polarization
map favour “MAD” models of M87* (Event Horizon Telescope Collaboration et al.,
2021b).

In addition to these recent studies of M87*, there exist unresolved
polarimetric measurements of Sgr A* and other low-luminosity AGN (Plambeck et
al., 2014; Kim et al., 2019). In particular, linear polarization measurements
of Sgr A* in the millimetre span two decades, motivating further studies of
time variable polarimetry (Aitken et al., 2000; Bower et al., 2003; Marrone,
2006; Muñoz et al., 2012; Bower et al., 2018). Moreover, although there is
only an upper limit on the total circular polarization of M87* so far
($\lesssim 0.8$ per cent Goddi et al., 2021), the EHT Collaboration plans to
one day produce images of circular polarization in addition to linear.
Circularly polarized images of SMBH accretion flows are less well studied than
their linearly polarized counterparts. For several observed sources, a
persistent magnitude and sign of total circular polarization has been
interpreted as stability in something inherent to the accretion flow such as
its magnetic field geometry, but the details are poorly understood (Wardle &
Homan, 2001; Beckert & Falcke, 2002; Ruszkowski & Begelman, 2002; Enßlin,
2003). One recent study of MAD accretion disks highlights the importance of
both Faraday rotation and conversion and points out a sign inversion in the
photon ring (Moscibrodzka et al., 2021).

There are two mechanisms that can generate circular polarization in these
systems: intrinsic emission of circular polarization, and Faraday conversion
of linear polarization (e.g., Wardle & Homan, 2003). Synchrotron emission is
the dominant emission mechanism in the millimetre, and it has an intrinsic
linear polarization fraction of about 70 per cent (e.g., Pandya et al., 2016).
In contrast, its emitted circular polarization fraction is only around 1 per
cent, but this may be significant enough to contribute to the modest circular
polarization fractions that have been measured for low-luminosity AGN. The
sign of intrinsically emitted circular polarization directly encodes the
direction of the magnetic field with respect to the photon wavevector.
Meanwhile, Faraday conversion exchanges linear and circular polarization, with
an efficiency depending on the temperature, density, and magnetic field of the
plasma through which the light is travelling (e.g., Kennett & Melrose, 1998).
While the intensity of intrinsic emission increases with temperature, Faraday
conversion is more efficient at lower (sub-relativistic) temperatures (Jones &
Odell, 1977). Both the intrinsic circular polarization fraction and the
strength of Faraday conversion are sensitive to the composition of the
underlying plasma. This allows circular polarization measurements to provide
some of the most direct constraints on plasma composition (Wardle et al.,
1998; Anantua et al., 2020; Emami et al., 2021).

Interestingly, Faraday conversion has no effect in a plasma with a uni-
directional magnetic field without an additional mechanism to rotate the
polarization plane. A synergy between Faraday conversion and Faraday rotation
is one mechanism to accomplish this, as discussed in-depth with GRMHD models
in Tsunetoe et al. (2020a, b). Alternatively, a line of sight twist in the
magnetic field can also effectively rotate the polarization, by instead
rotating the basis in which the polarization states are defined (Hodge, 1982;
Wardle & Homan, 2001). This phenomenon is believed to set the circular
polarization of jets on larger scales (e.g., Gabuzda et al., 2008). As we
shall show, a more complicated magnetic field structure, combined with light
bending effects, produces a rich circular polarization image on event horizon
scales.

In this work, we study in detail the mechanisms which generate circular
polarization in SMBH accretion flows using two GRMHD simulations, described in
§2. To better understand the connection between circular polarization and
magnetic field, we first study the structure of the magnetic field in these
models in §3. Then, in §4, we study each of the mechanisms for generating
circular polarization in turn. We discuss the more general implications of
this study in §5, and summarise our results in §6.

##  2 Methodology

We use as our starting point two GRMHD simulations of M87* that are included
in Event Horizon Telescope Collaboration et al. (2019e, 2021b), one
“Magnetically Arrested Disk” (MAD) and one “Standard and Normal Evolution”
(SANE), both run using harm (Gammie et al., 2003). Although a SANE model is
disfavoured for M87* based on linear polarization (Event Horizon Telescope
Collaboration et al., 2021b), it remains instructive to explore how such a
system differs in its circularly polarized image. Both models have
dimensionless spin parameter $a_{\bullet}=0.94$ and are run using a variation
of Kerr-Schild coordinates (“Funky Modified Kerr-Schild,” or FMKS) that
concentrates resolution at the mid-plane. These simulations are seeded with a
weak dipolar magnetic field parallel to both the spin of the BH and the
angular momentum of the accretion disk. The MAD simulation is performed using
a $384\times 192\times 192$ grid with a maximum radius of $10^{3}\
GM_{\bullet}/c^{2}$, while the SANE simulation uses a $288\times 128\times
128$ grid and a maximum radius of $50\ GM_{\bullet}/c^{2}$. (Throughout this
work, $G$ is the gravitational constant, $M_{\bullet}$ is the SMBH mass, and
$c$ is the speed of light.) As in Ricarte et al. (2020), we only evolve the
radiative transfer equations within a radius of $20\ GM_{\bullet}/c^{2}$,
inside of which we find the simulations exhibit inflow equilibrium. In
Appendix B, we demonstrate that this region encloses the relevant plasma for
producing circular polarization in these models. Both models are evolved to
$10^{4}\ GM_{\bullet}/c^{3}$, and we focus our presentation on the images of
their final snapshots.

To perform the General Relativistic Ray Tracing (GRRT), we use ipole, a ray-
tracing code which first solves the null geodesic equation from the camera
through the source, then integrates forward the fully polarized radiative
transfer equation (Mościbrodzka & Gammie, 2018). Unless otherwise noted,
images are created at a frequency of 228 GHz, with a field of view of 160
$\mu$as, and an angular resolution of 0.5 $\mu$as. As in many previous works,
the plasma is modelled as two-temperature, where the ratio of the ion
temperature to the electron temperature is a function of the plasma $\beta$
parameter $\beta=p_{\mathrm{gas}}/p_{\mathrm{mag}}$ via

$\frac{T_{i}}{T_{e}}=R_{\mathrm{high}}\frac{\beta^{2}}{1+\beta^{2}}+\frac{1}{1+\beta^{2}}\;,$
(1)

where $p_{\mathrm{gas}}$ and $p_{\mathrm{mag}}$ are the gas and electron
pressures respectively (Mościbrodzka et al., 2016). As a consequence,
$T_{i}=T_{e}$ in the highly magnetised funnel region, while $T_{i}/T_{e}\to
R_{\mathrm{high}}$ in the gas pressure supported mid-plane. For details of the
radiative transfer coefficients, we refer readers to Mościbrodzka & Gammie
(2018), and the modification to the Faraday rotation coefficient $\rho_{V}$ in
Ricarte et al. (2020).

