# Positron Effects on Polarized Images and Spectra from Jet and Accretion Flow
Models of M87* and Sgr A*

Razieh Emami  Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA  Richard Anantua  Center for Astrophysics $|$
Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Black Hole
Initiative at Harvard, 20 Garden Street, Cambridge, MA 02138, USA  Center for
Computational Astrophysics, Flatiron Institute, Simons Foundation, 162 Fifth
Avenue, New York, NY 10010, USA  Andrew A. Chael  Princeton Center for
Theoretical Science, Jadwin Hall, Princeton University, Princeton, NJ 08544,
USA  NASA Hubble Fellowship Program Einstein Fellow  Abraham Loeb  Center for
Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA

###### Abstract

We study the effects of including a nonzero positron-to-electron fraction in
emitting plasma on the polarized SEDs and sub-millimeter images of jet and
accretion flow models for near-horizon emission from M87* and Sgr A*. For
M87*, we consider a semi-analytic fit to the force-free plasma regions of a
general relativistic magnetohydrodynamic jet simulation which we populate with
power-law leptons with a constant electron-to-magnetic pressure ratio. For Sgr
A*, we consider a standard self-similar radiatively inefficient accretion flow
where the emission is predominantly from thermal leptons with a small fraction
in a power-law tail. In both models, we fix the positron-to-electron ratio
throughout the emission region. We generate polarized images and spectra from
our models using the general-relativistic ray tracing and radiative transfer
from GRTRANS. We find that a substantial positron fraction reduces the
circular polarization fraction at infrared and higher frequencies. However, in
sub-millimeter images higher positron fractions increase polarization
fractions due to strong effects of Faraday conversion. We find a M87* jet
model that best matches the available broadband total intensity and 230 GHz
polarization data is a sub-equipartition, with positron fraction of $\simeq$
10%. We show that jet models with significant positron fractions do not
satisfy the polarimetric constraints at 230 GHz from the Event Horizon
Telescope (EHT). Sgr A* models show similar trends in their polarization
fractions with increasing pair fraction. Both models suggest that resolved,
polarized EHT images are useful to constrain the presence of pairs at 230 GHz
emitting regions of M87* and Sgr A*.

M87*— Sgr A*— Active Galactic Nuclei — relativistic jets — GRMHD — Simulations

††journal: ApJ

##  1 Introduction

Supermassive black holes have an outsized influence on galactic dynamics –
catalyzing the radiatively inefficient inflow of billion-degree magnetized
plasmas (e.g. Quataert, 2003) and the formation of energetic relativistic jets
(e.g. Blandford & Znajek, 1977). The central supermassive black hole in the
giant elliptical galaxy M87 launches an extragalactic jet which extends from
several to tens of kiloparsecs across the EM spectrum. M87’s jet has been
studied for over a century from radio to $\gamma$-ray wavelengths (e.g.
Curtis, 1918; Palmer et al., 1967; Nagar et al., 2001; Doeleman et al., 2012;
Whysong & Antonucci, 2004; Madrid, 2009; Di Matteo et al., 2003; Aharonian et
al., 2006; EHT MWL Science Working Group et al., 2021). Very-Long-Baseline
Interferometry (VLBI) images at radio and sub-millimeter wavelengths show a
dynamical, polarized, limb-brightened forward jet and counter jet in the
central parsecs of M87 (e.g. Junor et al., 1999; Asada & Nakamura, 2012; Hada
et al., 2016; Walker et al., 2018; Kim et al., 2018; Kravchenko et al., 2020).
The jets terminate in a bright radio core that corresponds to the location of
the central black hole at frequencies $\gtrsim 86$ GHz (Hada et al., 2011).

Recently, the Event Horizon Telescope (EHT) observed M87* (the central black
hole in M87) at 230 GHz and produced the first resolved images of emission
within $\lesssim 10$ gravitational radii $r_{\rm g}$ of a black
hole111Throughout the text, we use the gravitational radius $r_{\rm
g}=GM/c^{2}$ and gravitational timescale $t_{\rm g}=GM/c^{3}$ for a black hole
of mass $M$. (Event Horizon Telescope Collaboration et al., 2019; EHT
Collaboration et al., 2019a, b, c, d, e). The EHT images show a ring of
synchrotron emission approximately $40\,\mu$as in diameter, consistent with
the expected lensed photon orbit size for a black hole of mass $6.5\times
10^{9}M_{\odot}$ (EHT Collaboration et al., 2019e). The emission ring in the
EHT images is brighter in its southern half, consistent with the effects of
Doppler beaming from emitting plasma rapidly co-rotating with the black hole
(EHT Collaboration et al., 2019d; Wong et al., 2021).

