First M87 Event Horizon Telescope Results. VIII.
Magnetic Field Structure near The Event Horizon
The Event Horizon Telescope Collaboration
(
See the end matter for the full list of authors.
)
Abstract
Event Horizon Telescope
(
EHT
)
observations at 230 GHz have now imaged polarized emission around the
supermassive black hole in M87 on event-horizon scales. This polarized synchrotron radiation probes the structure
of magnetic
fi
elds and the plasma properties near the black hole. Here we compare the resolved polarization
structure observed by the EHT, along with simultaneous unresolved observations with the Atacama Large
Millimeter
/
submillimeter Array, to expectations from theoretical models. The low fractional linear polarization in
the resolved image suggests that the polarization is scrambled on scales smaller than the EHT beam, which we
attribute to Faraday rotation internal to the emission region. We estimate the average density
n
e
∼
10
4
–
7
cm
−
3
,
magnetic
fi
eld strength
B
∼
1
–
30 G, and electron temperature
T
e
∼
(
1
–
12
)
×
10
10
K of the radiating plasma in a
simple one-zone emission model. We show that the net azimuthal linear polarization pattern may result from
organized, poloidal magnetic
fi
elds in the emission region. In a quantitative comparison with a large library of
simulated polarimetric images from general relativistic magnetohydrodynamic
(
GRMHD
)
simulations, we identify
a subset of physical models that can explain critical features of the polarimetric EHT observations while producing
a relativistic jet of suf
fi
cient power. The consistent GRMHD models are all of magnetically arrested accretion
disks, where near-horizon magnetic
fi
elds are dynamically important. We use the models to infer a mass accretion
rate onto the black hole in M87 of
(
3
–
20
)
×
10
−
4
M
e
yr
−
1
.
Uni
fi
ed Astronomy Thesaurus concepts:
Accretion
(
14
)
;
Black holes
(
162
)
;
Event horizons
(
479
)
;
Jets
(
870
)
;
Kerr
black holes
(
886
)
;
Magnetic
fi
elds
(
994
)
;
Magnetohydrodynamics
(
1964
)
;
Plasma astrophysics
(
1261
)
;
Polarimetry
(
1278
)
;
Radiative transfer
(
1335
)
;
Radio jets
(
1347
)
;
Relativistic jets
(
1390
)
1. Introduction
The Event Horizon Telescope
(
EHT
)
Collaboration has
recently published total intensity images of event-horizon-scale
emission around the supermassive black hole in the core of the
M87 galaxy
(
M87
*
; Event Horizon Telescope Collaboration et al.
2019a
,
2019b
,
2019c
,
2019d
, hereafter
EHTC I
,
EHTC II
,
EHTC III
,
EHTC IV
)
. The data reveal a 42
±
3
μ
as diameter ring-
like structure that is broadly consistent with the shadow of a black
hole as predicted by Einstein
’
s Theory of General Relativity
(
Event Horizon Telescope Collaboration et al.
2019e
,
2019f
;
hereafter
EHTC V
,
EHTC VI
)
. The brightness temperature of the
ring at 230 GHz
(
10
10
K
)
is naturally explained by synchrotron
emission from relativistic electrons gyrating around magnetic
fi
eld
lines. The ring brightness asymmetry results from light bending
and Doppler beaming due to relativistic rotation of the matter
around the black hole.
M87
*
is best known for launching a kpc-scale FR-I type
relativistic jet, whose kinetic power is estimated to be
∼
10
42
–
44
erg s
−
1
(
e.g., Stawarz et al.
2006
; de Gasperin et al.
2012
)
. The structure of the relativistic jet has been resolved and
studied at radio to X-ray wavelengths
(
e.g., Di Matteo et al.
2003
; Harris et al.
2009
; Kim et al.
2018
; Walker et al.
2018
)
.
The published EHT image of M87
*
together with multi-
wavelength observations are consistent with the picture that the
supermassive black hole in M87 is surrounded by a
relativistically hot, magnetized plasma
(
Rees et al.
1982
;
Narayan & Yi
1995
; Narayan et al.
1995
; Yuan &
Narayan
2014
; Reynolds et al.
1996
; Yuan et al.
2002
;Di
Matteo et al.
2003
)
. However, it is not clear whether the
compact ring emission is produced by plasma that is in
fl
owing
(
in a thick accretion
fl
ow
)
, out
fl
owing
(
at the jet base or in a
wind
)
, or both. Furthermore, the total intensity EHT observa-
tions also could not constrain the structure of magnetic
fi
elds in
the observed emission region. In order to
fi
nd out which
physical scenario is realized in M87
*
, additional information is
necessary.
Event Horizon Telescope Collaboration et al.
(
2021
,
hereafter
EHTC VII
)
reports new results from the polarimetric
EHT 2017 observations of M87
*
. The polarimetric images of
M87
*
are reproduced in Figure
1
. These images reveal that a
signi
fi
cant fraction of the ring emission is linearly polarized, as
expected for synchrotron radiation. The EHT polarimetric
measurements are consistent with unresolved observations of
the radio core at the same frequency with the Submillimeter
Array
(
SMA; Kuo et al.
2014
)
and the Atacama Large
Millimeter
/
submillimeter Array
(
ALMA; Goddi et al.
2021
)
.
They also provide a detailed view of the polarized emission
region on event-horizon scales near the black hole. Polarized
synchrotron radiation traces the underlying magnetic
fi
eld
The Astrophysical Journal Letters,
910:L13
(
43pp
)
, 2021 March 20
https:
//
doi.org
/
10.3847
/
2041-8213
/
abe4de
© 2021. The Author
(
s
)
. Published by the American Astronomical Society.
Received 2020 December 2; revised 2021 February 3; accepted 2021 February 8; published 2021 March 24
126
NASA Hubble Fellowship Program, Einstein Fellow.
127
EACOA Fellow.
128
UKRI Stephen Hawking Fellow.
Original content from this work may be used under the terms
of the
Creative Commons Attribution 4.0 licence
. Any further
distribution of this work must maintain attribution to the author
(
s
)
and the title
of the work, journal citation and DOI.
1
con
fi
guration and magnetized plasma properties along the line
of sight
(
Bromley et al.
2001
; Broderick & Loeb
2009
;
Mo
ś
cibrodzka et al.
2017
)
. These polarimetric measurements
allow us to carry out new
quantitative tests
of horizon-scale
scenarios for accretion and jet launching around the
M87
*
black hole. In this Letter we present our interpretation
of the
EHTC VII
resolved polarimetric images of the ring
in M87
*
.