In the two models considered in this work, we fix $R_{\mathrm{high}}=20$, a
modest value motivated by recent studies of plasma heating (Mizuno et al. in
prep.). For images with $R_{\mathrm{high}}=160$, we comment that the signals
we discuss in the paper remain largely unchanged, but there is additional
“noise” and depolarization due to the increased Faraday rotation. This is
because larger values of $R_{\mathrm{high}}$ decrease the electron temperature
in the gas pressure supported mid-plane, by construction, making Faraday
effects more efficient. Meanwhile, for $R_{\mathrm{high}}=1$, the MAD image is
largely unchanged, but the SANE image exhibits a stronger photon ring feature
discussed in §4.3.2, which is suppressed for $R_{\mathrm{high}}=20$ due to
Faraday rotation.

Since GRMHD simulations are scale free, the SMBH mass, distance, and accretion
rate are set during the GRRT step. To set the spatial and temporal scales, we
adopt a mass of $6.2\times 10^{9}\ \mathrm{M}_{\odot}$ and a distance of 16.9
Mpc, consistent with stellar dynamics and shadow size measurements of M87*
(Gebhardt et al., 2011; Event Horizon Telescope Collaboration et al., 2019f).
Then, we adopt the accretion rate scalings that are fit in Event Horizon
Telescope Collaboration et al. (2019e) to reproduce an average flux of 0.5 Jy
at 230 GHz. Such scaling is possible because ideal GRMHD simulations are
invariant under the transformation $\rho\mapsto\mathcal{M}\rho$ and
$\vec{B}\mapsto\mathcal{M}^{1/2}\vec{B}$, where $\rho$ is the mass density,
$\vec{B}$ is the magnetic field vector, and $\mathcal{M}$ is a scalar.

##  3 Magnetic Field Structure

###  3.1 How Circular Polarization Encodes the Magnetic Field

(a)

(b)

(c)

Figure 1: Schematics illustrating the mechanisms by which circular
polarization can be generated and the resultant signs of Stokes $V$. The sign
of intrinsically emitted circular polarization directly reflects the magnetic
field direction with respect to the photon wavevector (Top Left).
Intrinsically emitted circular polarization is positive if the magnetic field
is pointed towards the observer, and negative if it is pointed away. Faraday
conversion only operates on Stokes $U$, and thus Stokes $V$ inherits the local
sign of Stokes $U$ from mediating phenomena. Circular polarization generated
through Faraday conversion mediated by a small amount of Faraday rotation
(Bottom Left) again encodes the line of sight magnetic field direction such
that $\vec{k}\cdot\vec{B}>0$ leads to $V>0$. If the magnetic field twists
along the line of sight (Top and Bottom Right), then circular polarization
generated through Faraday conversion acquires a sign opposite to the sign of
the twist $\xi$, since it is the basis in which the Stokes parameters are
defined that is twisted, rather than the emission itself. Therefore, a
counter-clockwise (CCW) twist of $\xi>0$ generates $V<0$. Figure 2: Schematic
showing the definitions of the Stokes parameters in the local tetrad frame,
defined by the photon wavevector $\vec{k}$ and the magnetic field vector
$\vec{B}$. Synchrotron emission produces linearly polarized emission in the
$+Q$ orientation, perpendicular to both the photon wavevector and the magnetic
field.

There are three pathways to generate circular polarization, each of which
reflects a different aspect of the magnetic field: intrinsic emission, Faraday
conversion mediated by Faraday rotation, and Faraday conversion mediated by a
line of sight twist in the magnetic field (e.g., Wardle & Homan, 2003). These
processes are illustrated in Figure 1.

First, synchrotron emission intrinsically produces a small amount of circular
polarization. The sign of this circular polarization directly encodes the line
of sight direction of the magnetic field $\vec{B}$ with respect to the photon
wavevector $\vec{k}$. The emitted radiation has positive $V$ if the two
vectors are aligned ($\vec{k}\cdot\vec{B}>0$) and negative $V$ if they are
anti-aligned ($\vec{k}\cdot\vec{B}<0$).

At the same time, synchrotron emission produces a large linear polarization
fraction in these models, oriented as illustrated in Figure 2. In the local
plasma frame, linear polarization is initialised perpendicular to both the
magnetic field and the photon wavevector, which is locally defined as the $+Q$
direction (parallel to the electric field). As this linear polarization
propagates along $\vec{k}$, some of it can be converted to circular via
Faraday conversion. However, Faraday conversion only operates on Stokes $U$ in
the local plasma frame, a component neither parallel nor perpendicular to
$\vec{k}\times\vec{B}$. If $U>0$, the resulting circular polarization has
$V>0$, and vice versa. Since the intrinsic synchrotron emission has only $+Q$
(i.e., $U=0$), something has to recast a part of the emitted $Q$ into $U$ as
the radiation propagates through the plasma for Faraday conversion to occur.
This can happen in two ways, corresponding to two pathways through which
Faraday conversion can operate.

Faraday rotation can change the plane of polarization as the radiation
propagates by directly exchanging $Q$ and $U$. The sense of rotation operates
via the “right hand rule” on the direction of the line-of-sight component of
the magnetic field. For small amounts of Faraday rotation, if
$\vec{k}\cdot\vec{B}>0$, Faraday rotation converts $+Q$ partially to $+U$, and
Faraday conversion then creates circular polarization with $+V$. Similarly,
$\vec{k}\cdot\vec{B}<0$ leads to $-V$.

The other way in which Faraday conversion can operate is if the component of
the magnetic field perpendicular to $\vec{k}$ rotates because of a line of
sight twist in the magnetic field. Since the basis in which the Stokes
parameters are defined is tied to the local magnetic field, if the field
twists along the line of sight, radiation that was initially emitted entirely
as Stokes $+Q$ can be “recast” as partially Stokes $U$, with a sign depending
on the direction and degree of the twist. In this pathway, since it is the
polarization basis rather than the radiation itself that is rotating, a small
positive (counter-clockwise, CCW) twist in the magnetic field produces $-U$
and consequently through Faraday conversion $-V$.

We caution that the above discussion on the sign of $V$ resulting from Faraday
conversion applies only for small relative rotations of the plane of
polarization ($<90^{\circ}$). For larger rotations, the sign of $U$ generated
by either Faraday rotation and/or magnetic field twist flips each time the
relative polarization plane is rotated by an additional $<90^{\circ}$. In
practice, the large Faraday rotation depth in some of these models
(Mościbrodzka et al., 2017; Jiménez-Rosales & Dexter, 2018; Ricarte et al.,
2020) can also depolarize circular polarization in their images, by
essentially randomising the sign of $U$ which would be converted into $V$, as
we will further discuss in §4.3.

###  3.2 Generic Magnetic Field Properties

Figure 3: Cartoon illustrating the formation of a twisted magnetic field. A
vertically oriented field line is twisted by frame dragging and flux freezing.
Notice the opposite handedness of the twist generated above and below the mid-
plane. The colours of these curves simply denote relative position along the
field lines.

Since the mechanisms to generate circular polarization depend on the details
of the magnetic field, let us first explore its three-dimensional geometry.
Assuming an initial dipolar magnetic field structure, frame dragging and flux
freezing produce a generic helical magnetic field pattern, as illustrated in
Figure 3 (e.g., Semenov et al., 2004). Notice that the handedness of these
helices flips across the mid-plane, resulting in a sign flip in the radial and
tangential components of the magnetic field.