Notably, the near-horizon synchrotron emission from M87* is linearly and
circularly polarized (EHT Collaboration et al., 2021a, b; Goddi et al., 2021).
The 230 GHz linear polarization resolved by the EHT has a relatively low
resolved linear polarization fraction $|m|\sim 20\%$ at the $\approx
20\,\mu$as scale of the EHT beam and a predominantly azimuthal, spiral pattern
of electric vector position angles (EVPAs) around the emission ring (EHT
Collaboration et al., 2021a). The circular polarization fraction of the 230
GHz ring is low, $\lesssim 0.4$% (Goddi et al., 2021). These polarization data
are much more constraining on physical models of M87* than total intensity
data alone. While EHT Collaboration et al. (2019d) found that many general
relativistic magnetohydrodynamic (GRMHD) simulations of M87* could produce
images consistent with the observed emission ring, EHT Collaboration et al.
(2021b) found that only a few simulation images fit the polarimetric data;
these images were all from simulations with strongly ordered poloidal magnetic
fields in the magnetically arrested disk (MAD) state (e.g. Narayan et al.,
2003).

Despite these groundbreaking observational results, persistent questions on
the fundamental nature of jets connected to black holes remain, including:
“What physical mechanisms accelerate the emitting particles and what is their
energy distribution?" and “Is the particle composition of relativistic jets
near supermassive black holes ionic, leptonic, or mixed?" With regard to the
second question, most analytic and simulation models of the near-horizon
emission from low-luminosity AGN (LLAGN) like M87* and the galactic center
black hole Sagittarius A* (Sgr A*) assume a purely ionic plasma when comparing
observables to data. For instance, the library of GRMHD simulations compared
to the resolved polarimetric images of M87* in EHT Collaboration et al.
(2021b) all assumed positron emission does not contribute to the observed
image. While some groundbreaking studies have estimated the positron fraction
around M87* and Sgr A* from different models of pair production (e.g.
Mościbrodzka et al., 2011; Broderick & Tchekhovskoy, 2015; Wong et al., 2021),
their implications for resolved images of the sources have not been studied in
detail. In this work, be begin to assess the effects of a non-zero pair
fraction on submm images and broadband spectral energy distributions (SEDs) of
M87* and Sgr A* using semi-analytic models for both sources. GRMHD simulations
of an electron-proton plasma are the primary modern theoretical tools for
understanding the physics of jet/accretion flow/black hole (JAB). Numerical
GRMHD codes such as HARM (Gammie et al., 2003), Athena++ (White et al., 2016),
and BHAC (Porth et al., 2017) can accurately solve for fluid variables in most
regions of JAB systems. Such codes have served as a bridge between the
microphysics of particle acceleration, emission (including the conversion of
magnetic to particle energy) and discrete observational features in AGN, e.g.,
Sgr A* (e.g. Mościbrodzka et al., 2009; Dexter et al., 2010; Shcherbakov et
al., 2012; Chan et al., 2015; Ressler et al., 2017; Gold et al., 2017; Anantua
et al., 2020b; Chael et al., 2018; Dexter et al., 2020) and M87* (e.g. Dexter
et al., 2012; Mościbrodzka et al., 2017; Anantua et al., 2018; Ryan et al.,
2018; Chael et al., 2019; Davelaar et al., 2019), but all images produced from
GRMHD simulations to date have assumed a plasma free of electron-positron
pairs. GRMHD simulations reliably solve the flows in the Kerr metric in
relatively high-density disk regions, but they have difficulty in reliably
solving for the plasma temperature and dynamics in the low density, magnetized
jet – particularly where the magnetization ($\sigma=\frac{\mathrm{magnetic\
energy\ density}}{\mathrm{plasma\ density}}$) exceeds unity. The magnetic
field is more accurately evolved in jet regions in GRMHD simulations. If we
wish to produce emission models from near-horizon jets in M87* or Sgr A*, it
may be useful to begin by using GRMHD magnetic field results as the basis for
self-similar/semi-analytic models of the emission region.

While they cannot capture the dynamics and turbulence of GRMHD simulation
images of M87* and Sgr A*, semi-analytic models allow for a wider scan of the
parameter space, can extend self-consistently to large radii, and can avoid
pathologies in fluid variables in the jet region. The foundational semi-
analytic model of synchrotron emission from relativistic jets is from
Blandford & Königl (1979), where a helical magnetic field near equipartition
with the radiating particle population naturally accounts for the radio
spectral slope. Applications of semi-analytic models of jet emission to M87*
include (e.g. Reynolds et al., 1996; Broderick & Loeb, 2009; Prieto et al.,
2016), and the flat radio Sgr A* SED can also be naturally explained by a jet
model (e.g. Falcke & Markoff, 2000). A plausible alternative to a jet origin
for observed near-horizon submm emission in M87* and Sgr A* is that the
emission originates in a geometrically thick equatorial accretion flow. In the
regime in which gas accreting towards a black hole is unable to radiate
efficiently, Narayan & Yi (1994) produced self-similar advection-dominated
accretion flow (ADAF) solutions with positive Bernoulli constant (enabling
them to become outflows if they escape accreting onto the black hole). ADAF
and similar radiatively inefficient accretion flow (RIAF) models for Sgr A*
can fit the observed SED and constraints on the intrinsic submm size (e.g.
Narayan et al., 1995; Quataert et al., 1999; Özel et al., 2000; Broderick &
Loeb, 2006a). They predict a highly sub-Eddington accretion rate with a
quiescent X-ray spectrum dominated by bremsstrahlung near the Bondi radius,
sub-mm synchrotron emission from a mixed thermal- and nonthermal electron
distribution near the black hole, and and near-infrared and X-ray flares
originating $\lesssim$ 10$r_{g}$ from the black hole (e.g Quataert, 2003). In
addition to pure ADAF or pure jet models, hybrid jet+ADAF models where
different emitting regions contribute to different parts of the observed M87*
or Sgr A* SED have also been compared to observations (e.g. Yuan et al., 2002;
Feng et al., 2016).