Our analysis is presented as follows. In Section
2
we report
polarimetric constraints from M87
*
EHT 2017 and supplemen-
tal observations, and argue that they can be used for scienti
fi
c
interpretation, focusing on several key diagnostics of the degree
of order and spatial pattern of the polarization map. In
Section
3
we present one-zone estimates of the properties of
the synchrotron-emitting plasma. In the transrelativistic para-
meter regime relevant for the M87 core, a full calculation of
polarized radiative transfer using a realistic description of the
plasma properties is essential. To that end, in Section
4
we
describe a set of numerical simulations of magnetized accretion
fl
ows that we use to compare with our set of observables. In
Section
5
we show that only a small subset of the simulation
parameter space is consistent with the observables. The favored
simulations feature dynamically important magnetic
fi
elds. We
discuss limitations of our models in Section
6
, and discuss how
future EHT observations can further constrain the magnetic
fi
eld structure and plasma properties near the supermassive
black hole
’
s event horizon in Section
7
.
2. Polarimetric Observations and Their Interpretation
2.1. Conventions in Observations and Models
Throughout this Letter we use the following de
fi
nitions and
conventions for polarimetric quantities, following the Interna-
tional Astronomical Union
(
IAU
)
de
fi
nitions of the Stokes
parameters
(
)
,,,
(
Hamaker & Bregman
1996
;
Smirnov
2011
)
. The complex linear polarization
fi
eld
is
()
=+
i
.1
Then, the electric-vector position angle
(
EVPA
)
is de
fi
ned as
()
()
º
EVPA
1
2
arg .
2
The EVPA is measured east of north on the sky. Therefore,
positive
is aligned with the north
–
south direction and
negative
with the east
–
west direction. Positive
is at
a
+
45 deg angle with respect to the positive
axis
(
in the
northeast
–
southwest direction
)
. Positive Stokes
indicates
right-handed circular polarization, meaning in our convention
that the electric
fi
eld vector of the incoming electromagnetic
wave is rotating counter-clockwise as seen by the observer. In
the synchrotron radiation models that we consider, a positive
value of emitted Stokes
is associated with an angle
θ
B
between the wavevector
k
μ
and magnetic
fi
eld
b
μ
as measured
in the frame of the emitting plasma in the range
θ
B
ä
[
0, 0.5
π
]
.
Negative
corresponds to
θ
B
ä
[
0.5
π
,
π
]
.
The linear and circular polarization fractions at a point in the
image are de
fi
ned as
∣∣
∣∣
()
º
m
,3
∣∣
∣∣
()
º
v
.4
We also de
fi
ne the rotation measure between two wavelengths
λ
1
and
λ
2
()
( )
()
ll
ll
º
-
-
RM
EVPA
EVPA
.5
12
1
2
2
2
Unresolved observations measure the
net
(
image-integrated
)
polarization fractions
()()
∣∣
()
åå
å
=
+
m
,6
i
i
i
i
i
i
net
22
()
å
å
=
v
,7
i
i
i
i
net
Figure 1.
Top panel: 2017 April 11
fi
ducial polarimetric image of
M87
*
from
EHTC VII
. The gray scale encodes the total intensity, and ticks
illustrate the degree and direction of linear polarization. The tick color indicates
the amplitude of the fractional linear polarization, the tick length is proportional
to
∣
∣
º+
22
, and the tick direction indicates the electric-vector
position angle
(
EVPA
)
. Polarization ticks are displayed only in regions where
>
10%
max
and
∣
∣∣∣
>
20%
max
. Bottom row: polarimetric images of
M87
*
taken on different days.
2
The Astrophysical Journal Letters,
910:L13
(
43pp
)
, 2021 March 20
EHT Collaboration et al.
where the sums are over all pixels
i
in the resolved image. In
addition to the signed circular polarization fraction
v
net
, we also
frequently consider the absolute value
|
v
net
|
, as circular
polarization measurements of the M87
*
core at 230 GHz do
not constrain its sign
(
Goddi et al.
2021
)
.
In describing the
resolved
linear polarization in EHT images,
we de
fi
ne the image-average linear polarization fraction,
weighted by the total intensity of each image pixel, as
∣∣
()
å
å
áñ=
+
m
.8
i
ii
i
i
22
Note that
〈
|
m
|
〉
depends on the imaging resolution
(
beam size
)
,
while
|
m
|
net
is the usual unresolved linear polarization fraction
and does not depend on resolution.
2.2. Unresolved Polarization and Rotation Measure
Measurements toward M87
’
s Core from ALMA
As part of the EHT 2017 observation campaign, we obtained
ALMA array measurements of the unresolved, net, near
230 GHz, polarimetric properties of M87
’
s core and jet on
2017 April 5, 6, 10, and 11
(
hereafter these observations are
referred to as ALMA-only observations
)
. ALMA-only mea-
surements are given at
ν
=
221 GHz, a central frequency of
ALMA Band 6, which has four spectral windows, each
centered at 213, 215, 227, and 229 GHz. These results, along
with details on the observations and data calibration, are
presented in Goddi et al.
(
2021
)
; we summarize them here in
Table
1
. From the ALMA-only data, the net linear polarization
fraction
(
Equation
(
6
))
of the core is
|
m
|
net
;
2.7%. The data
also provide an upper limit on the net circular polarization
fraction
(
Equation
(
7
))
of the core of
|
v
|
net
0.3%, with a
magnitude and sign that vary over the four observed epochs.
Goddi et al.
(
2021
)
also measured an EVPA that varies with
wavelength across the ALMA band; the slope of EVPA with
wavelength is consistent with EVPA
∝
λ
2
, as expected for
Faraday rotation. The inferred rotation measure
(
Equation
(
5
)
,
for frequencies
/
wavelengths in ALMA Band 6
)
is also time
variable and changes sign between 2017 April 5 and 11, with a
maximum magnitude
|
RM
|
;
1.5
×
10
5
rad m
−
2
.
The ALMA-only measurements include extended
∼
arcse-
cond scale structures that are entirely resolved out of the EHT
maps of M87
’
s core region. As a result, the total 221 GHz
fl
ux
density of M87
*
measured by ALMA alone is a factor of
;
2
larger than that captured by the resolved EHT images
(
see
also
EHTC IV
)
. For that reason, we adopt a more conservative
upper limit of
|
v
|
net
<
0.8%, which would account for the case
where the large-scale emission is not circularly polarized.
2.3. Spatially Resolved Linear Polarization of M87
’
s Core in
EHT 2017 Data
The resolved polarimetric images of the M87 core reported
in
EHTC VII
display robust features between different image-
reconstruction algorithms and across four days of observations
(
2017 April 5, 6, 10, and 11
)
.At20
μ
as resolution, those
images consistently show a region of the highest linear
polarized intensity in the southwest portion of the ring, with
a fractional linear polarization
|
m
|
30% at its maximum. The
image-average linear polarization fraction takes on values
5.7%
〈
|
m
|
〉
10.7% across the different observation days
and image-reconstruction techniques. The range of the image-
integrated net polarization fraction is 1.0%
|
m
|
net
3.7%
(
see
EHTC VII
, their Table 2
)
. Because polarized emission
outside the EHT
fi
eld of view but inside the ALMA-only core
is unconstrained, we adopt the EHT
|
m
|
net
range when
evaluating models.