Figure 4: Schematic illustrating two field lines anchored at opposite sides of
the disk mid-plane. The $\bigodot$ and $\bigoplus$ symbols indicate whether
the magnetic field lines move out of or into the page. Note the four quadrants
generated by the sign flip in the tangential and radial components across the
mid-plane, as illustrated in Figure 3. At two locations chosen at opposite
sides of the mid-plane, the angle $\xi_{T}$ illustrates the magnetic twist
direction between the back and front sides, what we term the “transverse
twist.” Since the top and bottom helices have opposite handedness, $\xi_{T}$
flips sign across the mid-plane.

The outcome of this process is the structure illustrated in Figure 4, where
blue and orange curves represent magnetic field lines anchored on opposite
sides of the BH in the mid-plane. We have assumed that the BH angular
momentum, the disk angular momentum, and the direction of the dipolar magnetic
field are all aligned with what we illustrate as the $z$ axis. Applying
Ampere’s law, this configuration corresponds to an inward current towards the
BH. We comment that for a small sample of AGN with constrained jet magnetic
field geometries, there does appear to be a preference for inward currents
(Gabuzda, 2018b), which matches expectations from the cosmic battery mechanism
(Contopoulos & Kazanas, 1998; Koutsantoniou & Contopoulos, 2014). For a
dipolar field initialised in the positive $z$ direction, the field lines head
out of the page in red regions marked with the $\bigodot$ symbol, and into the
page in blue regions marked with the $\bigoplus$ symbol. Two sign flips occur
in the line of sight magnetic field direction, across both the $x$ and $z$
axes in this cartoon, which is also pointed out by Tsunetoe et al. (2020b).
This pattern, which we refer to as the “four quadrants,” reverses if the
dipolar field is instead initialised in the negative $z$ direction. The sign
flip across the $z$ axis is just a viewing angle effect due to rotation, while
the sign flip across the mid-plane is physical, generated as described in
Figure 3. The mid-plane sign flip implies the existence of an equatorial
current sheet, which would be subject to plasmoid instabilities and magnetic
reconnection in higher resolution simulations that include resistivity
(Ripperda et al., 2020).

###  3.3 Magnetic Twist

In the absence of physical effects to exchange Stokes $Q$ and $U$, the line of
sight twist in the magnetic field determines the sign of circular polarization
generated through Faraday conversion (Gabuzda et al., 2008). We write the
magnetic twist along a line of sight as $\xi$, where a right-handed (counter-
clockwise) twist has positive $\xi$, and a left-handed (clockwise) twist has
negative $\xi$.111To be more explicit, the magnetic twist along the photon
trajectory between points A and B is the angle, in the plane perpendicular to
the photon wavevector at point B, between the local magnetic field vector and
the vector resulting from parallel transport of the magnetic field vector from
point A to point B. Throughout this work, we always consider twist as defined
in the direction of the photon’s propagation, through the source to the
observer.

During our explorations, we found that circular polarization generated via
conversion mediated by magnetic twist could exhibit two different signs when
viewing the same side of the disk tilted at different angles. This is because
there are two different relevant twists, depending on one’s viewing
inclination, which we term the “transverse” twist (for edge-on viewing angles)
and “vertical” twist (for pole-on viewing angles). By examining Figure 4, we
can determine the sign of what we term the “transverse twist” of these
helices, $\xi_{T}$: the relative twist between the far- and near-side magnetic
field lines for this edge-on inclination. These are illustrated by the blue
and red arcs in this figure. If the angular momentum of the system is oriented
in the $+z$ direction, then $\xi_{T}>0$ above the mid-plane and $\xi_{T}<0$
below the mid-plane. Note that the magnetic twist is invariant to the sign of
the magnetic field; it would be the same if the arrows denoting the direction
of the field were reversed.

Figure 5: Schematic illustrating vertical twist, $\xi_{V}$, for a pole-on
viewing angle. Here, the angular momentum of the BH is in the positive $z$
direction, out of the page. Because parabolic jets radially expand more
rapidly near the base than at large distances, the radial component of a
magnetic field threading such a structure grows weaker relative to the
tangential component as $|z|$ increases. Interestingly, comparing $\xi_{V}$ to
$\xi_{T}$ in Figure 4, the vertical twist in a given physical region exhibits
the opposite sign as its transverse twist.

We distinguish the “vertical twist,” $\xi_{V}$, from the “transverse twist,”
$\xi_{T}$, as the twist of this structure when viewed from pole-on inclination
angles. The vertical twist is more relevant for M87* than its transverse
twist, since we view the system at approximately $20^{\circ}$ inclination
(Walker et al., 2018). The vertical twist direction can be deduced from
generic parabolic jet structures, which widen more rapidly at the base than at
larger distances (e.g., Asada & Nakamura, 2012; Chatterjee et al., 2019). This
implies that as $|z|$ increases, the radial component of a magnetic field
threading such a structure weakens relative to its tangential component. In
the funnel, it is actually the vertical field which dominates, which would go
into or out of the page, but we are concerned only with the part of the
magnetic field perpendicular to the propagation direction.

From this, we can infer the sign of the vertical twist, as illustrated in
Figure 5. In the $z>0$ region, $\xi_{T}>0$ but $\xi_{V}<0$, while the opposite
is true for the $z<0$ region. This implies that for generic accretion flows
near the event horizon, vertical and transverse twist have opposite sign in
the same region. When viewing the same side of the disk edge-on, the
transverse twist is more important for determining circular polarization,
while for pole-on inclination angles, the vertical twist is more important. We
confirm the sense of vertical twist in our GRMHD models in Appendix A.

###  3.4 Magnetic Fields of Our GRMHD Models

Figure 6: Azimuthally averaged magnetic field decompositions in the coordinate
frame for the two models explored in this study during their final snapshots.
Notice (i) the dominance of the azimuthal field component in these particular
models, (ii) the sign flip in the azimuthal and radial components across the
mid-plane, and (iii) the greater disorder in the SANE model compared to the
MAD.

Next, we examine the magnetic field structure of our two GRMHD models to learn
how they may depart from the simple pictures described in the previous
section. In Figure 6, we plot the magnetic fields in the coordinate frame in
Cartesian Kerr-Schild coordinates. These are decomposed into their radial,
azimuthal, and vertical components, then azimuthally averaged during the final
snapshot, time $t=10000\ GM_{\bullet}/c^{3}$. These decompositions are
normalised to the local magnetic field strength, such that $\langle
B_{\mathrm{tot}}\rangle^{2}=\langle B_{r}\rangle^{2}+\langle
B_{\phi}\rangle^{2}+\langle B_{z}\rangle^{2}$. There are three interesting
phenomena to notice in this figure. First, these two models generally have
stronger azimuthal fields than radial fields, which need not generally be
true. The azimuthal component dominates everywhere except in the funnel,
directly above and below the BH. Second, notice the sign flip in the radial
and tangential field components as discussed in the previous section. Third,
the SANE model is more turbulent and disordered than the MAD model,
complicating the simple picture drawn in Figure 4. This turbulence will appear
as noise in its circular polarization images.

|
---|---
|
Figure 7: Magnetic field line illustrations created with VisIt by integrating
magnetic field components in Cartesian Kerr-Schild coordinates. The MAD
$a_{\bullet}=+0.94$ simulation is visualised in the left column, while the
SANE $a_{\bullet}=+0.94$ simulation is visualised in the right column. Lines
coloured red are anchored to a sphere of radius $2GM_{\bullet}/c^{2}$, while
those coloured blue are anchored to a disk in the mid-plane. The fields in the
jet funnel are similar in both models, but the disk field is more ordered and
has a stronger vertical component in the MAD model.