In this paper, we provide examples of semi-analytic models of JAB systems
tuned to simulate the emission from M87* and Sgr A* while including the
effects of mixed pair and ionic plasmas. Positrons can be created through
various channels in JAB systems, including photon annihilation in the disk and
jet funnel walls, and spark gap acceleration (Mościbrodzka et al., 2011;
Broderick & Tchekhovskoy, 2015). We utilize the general relativistic ray
tracing and radiative transfer (GRRT) code GRTRANS (Dexter, 2016), which
computes synchrotron Stokes maps from observer-to-source rays in the Kerr
metric through an emitting plasma around a Kerr black hole. We focus on images
from semi-analytic models of M87* and Sgr A*. We analyze the effects of pairs
on GRMHD simulation images by replacing some proton positive charge carriers
with positrons in the post-processing step of our radiative transfer.
Incorporating the effects of pairs in radiative transfer of physical models of
M87* and Sgr A* is a key step towards understanding how EHT and broadband
measurements may constrain the composition and nature of plasma in such
sources.

This paper is organized as follows: Section 2 describes recent observations of
M87* and Sgr A* that we make use of in comparing our models to data. Section 3
summarizes two recent models for positron production in the context of JAB
systems and their implications for the positron-to-electron fraction $f_{\rm
pos}$ in M87* and Sgr A*. Section 4 describes the semi-analytic jet and disk
models we use. Section 5 describes our parameterization for incorporating the
effects of a mixed ionic and leptonic plasma on the images from these models.
Sections 6 and 7 present images and polarized spectra from these models and
compare to observations. Section 8 concludes and proposes future directions.

##  2 Observations of M87 and Sgr A*

Below we introduce observational conventions before we describe different
observations for M87* and Sgr A*.

###  2.1 Conventions

In this paper, we produce polarized images from semi-analytic models at
frequencies for which we can compare to VLBI data from M87* and Sgr A*. We
first establish our polarimetric conventions. Synchrotron radiation from a
single emitter is elliptically polarized. Denoting the ellipse orientation
angle $\psi$ and the ellipticity angle $\chi$ for an incoming electromagnetic
wave with electric vector projection $\vec{E}_{0}$ on the observer plane, we
use standard definitions of the Stokes parameters $I,Q,U$ and $V$:

$\displaystyle I$ $\displaystyle=E_{0}^{2},$ (1) $\displaystyle\tan 2\psi$
$\displaystyle=\frac{U}{Q},$ (2) $\displaystyle\sin 2\chi$
$\displaystyle=\frac{V}{I}.$ (3)

We measure the electric vector position angle (EVPA) $\psi$ in degrees East of
North.

Throughout this work, we consider two different types of fractional
polarization: an unresolved fractional polarization and an average, resolved
fractional polarization. For both M87* and Sgr A*, most of our information on
the polarimetric properties of the near-horizon accretion flow and jet
launching region is unresolved. To compare models against the data, we compute
the unresolved, net polarization fractions $|m|_{\rm net}$ and $|v|_{\rm
net}$:

$\displaystyle|m|_{\mathrm{net}}\equiv$
$\displaystyle\frac{\sqrt{\overline{Q}^{2}+\overline{U}^{2}}}{\overline{I}}$
(4) $\displaystyle|v|_{\mathrm{net}}\equiv$
$\displaystyle\frac{\overline{|V|}}{\overline{I}},$ (5)

where bars indicate averages over pixels in a source model image. For M87*,
for the first time, EHT results in (EHT Collaboration et al., 2021a) constrain
the resolved linear polarization near the horizon. They define the average
resolved polarization fraction as

$\langle|m|\rangle\equiv\frac{\overline{\sqrt{Q^{2}+U^{2}}}}{\overline{I}}.$
(6)

The average resolved polarization fraction $\langle|m|\rangle$ depends on the
observation resolution, while the unresolved fractions $|m|_{\rm net}$ and
$|v|_{\rm net}$ do not. In comparing results from 230 GHz theoretical images,
we simulate the effects of limited observational resolution by blurring the
image with a circular Gaussian filter at the EHT resolution of 20 $\mu$as.

We note that in addition to $|v|_{\rm net}$, $|m|_{\rm net}$ and
$\langle|m|\rangle$, in comparing GRMHD simulation images to EHT polarimetric
data, EHT Collaboration et al. (2021b) also made use of the second azimuthal
Fourier model of the complex linear polarization $\beta_{2}$ (Palumbo et al.,
2020). Here we do not consider this constraint on the model images of M87*, as
it is predominantly set by the field geometry in our model, which we do not
vary as a free parameter. Nonetheless, we note that the EVPA patterns we
recover in our jet model for M87* are predominantly azimuthal and are somewhat
qualitatively similar to the observed M87* image.