The EHT images on all four observing days reveal a
characteristic azimuthal pattern of the EVPA angle around the
emission ring. To quantify this pattern across the image, we
use a decomposition of the complex linear polarization
=+
i
into azimuthal modes with complex coef
fi
cients
β
m
(
Palumbo et al.
2020
, hereafter
PWP
)
. For a polarization
fi
eld in the image plane given in polar coordinates
(
ρ
,
j
)
, the
β
m
coef
fi
cients are
()
()
òò
brjrjr
=
r
rp
j
-
I
edd
1
,,9
m
im
ann
0
2
min
max
where
I
ann
is the Stokes
fl
ux density contained inside the
annulus set by the limiting radii
r
mi
n
and
r
max
. We take
r
=
0
min
and
r
max
to be large enough to include the entire EHT
image.
Within the library of polarized images from general
relativistic magnetohydrodynamic
(
GRMHD
)
simulations pre-
sented in
EHTC V
,
PWP
found that the
m
=
2 coef
fi
cient,
β
2
,
was the most discriminating in identifying the underlying
magnetized accretion model. The phase of
β
2
maps well onto
the qualitative behavior expected of polarization maps with
idealized magnetic
fi
eld con
fi
gurations. In our convention,
radial EVPA patterns have positive real
β
2
(
∠
β
2
=
0 deg
)
,
azimuthal EVPA patterns have negative real
β
2
(
∠
β
2
=
180 deg
)
, and left-
(
right-
)
handed spiral patterns have positive
(
negative
)
pure imaginary
β
2
(
∠
β
2
=
90 deg and
−
90 deg,
respectively
)
. These idealized EVPA pattern con
fi
gurations and
their corresponding
β
2
coef
fi
cients are summarized in
Appendix
A
and Figure 1 of
PWP
. The measured range of
the complex
β
2
coef
fi
cient across the different image-
reconstruction methods and observing days reported in
EHTC
VII
, their Table 2, is 0.04
|
β
2
|
0.07 for the amplitude and
[]
b
-
-
163 deg arg
129 deg
2
for the phase.
Appendix
A
demonstrates that the information in the
β
2
coef
fi
cient can be obtained in the visibility domain using
measurements of the linear polarization
E
−
(
gradient
)
and
B
−
(
curl
)
modes of the polarization
fi
eld
(
e.g., Kamionkowski
& Kovetz
2016
)
. Trends in
β
2
metric computed across the
GRMHD image library
(
Section
4
)
can be obtained in the
visibility domain using only
E-
and
B-
mode measurements
taken on EHT 2017 baselines, as long as the visibilities are
accurately phase calibrated. Because accurate phase calibration
of EHT data is non-trivial and requires fully modeling the
polarized source structure, we use image-domain comparisons
Table 1
ALMA-only Measurements of M87
*
’
s Unresolved Polarization Properties at
ν
=
221 GHz
(
Goddi et al.
2021
)
Day
F
|
m
|
net
|
v
|
net
RM
(
Jy
)(
%
)(
%
)(
10
5
rad m
−
2
)
April 5
1.28
±
0.13
2.42
±
.03
0.2
(
0.6
±
0.3
)
April 6
1.31
±
0.13
2.16
±
.03
0.3
(
1.5
±
0.3
)
April 10
1.33
±
0.13
2.73
±
.03
0.3
(
−
0.2
±
0.2
)
April 11
1.34
±
0.13
2.71
±
.03
0.4
(
−
0.4
±
0.2
)
3
The Astrophysical Journal Letters,
910:L13
(
43pp
)
, 2021 March 20
EHT Collaboration et al.
to the reconstructions presented in
EHTC VII
for the
constraints in this Letter.
As in total intensity, both the unresolved and resolved
polarimetric properties of the 230 GHz M87
*
image changed
over the week between 2017 April 5 and April 11. Notably, the
integrated EVPA in the EHT image changes by
≈
30
–
40 deg
(
while the ALMA-only EVPA changes by
10 deg
)
. We will
not interpret this variability further in this work; however,
Section
6
discusses expectations for time variability from
viable simulation models. The observational ranges of the key
parameters that we use in comparing theoretical models to data
in Section
5
—
namely
|
m
|
net
,
|
v
|
net
,
〈
|
m
|
〉
, and
β
2
amplitude and
phase
—
are summarized in Table
2
.
2.4. External and Internal Faraday Rotation
Faraday rotation in a uniform plasma with rotation measure
(
RM
)
rotates the EVPA away from its intrinsic value EVPA
0
according to Equation
(
5
)
. The change in EVPA from its
intrinsic value at 230 GHz
(
λ
;
1.3 mm
)
is
⎜⎟
⎛
⎝
⎞
⎠
()
D
-
EVPA 9.7
RM
10 rad m
deg.
10
52
Polarized light rays passing through a uniform medium are
subject to the same RM. The net source polarization angle is
then coherently rotated away from its intrinsic value without
any depolarization. This scenario of
“
external
”
Faraday rotation
has been used to infer the mass accretion rate for sources where
an RM is measured or constrained
(
e.g., Bower et al.
2003
;
Marrone et al.
2006
,
2007
; Kuo et al.
2014
)
, by assuming that
the observed radiation passes through the bulk of the accretion
fl
ow. Because relativistic electrons suppress the Faraday
rotation coef
fi
cient as
∝
T
1
e
2
(
e.g., Jones & Hardee
1979
)
,
these models assume that the RM is produced outside the
emission region at the radius where
Θ
e
=
kT
e
/
m
e
c
2
=
1, usually
r
∼
100
r
g
(
where
r
g
=
GM
/
c
2
is the gravitational radius
)
.
However, in accreting systems like M87
*
, it is unclear
whether this external Faraday rotation model is a good
approximation. As we estimate below, one-zone emission
models of M87
*
predict substantial RM within the emission
region itself at radii
r
5
r
g
. At its low viewing inclination, the
observed polarized radiation emitted near the horizon may not
pass through the bulk of the high-density, infalling gas.
Therefore,
“
internal
”
Faraday rotation, which can depolarize
the emission as well as rotate the EVPA
(
Burn
1966
)
, is also an
important effect to consider.
The observed
;
10% linear polarization of the ring at the EHT
scale of
∼
20
μ
as is much lower than the intrinsic synchrotron
polarization fraction
70% expected locally. This could result from
synchrotron self-absorption of th
e emitted radiation, but one-zone
estimates and theoretical models
(
e.g.,
EHTC V
, and references
therein
)
suggest that the 230 GHz emission is mostly optically thin.