By performing path integrals on the magnetic field (in Cartesian Kerr-Schild
coordinates), we produce three-dimensional visualisations using VisIt, shown
in Figure 7 (Childs et al., 2012). Magnetic field lines coloured blue are
anchored within the plane of the disk, while magnetic field lines coloured red
are anchored around the BH (on a sphere of radius $2GM_{\bullet}/c^{2}$). The
left column depicts the MAD simulation, while the right column depicts the
SANE simulation. In both cases, the magnetic field in the funnel region is
ordered and well-defined. However, the disk magnetic field in the MAD model is
more ordered and has a stronger vertical component than its SANE counterpart.
These visualisations also reveal a steep pitch angle in the funnel region,
which affects the circular polarization produced by the transverse twist
(e.g., Gabuzda et al., 2008).

##  4 Circular Polarization Images

###  4.1 Circular Polarization in Our Models

Figure 8: An introduction to the circular polarization of our MAD model. The
final snapshot ($t=10000\ GM_{\bullet}/c^{3}$) is shown. The top row displays
the system with a pole-on inclination of $5^{\circ}$, while the bottom row
displays the system at an edge-on inclination of $90^{\circ}$. In the first
and third columns, 501 images between $7500<GM_{\bullet}/c^{3}<10000$ have
been averaged. In the centre column, we display the final snapshot. For this
model, the photon ring dominates the circularly polarized image for pole-on
inclinations, while a sign flip across the mid-plane that reflects a sign flip
in the “transverse twist” is seen at edge-on inclinations.  Figure 9: As
Figure 8, but for the SANE model. At a given snapshot, the SANE model image is
messier than the MAD’s due to both its more turbulent disk and its higher
Faraday depths. Faraday rotation scrambles the photon ring more strongly in
this model. Note the different colour scalings for this figure compared to
Figure 8.  Figure 10: Depths of Faraday rotation ($\tau_{\rho_{V}}$) and
conversion ($\tau_{\rho_{Q}}$) in our models when viewed at $5^{\circ}$ (top
row) and $90^{\circ}$ (bottom row). The MAD model is more Faraday thin than
the SANE model. Both models are more thick to Faraday rotation than Faraday
conversion. Models are more Faraday thick for edge-on inclinations than face-
on ones. In the pixels marked with white circles, we study the emissivity and
rotativity along the line of sight in Appendix B.

In Figures 8 and 9, we introduce the circular polarization images of the MAD
and SANE models respectively. The top row plots a pole-on viewing angle of
$5^{\circ}$, while the bottom row plots an edge-on viewing angle of
$90^{\circ}$. For these figures alone, we have time-averaged 501 images, one
for each of the snapshots in the final quarter of the simulation,
$7500<GM_{\bullet}/c^{3}<10000$. Time-averaged circular polarization images
(Stokes $V$) are shown in the first column, the image of just the final
snapshot is shown in the second column, and the circular polarization fraction
(Stokes $V/I$) is shown in the third column. Immediately we can see that
images from both models and both viewing angles feature both positive and
negative regions, which will complicate the interpretation of the unresolved
Stokes $V$ measurements. The face-on MAD image is surprisingly dominated by
its photon ring (discussed at length in §4.3.2), while the edge-on images are
preferentially dominated by opposite signs of circular polarization on
opposite sides of the mid-plane.

Comparing the two, the MAD image appears more ordered than its SANE
counterpart. Furthermore, the photon ring, prominent in the face-on MAD
circular polarization image (and discussed at length in §4.3.2), is suppressed
in the SANE image. Some of the disorder in the SANE is due to its more
turbulent magnetic field structure, as shown in Figure 6. Moreover, the SANE
model is more scrambled by its much larger Faraday rotation and conversion
depths. Further discussed in §4.4, the much higher Faraday conversion depth in
the SANE model also leads to a much larger fractional circular polarization
(see also Event Horizon Telescope Collaboration et al., 2021b). We plot the
Faraday depths of these models in Figure 10, where the Faraday rotation depth
$\tau_{\rho_{V}}=\int\rho_{V}ds$ and the Faraday conversion depth
$\tau_{\rho_{Q}}=\int\rho_{Q}ds$. Here, $\rho_{V}$ and $\rho_{Q}$ are the
radiative transfer coefficients responsible for Faraday rotation and
conversion respectively (see Dexter, 2016; Mościbrodzka & Gammie, 2018, for
more details). In this figure, the brightness of each pixel encodes its total
intensity, while the colour encodes the appropriate Faraday depth as indicated
in the colour bar. The intensity-weighted image average of these values is
written on the bottom of each panel. The top row shows an inclination of
$5^{\circ}$, while the bottom row shows an inclination of $90^{\circ}$. In
Appendix B, we decompose the emissivity and rotativity along the lines of
sight marked by white circles in this figure.

At $5^{\circ}$, the MAD model is both Faraday rotation and conversion thin,
while the SANE model is strongly Faraday rotation thick and moderately Faraday
conversion thick. Faraday rotation is directly encoded in the rotation
measure, which for M87* has been observed to have a magnitude of $\approx
10^{5}\ \mathrm{rad}\;\mathrm{m}^{-2}$ and exhibit sign flips (Goddi et al.,
2021), consistent with GRMHD studies (Ricarte et al., 2020). At an inclination
of $90^{\circ}$, the MAD model becomes moderately Faraday rotation thick,
while the SANE becomes thick to both Faraday rotation and conversion. We
should thus expect that the circular polarization of the MAD model should more
closely reflect the simple and ordered magnetic field structure discussed in
§3.

Now, to understand the features of these images in detail, we will decompose
their circular polarization images into components originating from intrinsic
emission and Faraday conversion, and explore how the sign of the circular
polarization encodes the magnetic field geometry.

###  4.2 Intrinsic Emission

To generate images of purely intrinsic emission, we remove Faraday conversion
effects from IPOLE by setting the radiative transfer coefficient $\rho_{Q}=0$.
Without conversion, linear polarization cannot be transformed into circular,
and the only remaining mechanism for the production of circular polarization
is intrinsic synchotron emission. Recall that the sign of the intrinsic
emission encodes the direction of the line of sight magnetic field.

####  4.2.1 Edge-on — Four Quadrants

Figure 11: Intrinsic circularly polarized emission fractions (Faraday
conversion switched off) for our two GRMHD models, viewed edge-on. We see the
“four quadrants” pattern resulting from the magnetic field geometry shown in
Figure 4. Intrinsic emission dominates the circular polarization on large
scales, producing a diffuse background.