###  2.2 M87 Observations

The giant elliptical galaxy M87 hosts our first- and best-observed jet
emanating from a supermassive black hole, as well as the first resolved image
of a black hole’s near-horizon environment. This section briefly summarizes
some recent observations of M87* applicable to this work. The jet from M87 has
been observed from radio to $\gamma$ ray frequencies; a comprehensive SED
constructed as part of simultaneous and near-simultaneous observations to the
2017 EHT campaign was recently published in EHT MWL Science Working Group et
al. (2021).

VLBI images at frequencies between $24-86$ GHz (Walker et al., 2018; Hada et
al., 2016; Kim et al., 2018; Kravchenko et al., 2020, e.g.) show the jet
structure within the central $\sim 1$pc. The jet is dynamical and polarized,
with a wide opening angle $\sim 55^{\circ}$ at 43 GHz. Hada et al. (2016) fits
the opening angle with distance from the black hole as $s\sim z^{0.56}$. The
inclination angle toward the line of sight is $\theta_{o}\approx
17-30^{\circ}$ (Mertens et al., 2016). The forward jet is limb-brightened, and
a counter-jet is visible in some images. There is no core-shift with frequency
above $\approx 86$ GHz, which suggests that the radio core at these
frequencies is becoming optically thin and is fixed at the black hole’s
location (Hada et al., 2011).

Polarized images of the jet generally show low polarization fractions near the
core with stronger polarization fractions along the jet. In 43 GHz images from
Walker et al. (2018), the polarization fraction peaks at $4\%$ in these images
along the edge of the jet. The EVPA structure in the 43 GHz images suggests
that the jet magnetic field is predominantly toroidal. Hada et al. (2016)
found the linear polarization fraction at 86 GHz is of the order $4\%$ near
the core, but features further down the jet rise to a level $\sim 20\%$; they
also infer a poloidal field structure in the jet. Kravchenko et al. (2020)
observed the jet from 24-43 GHz and found that the global polarization pattern
was stable over 11 years of observations, with some variability on top of the
global pattern on $\sim$ month timescales. The polarization pattern again
indicates a predominantly toroidal jet magnetic field near the jet launching
site. They estimate a rotation measure (RM) toward the core of $\sim 10^{4}$
rad m-2 between 24 and 86 GHz; they interpret this RM as due to an external
screen, possibly in a non-relativistic wind outside of the jet. As part of the
2017 EHT campaign, Goddi et al. (2021) performed observations of M87* with
ALMA at a central frequency $\nu=221$ GHz; these observations do not resolve
the inner jet or core, but they were used to estimate the unresolved linear
and circular polarization $|m|_{\rm net}\approx$ 2.7%, $|v|_{\rm net}\leq
0.4$%. (Goddi et al., 2021) also measured the core RM at 221 GHz; they found a
rotation measure that is time variable with a maximum magnitude of $10^{5}$
rad m-2. This RM varies in magnitude and sign by order $\sim$1 over the week
of observations in 2017. While substantial RM can be produced close to or
internal to the submm emission region Ricarte et al. (e.g. 2020), it is still
uncertain whether the observed RM occurs close to or far away from the
emission region. In this paper, we only consider the effects of RM internal to
our models and do not impose an external Faraday screen.

The EHT has released 230 GHz M87* image data from its 2017 observing campaign
in eight papers (EHT Collaboration et al., 2019f, a, b, c, d, e, 2021a,
2021b). The 230 GHz images feature a ring of emission of mean radius $42\mu$as
$\pm 3\mu$as that is brighter in the South than the North (EHT Collaboration
et al., 2019f). Proto-EHT observations with three stations from 2009-2013 also
show evidence for a ring morphology with a consistent radius (Wielgus et al.,
2020). The 2017 EHT 230 GHz polarized intensity maps of M87 (EHT Collaboration
et al., 2021a, b) show azimuthal EVPA patterns and a low level of resolved
linear polarization ($\langle|m|\rangle\lesssim 11\%$). The only GRMHD
simulations from the EHT simulation library that pass polarimetric constraints
derived from these images are of Magnetically Arrested Disk (MAD) systems,
which feature dynamically important poloidal magnetic fields. These passing
simulations prefer near-horizon magnetic fields of 7-30 G, densities in a
range $10^{4-5}$ cm-3 and electron temperatures in a range from $(5-35)\times
10^{10}$ K. These results imply a value of plasma-$\beta_{\rm e}$ (where
$\beta_{\rm e}=\frac{\rm plasma\ pressure}{\rm magnetic\ pressure}$) in the
range $10^{-3}\lesssim\beta_{\rm e}\lesssim 1$ in the 230 GHz emission region
(EHT Collaboration et al., 2021b).