It is more likely that the observed depolarization of the resolved
emission could be the result of polarization structure that is
scrambled at resolutions
fi
ner than the EHT beam. Turbulent
magnetic
fi
elds and Faraday rotation internal to the emission region
could produce this scrambling. In Section
4.3
we show that
turbulence in GRMHD models alone is insuf
fi
cient to produce this
level of depolarization. Signi
fi
cant internal Faraday rotation of
polarization vectors on different rays by different amounts can
produce a suf
fi
ciently scrambled image that is depolarized when
spatially averaged over a telescope resolution element
(
beam,e.g.,
Burn
1966
; Agol
2000
; Quataert & Gruzinov
2000
; Beckert &
Falcke
2002
; Ruszkowski & Begelman
2002
; Ballantyne et al.
2007
)
.
From the simultaneous ALMA-only M87
*
observations, the
RM implied by changes in the EVPA across the ALMA band is
|
RM
|
1.5
×
10
5
rad m
−
2
. These values are consistent with,
but much more constraining than, the range determined from
past SMA observations
(
−
3.4
–
7
×
10
5
rad m
−
2
, Kuo et al.
2014
)
. The ALMA-only EVPA difference varies by order unity
in magnitude and sign over the observing campaign, and
includes a large
fl
ux contribution from extended emission not
captured by EHT 2017 imaging
(
EHTC IV
)
. Using a two-
component model, Goddi et al.
(
2021
)
show that the RM
toward the core emission in the EHT
fi
eld of view could exceed
the RM computed from the ALMA-only data, with allowed
values as large as
|
RM
|
10
6
rad m
−
2
. Because of this
uncertainty, we do not use the observed RM as an observa-
tional constraint in our analysis. We account for uncertainty
related to the observed time variability by using reconstructed
polarized EHT images from both 2017 April 5 and 11 to de
fi
ne
the acceptable ranges
(
see Table
2
)
of the observational
parameters used to score theoretical models in Section
5
.
3. Estimates and Phenomenological Models
In this Section, we take a
fi
rst look at the importance of
internal Faraday rotation and magnetic
fi
eld structure in
determining the characteristics of the 230 GHz EHT image. In
Section
3.1
we obtain order-of-magnitude estimates of the
plasma properties in M87
*
by interpreting the observed
depolarization as entirely due to the effect of internal Faraday
rotation on small scales. In Section
3.2
we explore the effects of
different idealized magnetic
fi
eld con
fi
gurations on the
observed polarization pattern from plasma orbiting a black
hole in the absence of Faraday effects.
3.1. Parameter Estimates from One-zone Models
Based on a one-zone isothermal sphere model,
EHTC V
derived order-of-magnitude estimates of the plasma number
density
n
e
and magnetic
fi
eld strength
B
in the emitting region
around M87
*
as constrained by the Stokes
image
’
s bright-
ness, size, and total
fl
ux density:
()
́
-
n
2.9 10 cm ,
11
e
43
Table 2
Parameter Ranges for the Quantities Used in Scoring Theoretical Models in this
Letter
Parameter
Min
Max
|
m
|
net
1.0%
3.7%
|
v
|
net
0
0.8%
〈
|
m
|
〉
5.7%
10.7%
|
β
2
|
0.04
0.07
∠
β
2
−
163 deg
−
129 deg
Note.
The ranges for
|
m
|
net
,
〈
|
m
|
〉
, and
β
2
were taken from
EHTC VII
Table 2.
These ranges represent the minimum
−
1
σ
error bound and maximum
+
1
σ
error bound across
fi
ve different image-reconstruction methods, and incorpo-
rate both statistical uncertainty in the polarimetric image reconstruction and
systematic uncertainty in the assumptions made by different reconstruction
algorithms. The upper limit on
|
v
|
net
was taken as
;
2
×
the maximum value
found by Goddi et al.
(
2021
)
.
4
The Astrophysical Journal Letters,
910:L13
(
43pp
)
, 2021 March 20
EHT Collaboration et al.
()
B
4.9 G.
12
In this model, the emission radius was assumed to be
r
;
5
r
g
,
and the electron temperature was assumed to be
T
e
=
6.25
×
10
10
K, based on the observed brightness temperature
of the EHT image. This temperature corresponds to
Θ
e
=
kT
e
/
m
e
c
2
=
10.5, so the emitting electrons have moderately
relativistic mean Lorentz factors
̄
g
»Q»
330
e
. The angle
between the magnetic
fi
eld and line of sight is set at
θ
=
π
/
3.
This model ignores several physical effects that are included in
more sophisticated models and simulations and which are
necessary to fully describe the emission from M87
*
. The
plasma is considered to be at rest and so there is no Doppler
(
de
)
boosting of the emitted intensity from relativistic
fl
ow
velocities. Redshift from the gravitational potential of the black
hole is also not included.
Given
n
e
,
B
, and
T
e
, we can estimate the strength of the
Faraday rotation effect at 230 GHz quanti
fi
ed by the optical
depth to Faraday rotation
t
r
V
:
⎛
⎝
⎜
⎞
⎠
⎟
()
tr
» ́
r
r
r
r
5.2
5
,13
V
g
V
where
ρ
V
is the Faraday rotation coef
fi
cient
(
e.g., Jones &
Hardee
1979
)
. For emission entirely behind an external
Faraday screen,
t
r
V
is related to the rotation measure RM via
tl
=
r
2RM
2
V
, which follows from the radiative transfer
equations for spherical Stokes parameters in the absence of
other effects
(
see e.g., Appendix A of Mo
ś
cibrodzka et al.
2017
)
and the fact that
ρ
V
∝
λ
2
.
Our estimated
t
r
V
indicates that Faraday rotation internal to
the emission region is an important effect and could thus
explain the depolarization observed in M87
*
. Faraday effects
are even more important for the case of polarized light emitted
by relativistic electrons that travel through a dense, colder
accretion
fl
ow
(
e.g., Mo
ś
cibrodzka et al.
2017
; Ricarte et al.
2020
)
. In addition, for the same parameters, Faraday conver-
sion of linear to circular polarization may also be important
(
t
r
0.5
Q
)
, while self-absorption is weak
(
τ
I
;
0.05
)
. Requir-
ing an internal Faraday optical depth
tp
>
r
2
V
(
large enough to
produce signi
fi
cant depolarization
)
provides an additional
constraint on one-zone models independent of those used
in
EHTC V
, which
fi
xed the electron temperature at an
assumed value. Assuming
tp
>
r
2
V
allows us to break the
degeneracy between magnetic
fi
eld strength, electron temper-
ature, and plasma number density.