The geometry imprinted onto the intrinsic circular polarization is most
clearly seen in the circular polarization fraction, $V/I$. This is plotted for
our two models in Figure 11 for an edge-on viewing angle. We see the “four
quadrants” pattern of the magnetic field geometry as described in Figure 4.
Recall that the sign flip across the $z$ axis is due to the helical twisting
of field lines, while the sign flip across the mid-plane is physical. The
intrinsic circular polarization fraction can reach a few per cent in some
areas. However, circular polarization generated via intrinsic emission is sub-
dominant in the bright parts of the image compared to that generated by
Faraday conversion in these particular models, as we will explore in more
detail in §4.4. In addition, we notice substantial cancellation due to the
symmetry of these images. The “four quadrants” pattern is also discussed in
Tsunetoe et al. (2020b), who find that the leftward offset of the vertical
sign-flip from the centre of the image is due to beaming effects. Ricarte et
al. (2020) discuss the potential impact of bandwidth depolarization in images
of Faraday thick models. To ensure that bandwidth depolarization does not
significantly impact our images, we average 129 individual images within a
bandwidth of 4 GHz in Figures 11, 12, and 13. We found that bandwidth
averaging has negligible effect on these images, owing in part to the
relatively small value of $R_{\mathrm{high}}$ adopted.

####  4.2.2 Face-on — $B_{z}$, plus $B_{r}$ on the Far Side

Figure 12: Intrinsic circularly polarized emission fractions (Faraday effects
switched off) for the MAD model, viewed face-on. One might expect a single
sign of the circular polarization, corresponding to the vertical field
direction. Instead, we find a sign flip for emission originating on the far
side of the mid-plane. This is due to the inward bending of photon orbits,
causing the circular polarization to be sensitive to the radial magnetic field
in addition to the vertical component. For the magnetic field geometry
described in §3, the radial field on the far side of the BH always has the
opposite sign as the vertical field.

For face-on inclinations, one might expect to see only a single sign of
intrinsic circular polarization, corresponding to the direction of the
vertical magnetic field, $B_{z}$. Instead, as plotted in Figure 12, we find
that the intrinsic emission from the far side of the emitting region, behind
the mid-plane, tends to exhibit the opposite sign. This unanticipated
phenomenon is due to the lensing of photon orbits. Behind the mid-plane,
photon orbits are bent towards the BH, causing the circular polarization to be
sensitive to not only the vertical magnetic field, but also its radial
component. Assuming the geometry sketched in Figure 4, $B_{r}$ on the far side
always has the opposite sign of $B_{z}$. Flips in the apparent line of sight
direction of the magnetic field have also been found in these models in the
context of Faraday rotation, which similarly encodes the direction of the
magnetic field relative to the photon wavevector (Ricarte et al., 2020).

###  4.3 Faraday Conversion

Faraday conversion exchanges linear and circular polarization states. On its
own, it is insensitive to the sign of the magnetic field, and depends on the
relative twist between the field producing the emission and the field
performing the conversion. We create images of circular polarization due only
to Faraday conversion by switching off the coefficient responsible for
intrinsic emission, $j_{V}=0$. In these particular models, Faraday conversion
contributes more to the unresolved circular polarization image than the
intrinsic emission, which we will further explore in §4.4.

####  4.3.1 Edge-on — Transverse Twist

Figure 13: Circular polarization images of our two models, purely from Faraday
conversion, $j_{V}=0$, from an edge-on viewing angle. These images are very
similar to the lower central panels of Figures 8 and 9, because Faraday
conversion dominates the circular polarization in the bright parts of the
image. Both images are predominantly negative on top and positive on the
bottom, directly reflecting the direction of transverse twist shown in Figure
4.

For edge-on inclinations, the sign of the circular polarization encodes the
transverse twist, $\xi_{T}$. As illustrated in Figure 4, $\xi_{T}$ flips sign
at the mid-plane and depends on the direction that the magnetic field is
twisted. Circularly polarized intensity images created only by Faraday
conversion are shown in Figure 13. These Stokes $V$ images are predominantly
negative on top and positive on the bottom, as expected from the sign of
$\xi_{T}$. The SANE image is much messier due to both turbulence in the
magnetic field and its thickness to both Faraday rotation and conversion.
Given a circularly polarized image of an edge-on system, we expect a sign flip
at the mid-plane and for the signs to directly encode the handedness of the
magnetic field on either side.

####  4.3.2 Face-on — Vertical Twist and a Photon Ring Signature

Figure 14: Circular polarization images of the MAD, $a_{\bullet}=0.94$,
$R_{\mathrm{high}}=20$ model, broken into sub-ring components, revealing that
alternating sub-rings flip sign. We scale each panel’s colour map individually
such that the colours span $\pm\mathrm{max}|V|$. On the top row, only circular
polarization originating from Faraday conversion through the magnetic twist
pathway is plotted, by setting the radiative transfer coefficients
$\rho_{V}=j_{V}=0$. All coefficients are active in the second row,
demonstrating that Faraday conversion mediated by magnetic twist causes this
signal. Images are blurred with a Gaussian with a full-width at half-maximum
of 1 $\mu$as to better visualise the narrow and faint sub rings. Rings of
increasing order alternate sign due to parallel transport and generic
properties of the magnetic field geometry. Under the right conditions, the
circular polarization of the photon ring directly encodes the handedness
(vertical twist) of the magnetic field. All sub-images flip sign if the system
is viewed from the opposite inclination, where the twists of the helices are
reversed.

For face-on viewing angles, circular polarization is sensitive to the vertical
twist, $\xi_{V}$, which happens to have the opposite sign as the transverse
twist. In the MAD model, we notice the remarkable feature that the photon ring
has the opposite sign compared to the rest of the image, a feature also
discussed in Moscibrodzka et al. (2021) for a different set of MAD
simulations. In fact, we find that photon sub-rings of increasing order
alternate sign. In Figure 14, we break down the image of the final snapshot of
this model into its photon ring components (George Wong, private
communication). To isolate the effect of Faraday conversion mediated by
magnetic twist, Faraday rotation and intrinsic emission are turned off in the
top row, but normal radiative transfer is performed in the bottom row. In
these panels, $n$ corresponds to the number of times the photon orbit has
reached an extremum in the $\theta$ coordinate, counting the number of half-
orbits the photon completes. The total integrated Stokes $V$ for each sub-
image is written on the bottom left. Notice that the $n=0$ direct image has
about a factor of two greater circular polarization when all coefficients are
turned on compared to the pure conversion and twist image. This behaviour
matches expectations from a magnetic field pointed towards the observer, as we
will further explore in §4.4. For the photon ring and its sub-rings, $n\geq
1$, Faraday conversion mediated by magnetic twist completely explains the
signal. Remarkably, the $n=1$ photon ring dominates the circularly polarized
image in this model, even though it contributes to only 16 per cent of the
total (unpolarized) flux, and we confirm this to be generally true for
snapshots in times $7500<GM_{\bullet}/c^{3}<10000$. As has been shown in
previous works, photon rings of increasing order have intensities which
decrease by about an order of magnitude with each half-orbit (Johnson et al.,
2020; Himwich et al., 2020), but the flips in the sign of Stokes V are
unanticipated.