We take the full SED of M87 from radio to X-ray frequencies from the table
compiled by Prieto et al. (2016), focusing on their results on scales less
than of 0.4 as (32 pc).222Note that a more recent and complete M87* SED is
presented in EHT MWL Science Working Group et al. (2021). Because we
calibrated our M87* images to the 230 GHz flux density from Akiyama et al.
(2015), which is a factor of 2 larger than the 2017 value observed by the EHT,
we use the Prieto et al. (2016) compiled SED, which includes this value, for
consistency. Prieto et al. (2016) compile observations previously published at
various frequencies in Aharonian et al. (2006); Di Matteo et al. (2003);
Perlman et al. (2001); Whysong & Antonucci (2004); Nagar et al. (2001);
Doeleman et al. (2012); Lonsdale et al. (1998); Lee et al. (2008); Junor &
Biretta (1995); Morabito et al. (1986, 1988). In addition, we make use of
updated estimates of the core compact flux density first compiled in from data
originally published in Lister et al. (2018); Hada et al. (2017); Walker et
al. (2018); Kim et al. (2018); Doeleman et al. (2012); Akiyama et al. (2015).
The total flux densities from these VLBI observations (from 15-230 GHz) were
re-measured and compiled in Chael et al. (2019) using a consistent strategy
across all data sets.

A synchrotron peak in M87’s SED occurs at $\nu_{\mathrm{syn}}=10^{11.5}$ Hz; a
secondary peak, which may be interpreted as due to synchrotron self-Compton or
from a secondary nonthermal synchrotron-emitting population, occurs at
$\nu_{\mathrm{SSC}}=10^{14.5}$ Hz. Between the infrared and X-ray bands, both
the observed quiescent and loud states have spectral flux density
$S_{\nu}=\nu^{-\alpha}$ with a slope $\alpha\sim 1.1$. If we assume this slope
originates from synchrotron emission from power-law electrons, we can estimate
the power-law index of the emitting distribution as $p\approx 3.5.$

The M87* core flux density we use to normalize our 230 GHz images throughout
this paper is 1.5 Jy (Akiyama et al., 2015). Note that in the 2017 EHT
observations, the core flux at 230 GHz had decreased to 0.6 Jy. While we
compare our M87* jet models to the full Stokes $I$ SED, note that when
considering polarimetric constrains on the models in this paper, we only
consider the 2017 EHT polarimetric constraints and do not apply constraints
from polarimetric observations at longer wavelengths. This is because EHT
Collaboration et al. (2021b) provide a straightforward framework for
comparison in terms of $|m|_{\rm net}$, $|\langle|m|\rangle$, and $|v|_{\rm
net}$, since we are most interested in modeling the 230 GHz source structure,
and as we find the 230 GHz polarimetric constraints to be sufficiently
constraining (in fact, they reject most of our models.) In a future work we
will consider a wider range of polarimetric constraints from multi-wavelength
data.

###  2.3 Sgr A* Observations

Sgr A* is the most extensively studied accreting supermassive black hole. Its
proximity ($D=8.178\pm 22$ kpc) and mass of $4.154\pm 0.014\times
10^{6}M_{\odot}$ (Gravity Collaboration et al., 2019) make it the largest
angular size black hole known. However, it is situated in a bath of dust and
gas which– along with its intrinsic variability– make it particularly
challenging to observe and to model. Sgr A* has been observed from radio to
gamma-ray wavelengths. At near-infrared and X-ray wavelengths, the continual
flaring behavior of Sgr A* has attracted particular interest, as it promises
to help elucidate the rapid dynamics and microphysical processes of hot plasma
near the event horizon (e.g. Yusef-Zadeh et al., 2006; Gillessen et al., 2006;
Marrone et al., 2008; Neilsen et al., 2013; Ponti et al., 2017; Gravity
Collaboration et al., 2018; Do et al., 2019). Sgr A*’s emitting region at
submm wavelengths is compact. At frequencies $<230$ GHz, strong interstellar
scattering prevents the direct imaging of Sgr A*’s intrinsic structure from
VLBI observations. Instead, most observations constrain the size of the
scattering kernel that blurs the source image (e.g. Johnson et al., 2018).
Issaoun et al. (2019) produced the first directly resolved VLBI images of Sgr
A* at 86 GHz from GMVA+ALMA data that constrain the intrinsic structure of the
source. Their 86 GHz estimates for Sgr A*’s axial sizes assuming an elliptical
profile are ($100\pm 18)\mu$as, or ($20.2\pm 3.6)r_{g}$. The comparison of
these size constraints in Issaoun et al. (2019) to theoretical models from
GRMHD simulations ruled out jet-dominated emission models at 86 GHz unless the
jet is nearly face-on.

Sgr A* has been observed by “proto-EHT" arrays (with 3-4 sites) from
2007-2013. The intrinsic source size at 230 GHz was first measured by Doeleman
et al. (2008) using proto-EHT data as $\approx 40\mu$as; this size was
confirmed with updated proto-EHT measurements including longer baselines to
APEX in Chile by Lu et al. (2018). De-scattered image moments were calculated
from 2013 proto-EHT observations by Johnson et al. (2018), who report an major
axis angular width $9.9r_{g}<\theta_{\mathrm{maj}}<12.7r_{g}$ (Johnson et al.,
2018). (Broderick & Loeb, 2006a) modeled proto-EHT observations of Sgr A* from
2007 to 2013 modeled with semi-analytic RIAF models find $\theta\approx
60^{\circ}$ and a low black hole spin.