Hence, we consider the same model as in
EHTC V
at several
different values of
β
e
=
8
π
n
e
kT
e
/
B
2
, constrained by the
requirement that the Faraday optical depth
tp
>
r
2
V
.Tobe
consistent with the 230 GHz EHT data, we also require that the
observed image have a total
fl
ux
F
ν
between 0.2 and 1.2 Jy, and
that the model has a maximum total intensity optical depth
τ
I
<
1. Figure
2
shows what constraints these requirements put
on the electron number density
n
e
and the dimensionless
electron temperature
Θ
e
at three different values of
β
e
. For
values of 0.01
<
β
e
<
100, in this simple model the electron
temperature is constrained to lie in a mildly relativistic regime
2
Θ
e
20
(
10
10
<
T
e
<
1.2
×
10
11
K
)
, and the magnetic
fi
eld
strength is 1
B
30 G. The number density of the emitting
electrons depends more sensitively on the assumed value of
β
e
,
taking on values between 10
4
cm
−
3
and 10
7
cm
−
3
.
The one-zone model estimates suggest that both the total
intensity and polarized emission can be produced in a mildly
relativistic plasma in a magnetic
fi
eld of relatively low strength
B
30 G. For higher values of
B
, the electron temperature
would be too small to explain the observed maximum
brightness temperature
(
;
10
10
K
)
in the M87
*
EHT image
(
EHTC IV
)
. Very high values of
B
are independently
disfavored by the small degree of circular polarization
|
v
|
net
1% seen in M87
*
. For
B
;
100 G, the ratio of the
Stokes
emissivity to the Stokes
emissivity
j
V
/
j
I
;
1%. For
B
;
10
3
G,
j
V
/
j
I
;
10%, for all temperatures
>
10
10
K. We also
note that magnetic
fi
elds of
B
5 G are suf
fi
cient to produce jet
powers of
P
jet
10
42
erg s
−
1
(
e.g.,
EHTC V
)
via the Blandford
& Znajek
(
1977
)
process.
3.2. EVPA Pattern and Field Geometry
To demonstrate how the intrinsic magnetic
fi
eld structure in
the emission region in
fl
uences the observed polarization
pattern, in this section we present the polarization con
fi
gura-
tions from three idealized magnetic
fi
eld geometries around a
black hole
—
a purely toroidal
fi
eld, a purely radial
fi
eld, and a
purely vertical
fi
eld
—
as seen by a distant observer. In Figure
3
Figure 2.
Allowed parameter space in number density and dimensionless electron temperature
(
n
e
,
Θ
e
)(
red region
)
for the one-zone model described in Section
3.1
for
three constant values of
β
e
=
8
π
n
e
m
e
c
2
Θ
e
/
B
2
. We require that the optical depth
τ
I
<
1
(
green region
)
, the Faraday optical depth
tp
>
r
2
V
(
blue region
)
, and the total
fl
ux density 0.2
<
F
ν
<
1.2 Jy
(
black region
)
. Contours of constant magnetic
fi
eld strength are denoted by labeled dashed lines.
5
The Astrophysical Journal Letters,
910:L13
(
43pp
)
, 2021 March 20
EHT Collaboration et al.
we show polarimetric images from these simple
fi
eld con
fi
gura-
tions computed with two methods: a numerical model of an
optically thin emission region around the black hole
(
top row of
Figure
3
)
, and an analytic treatment of th
e parallel transport of the
polarization vector that is origin
ally perpendicular to the magnetic
fi
eld
(
R. Narayan et al. 2021, in preparation, middle row of
Figure
3
)
. We show the polarization maps from both methods for
the three idealized magnetic
fi
eld con
fi
gurations viewed at an
inclination angle of
i
=
163 deg. Both the analytical and
numerical calculations assume a zero-spin black hole
(
Schwarzs-
child metric
)
, though we have found that the main features of
these polarization patterns are insensitive to spin.
Figure 3.
(
a
)
Numerical calculations of the polarization con
fi
guration generated by an orbiting emission region in the shape of a torus at 8
r
g
in three imposed magnetic
fi
eld geometries and viewed at
i
=
163 deg
(
with material orbiting clockwise on the sky
)
. The orbital angular momentum vector is pointing away from the observer
and to the east
(
to the left
)
. Total intensity is shown in the background with higher brightness temperature regions shown as being lighter in color. In the foreground,
the observed EVPA direction is shown with white ticks, with the tick length proportional to the polarized
fl
ux.
(
b
)
Analytic calculations of the polarization
con
fi
guration from a thin ring of magnetized
fl
uid at 8
r
g
inclined by 163 deg to the observer in the same magnetic
fi
eld geometries as in
(
a
)
. While the distribution of
emitting material is different in the two models, both the sense of asymmetry in the brightness distributions and the polarization patterns match tho
se from the
numerical calculations.
(
c
)
Schematic cartoons showing the emitting frame wavevector
k
^
, magnetic
fi
eld direction
B
^
, and polarization vector
= ́
PkB
^
^
^
for each case.
In the bottom-right panel,
ˆ
¢
k
denotes the approximate light bending contribution to the wavevector.
6
The Astrophysical Journal Letters,
910:L13
(
43pp
)
, 2021 March 20
EHT Collaboration et al.
In the top row of Figure
3
we show the result of numerical
calculations performed with the general relativistic ray tracing
code
grtrans
(
Dexter & Agol
2009
; Dexter
2016
)
of
polarized emission from an optically and Faraday-thin compact
emission region, or
“
hotspot
”
, in Keplerian orbit around a black
hole in the equatorial plane. The hotspot has a radial extent of
3
r
g
and moves in an imposed and idealized magnetic
fi
eld
geometry in a circular orbit at a radius of 8
r
g
(
following
Gravity Collaboration et al.
2018
,
2020
)
. We construct a
phenomenological model of a torus of emitting, rotating plasma
by studying the time-averaged polarized emission images from
one revolution of this hotspot around the black hole. We have
veri
fi
ed that a semi-analytic implementation
(
Broderick &
Loeb
2006
)
of a hot accretion
fl
ow model
(
Yuan et al.
2003
)
produces consistent polarization patterns when using the same
fi
eld geometry.
In the second row of Figure
3
, we compare these numerical
results to results from an analytic calculation of the observed
polarization pattern generated by the emission of polarized
light on a thin ring of radius 8
r
g
in the equatorial plane. In this
model
(
R. Narayan et al. 2021, in preparation
)
the polarization
vectors are emitted perpendicular to the imposed magnetic
fi
eld
geometry in the
fl
uid rest frame; they are transformed on their
way to the observer using an approximate, analytic treatment of
the effects of light bending, parallel transport, and Doppler
beaming. This calculation includes radial in
fl
ow as well as
rotation in the velocity
fi
eld; the models shown use purely
toroidal motion
(
clockwise on the sky
)
with the same idealized
magnetic
fi
eld geometries as in the numerical case. The models
match the asymmetric brightness distributions and polarization
patterns of the numerical calculations. In particular, both
models produce consistent helical EVPA pattern in the case of
a vertical magnetic
fi
eld.