These sign flips can be explained by a combination of symmetries of the
magnetic field, the dominance of its azimuthal component, and features of
parallel transport. We illustrate how this results in a sign flip in the
photon ring in Figure 15. We consider a pole-on viewing angle and a pixel
which appears in the right side of the photon ring (at the intersection of the
arrows). By construction, intensity in the $n=1$ photon ring component
originates from the opposite side of the SMBH, having bent around the back
side (behind the page). The field in the mid-plane is represented by red
arrows, while the field at $z\gg 0$ (towards the observer) is represented by
the orange arrow. The field at $z\gg 0$ is twisted following the rules
described in Figure 5. In the left panel, the magnetic field is toroidally
dominated (as in our models), while in the right panel, the field is radially
dominated.

To determine the magnetic twist along this bent trajectory, we must parallel
transport the emitting field (on the left, moving in the $\bigotimes$
direction) along the geodesic to the converting field (on the right, moving in
the $\bigodot$ direction). When parallel transported around the back of the
SMBH, the azimuthal component of the magnetic field flips sign, but the radial
component does not.222For spinning SMBH, a photon will also move in the
azimuthal direction, but due to the axissymmetry of the problem, this has no
effect on our argument. The relative twist between photons loaded on the left
side of the SMBH and the Faraday converting material on the right side can be
inferred by comparing the parallel transported field line (as shown in blue)
with the field orientations both in the mid-plane (red) and at larger
distances towards the camera (orange). We can see that this brackets the twist
angles $\theta_{\mathrm{min}}$ (at the mid-plane) and $\theta_{\mathrm{max}}$
(at infinity).

Next, recall that Stokes $V$ inherits the sign of Stokes $U$, and Stokes $U$
depends on the relative twist between the emitting field and the converting
field. In the plasma frame, synchrotron emission produces light polarized in
the $+Q$ orientation, but a twist in the magnetic field in the propagation
direction can recast what was once entirely $+Q$ into some parts $\pm U$. Most
importantly, the resultant sign of $U$ flips each time the twist increases by
$90^{\circ}$. For the toroidally dominated field on the left, the twist angle
is always between 90∘ and 180∘. Therefore, the $n=1$ photon ring should always
exhibit the opposite sign of Stokes $V$ compared to the $n=0$ image overall.
Generalizing to $n>1$, emission in odd numbered sub-rings is generated on the
left side of this schematic, while emission in even numbered sub-rings is
generated on the right side. Consequently, odd numbered and even numbered sub-
rings should each share signs.

Figure 15: Schematic explaining the sign flips in the photon ring for a face-
on viewing angle due to magnetic field symmetries and parallel transport.
Here, we consider a pixel on the right side of the photon ring (where the
arrows intersect). The magnetic field in the mid-plane is shown in red, while
the field at $z\gg 0$ (towards the camera) is shown in orange. (These are the
magnetic fields depicted in Figure 5.) Emission in the $n=1$ photon ring is
loaded on the opposite side of the BH, so to determine the line of sight
twist, we parallel transport the emitting field along a photon geodesic around
the SMBH, as illustrated in blue. The sign of Stokes $V$ depends on the angle
that this emitting field makes with the converting field ahead of it, and this
is bracketed by $\theta_{\mathrm{min}}$ and $\theta_{\mathrm{max}}$. For a
toroidally dominated field (left), these angles are always confined to the
quadrant between $90^{\circ}$ and $180^{\circ}$, which guarantees a sign flip
in the $n=1$ ring. This is not the case for a radially dominated field
(right), so the existence of a sign flip depends on exactly where along the
line of sight the Faraday conversion occurs. Faraday rotation can also disrupt
these inequalities and corrupt this signal.

If the field is radially dominated, as shown on the right, then unfortunately
the sign of the photon ring depends on the details of the plasma. Parallel
transported around the SMBH, the magnetic field would intersect the magnetic
field near the mid-plane at an angle $\theta_{\mathrm{min}}<90^{\circ}$, but
would eventually encounter magnetic field where the relative angle
$\theta_{\mathrm{min}}>90^{\circ}$. The resultant sign of the circular
polarization then depends on where exactly the Faraday conversion occurs along
this trajectory.

To summarise, for face-on viewing angles, the sign of circular polarization of
the photon ring can directly encode the handedness (the vertical twist) of the
magnetic field in the approaching helix. If viewed from the opposite
inclination, all sub-images would flip sign, since the handedness of the field
flips across the mid-plane. As long as Faraday rotation is weak, sign flips in
Stokes $V$ are guaranteed in sub-images of increasing sign for toroidally
dominated fields, and may also occur in radially dominated fields depending on
the plasma details. This signal can be explained by generic properties of the
magnetic field geometry, the sensitivity of circular polarization to both the
direction and degree of magnetic twist, and features of parallel transport
within the photon ring. EHT imaging constraints favour clockwise rotation of
material in the accretion flow of M87* (Event Horizon Telescope Collaboration
et al., 2019e). If the accretion flow is produced by a prograde model with
$|B_{\phi}|>|B_{r}|$, then the photon ring should have $V>0$. If the accretion
flow is retrograde, we anticipate additional complications, as we discuss in
§5.3.

If Faraday rotation is strong, it can corrupt this signal in two ways. First,
Faraday rotation along the line of sight can further rotate the plane of
polarization based on the line of sight magnetic field direction, potentially
disrupting the inequalities in Figure 15. Indeed, we caution that for our
Faraday rotation thick SANE model, we only see the signal exactly as described
in this section if Faraday rotation is switched off. Second, strong enough
Faraday rotation can scramble the circular polarization by randomising the
sign of Stokes $U$ that gets converted to $V$, which especially affects photon
ring geodesics, since they have larger than usual path lengths through the
system. Complementary linearly polarized images, especially at multiple
frequencies, would help assess if Faraday rotation in the photon ring is
important.

###  4.4 Unresolved Circular Polarization Measurements

Figure 16: Image-integrated circular polarization fractions as a function of
inclination for our two models. Normal radiative transfer is plotted in blue,
images created without Faraday rotation are plotted in orange, and images
created without Faraday rotation or intrinsic emission are plotting in green.
We include multiple snapshots in this plot, and the filled regions enclose the
5th to 95th percentiles of the values spanned between times $t\in[7500,10000]\
GM_{\bullet}/c^{3}$, while the solid lines mark the median. This reveals that
constancy of the sign of $V/I$ occurs at some, but not all, inclinations.
These models exist in two different regimes of Faraday rotation. In the SANE
model, Faraday rotation is strong enough to significantly depolarize the
circularly polarized image, while in the MAD model Faraday rotation is weak
enough to imprint the magnetic field direction on the sign of $V/I$, pushing
it towards positive values for inclinations $<90^{\circ}$ and towards negative
values for inclinations $>90^{\circ}$. Intrinsic emission imprints the line of
sight direction of the magnetic field in a similar fashion.