The polarized emission from Sgr A* has also been extensively studied at sub-mm
wavelengths (e.g. Bower et al., 1999; Marrone et al., 2006, 2008; Yusef-Zadeh
et al., 2007; Muñoz et al., 2012; Johnson et al., 2015; Bower et al., 2018a).
At 230 GHz, Sgr A* is significantly linearly polarized with mean polarization
fractions ranging from $|v|_{\rm net}\approx 4-8$% (Bower et al., 2018a). Like
the total intensity at submm-wavelengths, the linear polarization exhibits
variability on $\sim$hour timescales (Marrone et al., 2006; Yusef-Zadeh et
al., 2007). Bower et al. (2018a) also provides the latest measure of the
rotation measure at 230 GHz $RM\approx-5\times 10^{5}$ rad m-2, consistent
with its value measured over 20 years (e.g. Marrone et al., 2007). The
measured RM can be used to infer an accretion rate onto Sgr A* of $\approx
10^{-8}M_{\odot}$ yr-1 (though note that the accretion rate measured via the
RM can produce inaccurate results for face-on systems (Ricarte et al., 2020)).
Muñoz et al. (2012) found that Sgr A* is also significantly circularly
polarized at 230-345 GHz, with $|v|_{\rm net}\approx 1-2$% GHz. The (negative)
sign of $V$ is consistent with that measured at longer wavelengths, suggesting
that it arises from Faraday conversion in the accretion flow with a stable
magnetic field orientation. Johnson et al. (2015) provided the first
_resolved_ look at Sgr A*’s polarimetric structure from proto-EHT
observations; they found that the EVPA pattern on event-horizon scales (and
hence the magnetic field configuration) exhibits an intermediate degree of
spatial order with coherent structures on scales of $\sim 10r_{\rm g}$. The
average 230 GHz resolved polarization fraction from these measurements is
$10\%\leq\langle|m|\rangle\leq 30\%$.

Intriguingly, GRAVITY, a near-infrared interferometer, has observed Sgr A*
(Gravity Collaboration et al., 2018) at 2.2 microns and detected circular
motions that potentially arise from semi-coherent “hotspots" orbiting in a
near face-on accretion flow between $6r_{g}$ and $10r_{g}$. These circular
motions arise during infrared flares from Sgr A*; in addition to signatures of
a circular motion from interferometric data. GRAVITY observed “loops" in the
near-infrared EVPA which suggests that the emitting hot spots are situated in
a vertical magnetic field. Similar “loops" have been seen in submm
observations (Marrone et al., 2006). These hotspots may correspond to the
formation of magnetic flux tube bundles, or plasmoids, which seen in GRMHD
simulations with resistivity enabling plasmoid formation through magnetic
reconnection (Ripperda et al., 2020; Ball et al., 2020).

The Stokes $I$ Sgr A* spectrum we use is compiled in the Appendix of Ressler
et al. (2017); it spans from $10^{10.5}$ Hz radio to $10^{10.5}$ Hz X-rays. It
includes radio and millimeter data from Falcke et al. (1998); An et al.
(2005); Bower et al. (2015); Liu et al. (2016a, b); Doeleman et al. (2008) and
Johnson et al. (2015). Infrared data were taken from Cotera et al. (1999);
Genzel & Eckart (1999); Genzel et al. (2003); Schödel et al. (2007, 2011) and
Witzel et al. (2012). X-ray data over the range 2-10 keV were taken from
Neilsen et al. (2013); Baganoff et al. (2003). In this paper, we normalize our
230 GHz Sgr A* images so that the total flux is 3.5 Jy from Bower et al.
(2015). Note that earlier, proto-EHT visibility amplitude versus baseline
length measurements gave a value of 2.4 Jy (Doeleman et al., 2008).

##  3 Pair Production in Accretion/Jet systems

Particles near the supermassive black hole event horizon in LLAGN systems like
M87* and Sgr A* are relativistically hot, with temperatures exceeding
$10^{10}$ K $\approx 1$ MeV. The near-horizon environment thus features many
particles with energies above the pair creation threshold. In principle,
electron-positron pairs can be produced by several different channels
including: particle-particle collisions ($e^{-}p$ or $e^{-}e^{-}$), photon-
particle collisions (the Bethe-Heitler process, $\gamma e^{-}$ or $\gamma p$)
and photon-photon collisions (the Breit-Wheeler process, $\gamma\gamma$). In
practice, because plasma density is extremely low in LLAGN (with mean free
paths exceeding the system size) and because photon-photon collisions have the
greatest cross section near the pair-production threshold (e.g. Stepney &
Guilbert, 1983), photon-photon collisions dominate pair production in
astrophysical models of these sources (see also Mościbrodzka et al., 2011).
That is, in this work we only consider models where pair production arises
from:

$\gamma+\gamma\to e^{-}+e^{+}.$ (7)