The linear polarization direction
̄
P
of synchrotron radiation
in the emitted frame is perpendicular to the wavevector
ˆ
k
and
the magnetic
fi
eld vector
̄
B
.Wede
fi
ne the toroidal magnetic
fi
eld as consisting only of magnetic
fi
eld components in the
azimuthal direction, while the poloidal magnetic
fi
eld consists
of the remainder, including both radial and vertical compo-
nents. In a purely toroidal
fi
eld case, the EVPA shows a radial
pattern
(
left column in Figure
3
)
. Purely radial magnetic
fi
elds
(
middle column
)
give a complementary result; the polarization
has a toroidal con
fi
guration, similar to a 90 deg rotation of the
linear polarization ticks from the toroidal case.
In a vertical magnetic
fi
eld
(
right column in Figure
3
)
,we
might expect that
̄
P
should be vertical
(
north
−
south
)
every-
where since a vertical
̄
B
is tilted east
−
west for this viewing
geometry. We might also expect that
̄
P
0
when the black
hole is viewed face on, because
ˆ
∣∣
̄
k
B
. Instead, the linearly
polarized emission from a purely vertical
fi
eld shows a twisting
pattern that wraps around the black hole. The twist results from
a combination of light bending and relativistic aberration. Light
bending in the emitting region near the black hole contributes a
radial component
ˆ
¢
k
to the emitted wavevector
ˆ
k
that initially
points away from the black hole
(
see the schematic cartoon in
the bottom-right panel of Figure
3
)
. As a result, close to the
black hole, the total wavevector
ˆˆˆ
=+
¢
k
kk
emit
and the
magnetic
fi
eld
̄
B
are no longer parallel, the polarization is
non-zero, and the resulting EVPA pattern is north
−
south
symmetric. Relativistic motion of the emitting material
(
aberration
)
breaks the symmetry and gives the twisting pattern
a handedness corresponding to the orbital direction. For the
pure vertical
fi
eld considered here, the handedness depends on
the rotation direction and the observed pattern is consistent
with clockwise rotation. The dependence on direction of
motion and magnetic
fi
eld con
fi
guration are discussed in more
detail in a forthcoming paper
(
R. Narayan et al. 2021, in
preparation
)
. The EVPA patterns in these images do not show a
strong dependence on the black hole spin.
In a rotating
fl
ow, weak magnetic
fi
elds are sheared into a
predominantly toroidal con
fi
guration
(
e.g., Hirose et al.
2004
)
.
In the absence of other effects
(
e.g., external Faraday rotation
)
,
the observed azimuthal EVPA pattern suggests the presence of
dynamically important magnetic
fi
elds in the emission region,
which can retain a signi
fi
cant poloidal component in the
presence of rotation. In the next sections, we compare
numerical simulations of the accretion
fl
ow and jet-launching
region in M87
*
with different
fi
eld con
fi
gurations to the
EHT2017 data to better constrain the magnetic
fi
eld structure.
4. M87
*
Model Images from GRMHD Simulations
The low resolved fractional linear polarization observed by
the EHT contradicts the results from an idealized magnetic
fi
eld
structure with no disorder. For typical parameters of the
230 GHz emission region, Faraday rotation and conversion are
expected to be important. Magnetic
fi
eld structure, plasma
dynamics and turbulence, and radiative transfer effects
including Faraday rotation can be realized in images from
three-dimensional general relativistic magnetohydrodynamic
(
3D GRMHD
)
simulations of magnetized accretion
fl
ows. We
use 3D GRMHD simulations
(
described in Section
4.1
)
in
combination with polarized general relativistic radiative
transfer
(
GRRT
)
models
(
described in Section
4.2
)
to model
polarized images of M87
*
. In Section
4.3
, we describe trends of
the key observables
(
|
m
|
net
,
|
v
|
net
,
〈
|
m
|
〉
, and
β
2
)
in our
GRMHD polarimetric image library.
4.1. GRMHD Model Description
The simulation library generated for the analysis of the
EHT 2017 total intensity data in
EHTC V
consists of a set of
3D GRMHD simulations that were postprocessed to generate
simulated black hole images via GRRT. For simulations using
black holes with non-zero angular momentum, we only
considered accretion
fl
ows in which the angular momentum
of the
fl
ow and the hole were aligned
(
parallel or anti-parallel
)
.
Because the equations of non-radiating
129
GRMHD are scale
invariant, each
fl
uid simulation was thus fully parameterized by
two values describing the angular momentum of the black hole
and the relative importance of the magnetic
fl
ux near the
horizon of the accretion system. A comparison of several
contemporary GRMHD codes, including those used to generate
the simulation library, can be found in Porth et al.
(
2019
)
and in
H. Olivares et al.
(
2021, in preparation
)
.
The black hole angular momentum
J
is expressed in terms of
the dimensionless black hole spin parameter
a
*
≡
Jc
/
GM
2
.In
this Letter, we consider simulations run with the
iharm
code
(
Gammie et al.
2003
; Noble et al.
2006
)
with
a
*
=
−
0.94,
−
0.5, 0, 0.5, and 0.94, where positive
(
negative
)
spin implies
alignment
(
anti-alignment
)
between the accretion disk and the
black hole angular momentum. Several studies of
“
tilted
”
disks
129
We assume that M87
*
can be described by models in which radiative
cooling is negligible so that it does not affect the dynamics of the plasma and
images can be generated in postprocessing.
7
The Astrophysical Journal Letters,
910:L13
(
43pp
)
, 2021 March 20
EHT Collaboration et al.
have been conducted
(
e.g., Fragile et al.
2007
; McKinney et al.
2013
; Morales Teixeira et al.
2014
; Liska et al.
2018
; White
et al.
2019
; Chatterjee et al.
2020
)
. As there does not yet exist a
full library of tilted disk simulations spanning a range of spins,
we limit our analysis to the aligned and anti-aligned
simulations considered in
EHTC V
.
The strength of the magnetic
fl
ux near the horizon
qualitatively divides accretion
fl
ow solutions into two cate-
gories: the Magnetically Arrested Disk
(
MAD
)
state
(
e.g.,
Bisnovatyi-Kogan & Ruzmaikin
1974
; Igumenshchev et al.
2003
; Narayan et al.
2003
)
in which the magnetic
fl
ux near the
horizon saturates and signi
fi
cantly affects the dynamics of the
fl
ow, and the contrasting Standard and Normal Evolution
(
SANE
)
state
(
e.g., De Villiers et al.
2003
; Gammie et al.
2003
;
Narayan et al.