The sign of unresolved circular polarization measurements has been proposed to
infer the sense of rotation in quasar engines (Enßlin, 2003). Using our two
models, we probe to what extent the unresolved circular polarization fraction
and its time variability can be used to infer the magnetic field geometry on
event horizon scales. In Figure 16, we compute unresolved circular
polarization fractions for each of our models stepping through inclination
from $5^{\circ}$ to $175^{\circ}$ in $5^{\circ}$ increments. In addition to
sampling different inclinations, we also make separate images for the times
$7500\ GM_{\bullet}/c^{3}$ to $10000\ GM_{\bullet}/c^{3}$, sampled in steps of
$100\ GM_{\bullet}/c^{3}$. For a given model at a given inclination, the
filled region encloses the 5th to 95th percentiles of the range spanned by
these models as a function of time, while the solid line marks the median.
This reveals that constancy of the sign of $V/I$ over time occurs at some, but
not all, inclinations for each model. In blue, we plot the results of normal
radiative transfer (Normal RT), in orange we switch off Faraday rotation (No
FR), and in green we switch off both Faraday rotation and intrinsic emission
of circular polarization (No FR or Intrinsic).

The surprising amount of structure in the circular polarization curves as a
function of inclination reflects the interplay of the three pathways to
generate circular polarization. Even when facing the same side of the disk, it
is possible to obtain both positive and negative circular polarization
fractions. The “Normal RT” curve is not perfectly anti-symmetric about
$90^{\circ}$ because not all physical effects depend on inclination in the
same way.

By comparing the blue “Normal RT” curves with the orange “No FR” curves, we
see two regimes in which Faraday rotation affects the circularly polarized
image. In the MAD model, which has mild values of $\tau_{\rho_{V}}$, Faraday
rotation shifts circular polarization fractions towards positive values for
inclinations $<90^{\circ}$, and towards negative values for inclinations
$>90^{\circ}$. This is because Faraday rotation is sensitive to the sign of
the magnetic field, which is oriented vertically upwards in these models. In
contrast, the SANE model is much more Faraday rotation thick, and hence
Faraday rotation strongly suppresses the circular polarization, which in its
absence could grow to values as large as 10 per cent. This is because, as
shown in Figure 10, the SANE model is much more Faraday conversion thick than
the MAD model. Such a large circular polarization fraction is strongly ruled
out by current constraints on M87* ($\lesssim 0.8$ per cent Kuo et al., 2014;
Goddi et al., 2021). For some inclinations, this internal Faraday rotation can
even change the sign of the unresolved circular polarization fraction.
Finally, intrinsically emitted circular polarization modifies these curves in
a similar direction by adding emission whose sign reflects the line of sight
magnetic field direction, just like a small amount of Faraday rotation.

In summary, we find that the sign of the unresolved circular polarization
fraction cannot easily be used to infer knowledge of the system without
additional constraints. All three pathways to generate circular polarization
are relevant and depend on inclination in different ways. Both models can
exhibit constancy of the sign of the $V/I$ with time at some, but not all,
inclinations.

##  5 Discussion

###  5.1 Dipolar vs. Quadrupolar Fields

The models considered in this work have dipolar magnetic fields, while a
quadrupolar magnetic field is also possible (e.g., Beckwith et al., 2008). A
spun-up quadrupolar magnetic field would not exhibit a sign-flip in the
tangential and radial components of the magnetic fields at the mid-plane.
While the twist of these magnetic field structures would be qualitatively
similar to that of a dipolar field, their field directions would not be.
Hence, if the field were quadrupolar rather than dipolar, the intrinsic
emission and Faraday rotation would be affected, but not the emission due to
Faraday conversion. Since Faraday conversion dominates our images, we expect
that circularly polarized images created with models with quadrupolar fields
should be qualitatively similar to those created with dipolar fields.

###  5.2 Electron-Positron Plasmas

A substantial population of positrons may exist within these accretion flows
via pair production (Begelman et al., 1984; Reynolds et al., 1996a; Wardle et
al., 1998), which would modify the equations of polarized radiative transfer.
As the ratio of positrons to electrons approaches unity, intrinsically emitted
circular polarization and Faraday rotation vanish, while Faraday conversion
doubles (Park & Blackman, 2010; Enßlin et al., 2019; Anantua et al., 2020;
Emami et al., 2021). Consequently, the existence of a positron population
would result in images which would more directly encode the twist in the
magnetic field geometry. In particular, the existence of positrons could
strengthen the signature of magnetic twist in the photon ring described in
§4.3.2.

###  5.3 Retrograde Models

In a retrograde model, frame dragging operates in the opposite direction as
the disk angular momentum, which produces more complicated magnetic field
geometries than sketched in Figure 5. In Figure 17, we plot the magnetic field
decompositions of two retrograde models as in Figure 6. In this plot, the
$z$-axis remains aligned with both the vertical magnetic field direction and
the disk angular momentum, but is now anti-aligned with the angular momentum
of the SMBH. The magnetic field in the funnel region is twisted along with the
SMBH angular momentum. In the MAD model, the field outside this region is
twisted along with the disk, producing opposite twist orientations and
directions of the toroidal field. Meanwhile, in the SANE model, we find that
the azimuthal field direction only consistently follows the disk angular
momentum in a thin shell around the funnel, beyond which the field becomes
more turbulent, and in this model in alignment with the SMBH angular momentum
again. We verify that this unexpected behaviour persists throughout times
$t\in[7500,10000]\ GM_{\bullet}/c^{3}$. As a result, we expect that the
imprint of this structure on circular polarization images would likely be more
sensitive to the details of the plasma properties—where these transitions in
magnetic field orientation occur, where the emission occurs, and where the
Faraday effects occur.

Interestingly, the asymmetry of the image of M87* combined with its known jet
orientation has constrained the motion of plasma on the sky as clockwise, and
the orientation of the SMBH spin direction as away from us (see Event Horizon
Telescope Collaboration et al., 2019e, for an extended discussion). If we can
assume that the evacuated inner funnel region does not contribute much to the
radiative transfer, then it is the angular momentum of the disk in the
boundary region that matters when determining the twist, rather than that of
the SMBH. Consequently, if M87*’s disk is prograde, we expect its photon ring
to exhibit positive $V$, while if the disk is retrograde, we expect negative
$V$ instead. Future studies thoroughly exploring a wider variety of GRMHD and
radiative transfer models would be useful in determining whether or not the
sign of the circular polarization of the photon ring is a clean signature of
prograde versus retrograde accretion.

Figure 17: Magnetic field decompositions as in Figure 6, but for retrograde
models. In the funnel region, the magnetic field aligns with the SMBH. Beyond
this region, in the MAD model, the field neatly switches azimuthal direction
to align with the disk instead. However, in the SANE model, the field lines
only align with the disk angular momentum in a narrow shell around the funnel.
These additional features may produce more complicated circular polarization
images depending on the details of how they are illuminated.

###  5.4 Lower Frequencies and Larger Spatial Scales

Farther from the horizon, it is more straightforward to infer magnetic field
properties free from the complications of event horizon scale magnetic field
structure and light bending. If measurable, the combination of a rotation
measure gradient across the jet, the pitch angle inferred from linear
polarization ticks, and the sign of circular polarization can be used to
determine the directions of both the toroidal and poloidal field components
(Gabuzda et al., 2008; Gabuzda, 2018a). Linear polarization maps exist for
M87* at 43 GHz with sub-milliarcsecond resolution, and a full rotation measure
analysis of this dataset is forthcoming (Walker et al., 2018). This
information could help us associate regions of a resolved circularly polarized
image in the millimetre to the photon ring and the different sides of the mid-
plane.