Whether or not there is a substantial pair population in the near-horizon
region of low luminosity AGN (LLAGN) like Sgr A* and M87* remains an open
question. On the one hand, in models where an accretion disk produces most of
the observed luminosity, a large pair fraction is disfavored. For instance, in
M87* the temperature of the synchrotron radiating particles at 1 mm is likely
in the range $5-30$ MeV (EHT Collaboration et al., 2021b). As a first
estimate, we can assume this temperature is close to the disk virial
temperature $T_{\rm vir}$, which for a disk with particles of mean mass
$\bar{m}$ and at the distance $r$ from the gravitating mass $M$ is set by

$k_{B}T_{\mathrm{vir}}=\frac{1}{3}\frac{GM\bar{m}}{r}.$ (8)

Equation 8 implies that at a radius e.g. $r\approx 5\,r_{\rm g}$, $k_{B}T_{\rm
vir}\approx 60$ MeV, within an order of magnitude of the observed temperature.
However, since the electron mass is a fraction 1/1,836 of the proton mass if
we replace each proton with a positron in an accretion disk, the virial
temperature would be two orders of magnitude below the observed value. While a
fully pair-dominated disk is thus unlikely, we could instead have a scenario
where a small population of pairs is produced in the disk-jet corona from
$\gamma\gamma$ (Laurent & Titarchuk, 2018; Mościbrodzka et al., 2011). This
pair population could be subdominant in determining the overall structure of
the observed image at millimeter wavelengths, but it could still affect the
polarized source structure. In contrast, the scenario may be quite different
if most of the emission at 1 mm originates from nonthermal particles in a
relativistic jet with substantial Doppler beaming and relativistic aberration.
Reynolds et al. (1996) used a Blandford & Königl (1979) model of nonthermal
emission in a magnetized jet to put constraints on the emitting particle
density $n$ and magnetic field $B$ of M87’s jet emission at 5 GHz. They find,
if they demand the observed 5 GHz core is just becoming optically thick, the
observed surface brightness and total jet power prefer an emitting particle
distribution in the jet dominated by pairs. Similarly, in the broader class of
Blazar jets, Ghisellini (2012) finds that the inner regions of observed Blazar
jets can efficiently produce positrons with densities an order of magnitude
greater than the ion density as long as the gamma-ray luminosity $\gtrsim
10^{44}$ erg s-1. We note, however that the observed gamma-ray luminosity of
M87*’s core is $\sim 10^{42}$ erg s-1 (EHT MWL Science Working Group et al.,
2021).

In this section, we review two classes of models that have been used to
estimate pair production efficiencies in near-horizon accretion flow and jets
models of Sgr A* and M87*; gap acceleration models, where pairs are produced
from gamma-ray photons originating in an inverse-Compton cascade from seed
particles accelerated by unscreened electric fields in the jet, and “pair-
drizzle" scenarios, where pairs are produced directly from photons emitted by
the bulk accretion flow and upscattered by inverse Compton collisions. The
models we summarize here provide some motivations for the range of positron-
to-electron fractions we explore in semi-analytic models of M87* and Sgr A*
discussed later in the paper. However, note that in our models in the
remainder of this work, we are intentionally agnostic about the physical
source of pairs; instead, we focus on the observational consequences for
polarized emission, if a fraction of pairs is present from any physical
production channel. Note in particular that while both of the production
models we discuss in this section produce non-negligible pair densities for
M87* (with positron-to-electron fractions from $10^{-3}-1)$, both models
produce near-zero positron fractions for Sgr A*. Nonetheless, in Section 5 we
consider RIAF models for Sgr A* and investigate their images under the same
sampling of $f_{\rm pos}$ as we do for M87*.

###  3.1 Gap acceleration and pair cascade

Gap acceleration occurs in magnetospheres where there are unscreened electric
fields ($\mathbf{E}\cdot\mathbf{B}\neq 0$), so particles directly accelerate
particles to high energies (Goldreich & Julian, 1969). These accelerated
particles inverse Compton scatter on ambient photons, which then collide with
other photons to produce pairs in a cascade. This is the canonical mechanism
for supplying pulsar magnetospheres with plasma; it has been applied to black
hole magnetospheres in analytic models by (e.g. Blandford & Znajek, 1977;
Beskin et al., 1992; Hirotani & Okamoto, 1998; Vincent & Lebohec, 2010;
Broderick & Tchekhovskoy, 2015). Recently, this mechanism has been applied to
full general relativistic particle in cell simulations of black hole jets by
(Chen et al., 2018; Levinson & Cerutti, 2018). Here we summarize the results
of Broderick & Tchekhovskoy (2015) and their implications for potential pair
fractions in M87* and Sgr A*.