2012
)
. The relative importance of magnetic
fl
ux
in a simulation is quantitatively described by the dimensionless
quantity
()
()
/
f
ºF
-
Mr c
,14
g
BH
2
12
where
Φ
BH
is the magnitude of the magnetic
fl
ux crossing one
hemisphere of the event horizon
(
see Tchekhovskoy et al.
2011
; Porth et al.
2019
)
and
M
is the mass accretion rate
through the event horizon. The
fl
ux saturates at values of
f
50, and the
fl
ow becomes MAD. The SANE simulations
that we consider have lower values of
f
≈
5.
130
Accreted
material supplied at large scales could, in principle, supply any
value of net vertical
fl
ux. Here, we do not explore cases with
small or zero net vertical
fl
ux
f
1. We also do not consider
values in the relatively narrow intermediate range 5
f
50.
The SANE simulations considered here used a grid
resolution of 288
×
128
×
128, a
fl
uid adiabatic index
γ
=
4
/
3, and an outer simulation domain of
r
out
=
50
r
g
. The MAD
simulations used a grid resolution of 384
×
192
×
192, an
adiabatic index
γ
=
13
/
9, and an outer simulation domain of
r
out
=
10
3
r
g
. The simulations were carried out in modi
fi
ed
spherical polar Kerr
–
Schild coordinates, where grid resolution
is concentrated toward the midplane to help resolve the
magnetorotational instability. All models in the EHT library are
evolved for at least
t
=
10
4
r
g
/
c
and their accretion
fl
ows reach
steady state within
r
=
10
–
20
r
g
.
4.2. Ray-traced Polarimetric Images from GRMHD
Simulations
Unlike the equations of GRMHD, the equations of radiative
transfer are not scale invariant, and so we must introduce two
scales to the simulation when we ray-trace images from the
numerical
fl
uid data. The length
(
and time
)
scale is set by the
mass of the black hole, assumed to be
M
BH
=
6.2
×
10
9
M
e
in
accordance with the value used to generate the
EHTC V
simulation library. For our models, we also adopt the
D
=
16.9
Mpc distance to M87
*
used in
EHTC V
. The density scale of
the accreting plasma
(
equal to the scale of the magnetic
pressure
)
is chosen so that on average the simulated images
reproduce the observed 230 GHz compact
fl
ux density,
F
ν
;
0.5 Jy.
Images were generated from the set of simulations for
several values of the polar inclination angle
i
chosen to be
broadly consistent with observational estimates of the inclina-
tion angle of the M87 jet
(
e.g., Walker et al.
2018
)
. The
position angle on the sky can be changed after image
generation by rotating both the image and the Stokes
and
components appropriately. Each image has a 320
×
320
pixel resolution over a 160
μ
as
fi
eld of view, where each pixel
contains full Stokes
,,,
intensities.
In GRMHD simulations, we make the approximation that the
plasma is thermal, i.e., that the electrons and ions are described
by a Maxwell-Jüttner distribution function
(
Jüttner
1911
)
.
However, the plasma around M87
*
and in other hot accretion
fl
ows is most likely collisionless, with electrons and protons
that are unable to equilibrate their temperatures
(
e.g., Shapiro
et al.
1976
; Ichimaru
1977
)
. We mimic collisionless plasma
properties in producing images from the GRMHD simulations
by allowing the electron temperature
T
e
to deviate from the
proton temperature
T
i
. The simulations used in this work only
track the total internal energy density
u
gas
, not the distinct
electron and ion temperatures. We set
T
e
after running the
simulation according to local plasma parameters following the
parameterization introduced by Mo
ś
cibrodzka et al.
(
2016
; see
also Mo
ś
cibrodzka et al.
2017
and
EHTC V
)
. The ratio
between the ion and electron temperatures
R
is determined by
the local plasma
β
=
p
gas
/
p
mag
, where
p
gas
=
(
γ
−
1
)
u
gas
, and
p
mag
=
B
2
/
8
π
. The temperature ratio is then taken to be
()
b
bb
==
+
+
+
R
T
T
RR
1
1
1
,15
i
e
high
2
2
low
2
where
R
high
(
R
low
)
are the free parameters of the model and give
the approximately constant temperature ratio at high
(
low
)
β
.
This approach allows us to associate the electron heating with
magnetic properties of the plasma.
In calculating the electron temperature, we further assume
that the plasma is purely ionized hydrogen and that ions are
nonrelativistic with an adiabatic index
γ
p
=
5
/
3 and electrons
are relativistic with
γ
e
=
4
/
3. Then, given
u
gas
from the
simulation and
R
from Equation
(
15
)
,
(
EHTC V
)
:
()
()
r
=
+
T
mu
kR
2
32
.16
e
p
gas
We note that this procedure is not entirely self-consistent, as the
γ
of the combined electron-ion
fl
uid will change depending on the
relative pressure contributions of electrons and protons while we
assume it is
fi
xed throughout the simulation domain. See
S
ą
dowskietal.
(
2017
)
for an alternative, self-consistent approach.
In this Letter, we consider a library of 72,000 simulated
images composed of sets of 200 realizations of the same
accretion system described by a
fi
xed set of heating and
observation parameters. Each set of 200 images is drawn from
output
fi
les spaced by 25
–
50
r
g
/
c
from the set of 10 GRMHD
simulations spanning
fi
ve spin values in both MAD and SANE
fi
eld con
fi
gurations. The inclination angle for each image is set
to one of either
i
=
12, 17, 22 deg
(
retrograde models,
a
*
<
0
)
or
i
=
158, 163, 168 deg
(
prograde models,
a
*
0
)
, according
to which parity is required to orient the brightest portion of the
ring in the southern part of the image while ensuring the
orientation of the approaching jet is consistent with large-scale
observations.
We use electron heating parameters
R
low
=
1, 10 and
R
high
=
1, 10, 20, 40, 80, or 160 in Equation
(
15
)
.
EHTC V
only considered models with
R
low
=
1. Larger values of
R
low
130
Note that the MAD threshold
f
50 is given in Gaussian units where
[
Φ
]
=
Gcm
2
. If the
fi
eld strength is given in the Lorentz
–
Heaviside units
typically used in simulations
(
p
=
B
B
4
LH
G
)
, the MAD threshold on the
dimensionless
fl
ux is
f
;
15.
8
The Astrophysical Journal Letters,
910:L13
(
43pp
)
, 2021 March 20
EHT Collaboration et al.
correspond to lower electron-to-proton temperature ratios in the
low
β
regions
(
e.g., the jet funnel
)
. This choice is physically
motivated for M87
*
, where radiative cooling of the electrons
may keep
T
e
<
T
i
even in magnetized regions where electron
heating is ef
fi
cient
(
e.g., Mo
ś
cibrodzka et al.
2011
; Ryan et al.
2018
; Chael et al.