###  5.5 Non-thermal Electron Distributions

This study was performed using only a single model of the electron
distribution function (Mościbrodzka et al., 2016), distributed thermally.
Meanwhile, the SEDs of these sources imply the existence of non-thermal
electron populations to produce flux particularly in the near infrared (Özel
et al., 2000; Yuan et al., 2003). Images including a non-thermal component to
the electron distribution function in the form of a high-energy tail exhibit
wider halos of emission than purely thermal models (Özel et al., 2000; Mao et
al., 2017). Since we find that images of circular polarization on large scales
are dominated by intrinsic emission, we hypothesise that the intrinsic
emission signatures discussed in §4.2 may be strengthened. For face-on
inclinations, this would preferentially add circular polarization with the
line of sight sign of the magnetic field, while for edge-on inclinations, this
would strengthen the “four quadrants” pattern. We expect that additional
Faraday effects would be weaker than the additional intrinsic emission, since
Faraday rotation and conversion are inefficient for high-energy electrons.
Light bending complications should also be weaker for a diffuse halo
component. An additional non-thermal component at lower energies would
increase Faraday effects, however. Overall, the effect of non-thermal
electrons on circular polarization merits further study, and tractable
prescriptions have been developed for the electron distribution function as a
function of plasma $\beta$ and magnetization $\sigma$ that can be explored
(Ball et al., 2018; Davelaar et al., 2018; Davelaar et al., 2019).

##  6 Conclusions

Using two GRMHD simulations of prograde accretion flows, we have studied the
magnetic field structure and its imprint on circular polarization images. Our
results are summarised as follows:

  * •

First, we explore the line of sight direction and twist of the magnetic field
in these models, which become imprinted on the circular polarization signal.
The handedness of the magnetic twist flips across the mid-plane. In addition,
we find that sign of the magnetic twist is different depending on whether the
helix is viewed at edge-on versus pole-on viewing angles.

  * •

Intrinsically emitted circular polarization carries the sign of the line of
sight direction of the magnetic field. For edge-on viewing angles, this
results in a “four quadrants” pattern, also reported by Tsunetoe et al.
(2020b). For face-on viewing angles, this mainly traces the vertical magnetic
field direction, but is also sensitive to the radial component of the magnetic
field behind the BH due to the lensing of photon orbits. Intrinsic emission
produces relatively diffuse emission on large scales in these models.

  * •

Circular polarization from Faraday conversion inherits the sign of Stokes $U$,
which we find is sensitive to the line of sight twist in the magnetic field.
For edge-on viewing angles, we expect a sign flip across the mid-plane due to
the switch in handedness of the magnetic twist. For pole-on viewing angles, we
find that alternating sub-images in the photon ring can flip sign. This is due
to parallel transport and symmetries of the magnetic field.

  * •

As discussed in previous works, a mild amount of Faraday rotation will imprint
the line of sight direction of the magnetic field on the circular polarization
(Tsunetoe et al., 2020a, b). However, too large an amount of Faraday rotation
will scramble the circular polarization, even if the model is “thin” to
Faraday conversion. We notice the former in our MAD model, and the latter in
our SANE model.

  * •

The circular polarization fraction and sign changes non-trivially with the
inclination due to the complicated magnetic field geometry, different
competing physical processes, and light bending. For a given model, it is
possible to obtain either sign of circular polarization while viewing the same
side of the disk.

This work represents an early step in decomposing and understanding circularly
polarized images on event horizon scales. In future work, it will be useful to
explore to what extent our findings generalise using different GRMHD and
radiative transfer models. In particular, we plan on expanding our analysis to
study the impact of non-thermal electron distributions and positron
populations.

##  7 Acknowledgements

We thank Charles Gammie, George Wong, and Ben S. Prather for support using
their GRMHD simulations, ipole, and VisIt. We also thank Michael Johnson,
Daniel Palumbo, Elizabeth Himwich, and Zachary Gelles for many fruitful and
invigorating discussions about polarimetry, general relativity, and magnetic
fields.

This material is based upon work supported by the National Science Foundation
under Grant No. OISE 1743747. Computations at the Black Hole Initiative (BHI)
were enabled by grants from the Gordon and Betty Moore Foundation and the John
Templeton Foundation.

##  8 Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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##  Appendix A Confirming Vertical Twist in our GRMHD Models

Figure 18: By examining the angle of the magnetic field projected onto the
$xy$-plane, we can affirm the sense of vertical twist illustrated in Figure 5.
Here, an angle of 45∘ implies that $|B_{\phi}|=|B_{r}|$, while an angle of 90∘
implies that $|B_{\phi}|\gg|B_{r}|$. At least in the funnel region, this angle
increases as $|z|$ increases. This is not true in the disk of the turbulent
SANE, where indeed our time-averaged $V$ image reveals weak edge-on
polarization.

In Figure 18, we confirm the sense of “vertical twist” sketched in Figure 5.
Ignoring the vertical component, we plot the angle made by the magnetic field
in the $xy$ plane, $\arctan{|B_{\phi}|/|B_{r}|}$. An angle of $45^{\circ}$
implies that the radial and azimuthal fields are equal in magnitude, while an
angle of $90^{\circ}$ implies that the toroidal field is much stronger. At
fixed $r$, this angle increases as one moves away from the mid-plane,
resulting in the vertical twist shown in Figure 5. However, this is only clear
in the funnel region of the SANE model, outside of which the field remains
consistently toroidally dominated and turbulent. The vertical twist can also
be inferred by close inspection of the pole-on viewing angles in Figure 7. The
red curves that are anchored around the BH, which tend to extend to larger
$|z|$, sweep more circular arcs than the blue curves anchored in the disk.

##  Appendix B Emissivity and Rotativity in Representative Pixels

Figure 19: Frame-invariant emissivities (upper panels) and rotativities (lower
panels) for representative lines of sight marked with white circles in Figure
10. We find that both emission and Faraday conversion are enclosed well within
the outer integration radius of $20\ GM_{\bullet}/c^{2}$, although Faraday
rotation continues out to large radius at $90^{\circ}$ inclinations.

Here, we examine representative pixels in each of the models during their
final snapshot, marked with white circles in Figure 10. Along each of these
lines of sight, we plot frame-invariant emissivities ($j/\nu^{2}$; upper
panels) and rotativities ($\nu\rho$; lower panels) as a function of radius in
Kerr-Schild coordinates, $r_{\mathrm{KS}}$ (see Mościbrodzka & Gammie, 2018,
for more details). In the rotativity panels, we omit sections of the geodesic
where less than 1 per cent of the final emission along the line of sight has
been produced. This allows us to ignore Faraday rotation and conversion in
areas behind the bulk of the emission that do not affect the observables. This
analysis reveals that both the emission and Faraday conversion are enclosed
within the outer integration radius of $20\ GM_{\bullet}/c^{2}$ chosen for the
radiative transfer. For $90^{\circ}$ inclinations, Faraday rotation continues
to large radius, as also shown in Ricarte et al. (2020), but this does not
affect the circular polarization.