2019
)
. Lower electron temperatures in
R
low
=
10 models increase the Faraday rotation depth and can
result in increased depolarization in parts of the image.
GRMHD simulations produce a highly magnetized jet funnel
above the black hole
’
s poles, away from the accretion disk. In
the funnel, where the plasma magnetization parameter
σ
≡
B
2
/
4
πρ
c
2
?
1, our numerical methods typically fail to
accurately evolve the plasma internal energy. In the image
library, we cut off all emission in regions where
σ
>
1 to ensure
that we limit the emitting region to plasma whose internal
energy is safely evolved without numerical artifacts
(
as
in
EHTC V
)
. We tested the importance of a
σ
>
1 electron
population by generating a supplementary set of images from
all models with a cut at
σ
=
10 and found that it did not change
the overall distribution of the derived metrics we use for model
scoring in Section
5
.
Each set of 200 model images with the same parameters in
the image library requires a unique density scaling factor that is
determined by matching the average
fl
ux density from the
model to the observed compact
fl
ux density of M87
*
measured
by the EHT. Hence, the mass accretion rates, radiative
ef
fi
ciencies, and jet powers will differ between two models
even if they are derived from the same underlying simulation
(
e.g., if
R
high
,
R
low
,or
i
are changed
)
. The additional models
discussed in Section
6
, which explore the effects of different
σ
cutoff values and the inclusion nonthermal electrons, also
require unique mass-scaling factors.
All of the polarimetric images from GRMHD simulations
that we analyze in this Letter were generated using the
ipole
code
(
Mo
ś
cibrodzka & Gammie
2018
)
, which has been tested
against analytic solutions and numerical ones produced by
other numerical GRRT codes
(
Dexter
2016
;Mo
ś
cibrodzka
2020
)
. A more comprehensive comparison of various GRRT
codes that perform total intensity transport and fully polarized
GRRT can be found in Gold et al.
(
2020
)
and B. Prather et al.
(
2021, in preparation
)
, respectively. Preliminary results from B.
Prather et al.
(
2021, in preparation
)
show that the tested codes
are consistent at the fraction of 1% in all Stokes parameters. All
calculated images in this work ignore light travel time delays
through the emission region
(
the so-called
“
fast light
”
approach
)
, and are calculated at a single frequency of
ν
=
230 GHz, neglecting the
fi
nite observing bandwidth of
the EHT. We con
fi
rm that neither of those effects are important
for models of interest for M87
*
.
4.3. Sample GRMHD Model Images and Polarization Maps
In Figures
4
and
5
we show images and polarization maps
for a subset of library models. In general, because the horizon-
scale magnetic
fi
elds in MAD models are strong enough not to
be advected with the accretion
fl
ow, they are more likely to
have a signi
fi
cant poloidal component and produce azimuthal
EVPA patterns
(
Figure
3
)
. In contrast, SANE models tend to
show more radial EVPA patterns. Some MAD
a
*
=
0.94 and
SANE
a
*
=
0 images are notable exceptions to this trend.
These trends are also apparent in the distributions of the
β
2
phase across the full image library that we consider later in
Figure
9
.
The GRMHD models at their native resolution include
notable disorder in the EVPA structure, resulting from both
magnetic turbulence and Faraday rotation. Models with larger
R
high
have lower electron temperatures and higher Faraday
rotation depths, resulting in the most disordered polarization
maps. Many of the EVPA patterns seen in the images blurred
with a 20
μ
as Gaussian kernel to simulate the limited EHT
resolution resemble those from the idealized magnetic
fi
eld
models in Figure
3
, indicating that the net EVPA pattern after
blurring may trace the intrinsic magnetic
fi
eld structure.
In Figure
6
we show a sample polarization map at full
resolution compared to the same map blurred with circular
Gaussian kernels of 10
μ
as and 20
μ
as FWHM. From tests with
synthetic data, blurring
(
convolving with a circular Gaussian
kernel
)
provides a reasonable approximation to image recon-
struction from the EHT data at a comparable resolution
(
EHTC
VII
)
. The resolved average fractional polarization in the blurred
images
〈
|
m
|
〉
traces the degree of order in the intrinsic
polarization map. In the blurred images, disordered polarized
structure on small scales produces beam depolarization. The
degree of depolarization decreases with increasing spatial
resolution
(
decreasing beam size
)
.
The bottom row of Figure
6
shows the same unblurred and
blurred polarization maps, but calculated without the effect of
Faraday rotation
(
ρ
V
=
0
)
. Those images show more coherent
EVPA structure, with much larger
|
m
|
net
and, particularly when
blurred, much larger
〈
|
m
|
〉
. Evidently, for this particular model,
the depolarization visible in the corresponding top panels is due
to Faraday rotation internal to the emission region. In addition,
the net EVPA pattern shifts by a signi
fi
cant amount. The
change in
β
2
by
;
80 deg would correspond to an apparent RM
of
;
−
4
×
10
5
rad m
−
2
. Our GRRT calculations include all
Faraday rotation occurring inside the GRMHD simulation
domain
(
r
out
=
50
–
100
r
g
)
, both external and internal to the
230 GHz emission region. The observables considered here, for
the low viewing inclination of M87
*
, do not depend strongly on
that outer radius, as long as it is at
r
40
r
g
. We cannot rule out
the presence of additional Faraday rotating material at larger
radii
100
r
g
, and its effects are not included in our models.
Appendix
B
discusses the origin of the RM in our models in
more detail.
4.4. GRMHD Model Theory Metrics
We compute the polarimetric observables
(
|
m
|
net
,
|
v
|
net
,
〈
|
m
|
〉
,
β
2
)
described in Section
2.3
from model images blurred
with a circular Gaussian kernel with a FWHM of 20
μ
as in
order to compare them to the ranges measured from EHT and
ALMA-only data. Both
〈
|
m
|
〉
and
β
2
depend on the resolution
and hence the size of the Gaussian blurring kernel. The value of
β
2
also depends on the choice of the image center. We do not
shift the library images before computing
β
m
coef
fi
cients for
comparison with the range inferred from the EHT image
reconstructions, which have been centered by aligning them to
the centered,
fi
ducial total intensity images in
EHTC IV
.As
discussed in Palumbo et al.
(
2020
)
, a centering offset
u
expressed as a fraction of the diameter of a
PWP
m
=
2 ring
causes a quadratic falloff in
β
2
power as
δβ
2
/|
β
2
|
≈
4
u
2
. Effects
on the
β
2
phase enter at similar order. In the case of the EHT
image,
u
is likely less than one-
fi
fth, meaning that centering
errors in
β
2
will be sub-dominant to other uncertainties, such as
the choice of the blurring kernel or the variation across methods
and days.
9
The Astrophysical Journal Letters,
910:L13
(
43pp
)
, 2021 March 20
EHT Collaboration et al.