Broderick & Tchekhovskoy (2015) assume that particles are accelerated to
initiate a pair cascade at the stagnation surface that divides the outer
region of the jet where matter accelerates outward from the region where it
falls into the black hole. The charge-starved region around such a surface
allows the build-up of an unscreened, coherent electric field (Broderick &
Tchekhovskoy, 2015, equation 1)

$E\approx\frac{s\Omega_{B}B}{c},$ (9)

where $s$ is the cylindrical radius and $\Omega_{B}$ is the angular velocity
of a magnetic field line (assumed to be some fraction of the horizon angular
velocity $\Omega_{\rm H}=ac/2r_{\rm H}$, where $a$ is the black hole spin and
$r_{\rm H}$ is the horizon radius). This unscreened electric field forms a
“spark gap" that accelerates leptons to to high energies with energy
$\gamma_{\rm max}m_{\rm e}c^{2}\approx eE\ell$, where $\ell$ is the length
scale over which particles are accelerated. After reaching their maximum
Lorentz factor $\gamma_{\rm max}$, accelerated electrons cool by inverse
Compton scattering off of ambient seed photons. If the emitted photons have
sufficient energy they will produce pairs in an inverse Compton cascade. The
limiting Lorentz factor below which the cascade halts,
$\gamma_{\gamma\gamma}$, is set by the density of photons with energies above
the pair production threshold: $\epsilon_{\rm th}\gtrsim 4m_{\rm
c}c^{2}/\gamma_{\rm max}$. The total number of leptons produced in the post-
gap cascade is then enhanced by a factor $\gamma_{\rm
max}/\gamma_{\gamma\gamma}$ from its value at the top of the gap.

In practice, electrons are accelerated to a distance $\ell\approx\ell_{\rm
IC}$ set by the inverse-Compton cooling length:

$\ell_{\rm IC}=\frac{3m_{\rm e}c^{2}}{4\sigma_{\rm T}u_{s}\gamma_{\rm max}},$
(10)

where $u_{s}$ is the energy density of seed photons and $\sigma_{\rm T}$ is
the Thomson cross-section (Broderick & Tchekhovskoy, 2015, Equation 6). For
M87* and Sgr A*, they assume $u_{s}$ is the same as the the peak observed flux
density (Broderick & Tchekhovskoy, 2015, Equation 5),

$u_{s}=\frac{3L_{s}}{4\pi r_{s}^{2}c}$ (11)

where $L_{s}$ is the observed bolometric luminosity and $r_{s}\approx 10r_{\rm
g}$. For M87*, $L_{s}\approx 10^{41}$ erg s-1, so $u_{s}\approx 10^{-2}$ erg
cm-3. Assuming a black hole spin $a=0.9$ and requiring the jet power $P\approx
10^{44}$ erg s-1 to arise from the BZ mechanism, Broderick & Tchekhovskoy
(2015) calculate a field strength $B\approx 35$ G at the stagnation surface
$r\approx 10r_{\rm g}$. Using $\ell_{\rm IC}$ as the cutoff acceleration
length, they find $\gamma_{\rm max}\approx 1.6\times 10^{9}$, so the (lab
frame) seed photon pair production threshold is at $\epsilon_{\rm th}\approx
1.2$ meV. Critically, this is slightly below the spectral break, so the
density of available photons for the post-gap cascade is maximized. As a
result, while Broderick & Tchekhovskoy (2015) find a pair density of only
$n_{g}\approx 2.2\times 10^{-2}$ cm-3 at the top of the gap, for M87*, this
initial density is enhanced by a factor $\gamma_{\rm
max}/\gamma_{\gamma\gamma}\approx 670$ in the post-gap cascade to a final
lepton density of $n_{\rm lep}\approx 15$ cm-3. (Broderick & Tchekhovskoy,
2015) also provide an estimate for the number density of emitters from
observations of M87*, assuming $dn/d\gamma\sim\gamma^{-3.4}$, a source size of
$40\,\mu$as, a 230 GHz flux of 1 Jy, and a magnetic field $B\approx 35$ G;
this value is also $n_{\rm emit}\sim 15$ cm${}^{-3}.$333Note that the
Broderick & Tchekhovskoy (2015) estimate of the number density of 230 GHz
emitters is substantially lower than the estimate of $10^{4-5}$ $cm^{-3}$ from
EHT Collaboration et al. (2021b). his is because they assume a nonthermal
energy distribution beginning at $\gamma_{\rm min}\approx 100$, while the EHT
simulation modeling work assumes thermal particle distributions with lower
mean energies. Since $n_{\rm lep}=2n_{e^{+}}$ and $n_{\rm
emit}=n_{e^{-}}+n_{e^{+}}$, for the ratio of electrons to emitters we have:

$f_{\rm pos}=\frac{n_{e^{+}}}{n_{e^{-}}}=\frac{n_{\rm lep}}{2n_{\rm
emit}-n_{\rm lep}},$ (12)

which gives an estimate of:

$f_{\rm pos}\approx 1\;\;\;(\mathrm{M87^{*}}).$ (13)

In contrast, for Sgr A*, Broderick & Tchekhovskoy (2015) use a luminosity
$L\approx 10^{36}$ erg s-1, estimate a magnetic field of $B\approx 10^{2-3}$
G, and an initial seed photon energy density $u_{s}\approx 1.5\times 10^{-2}$
erg cm-3. The maximum Lorentz factor from gap acceleration is then
$\gamma_{\rm max}\approx 7.6\times 10^{8}$. As a result, the threshold energy
for inverse Compton scattering is at a higher energies than in M87*
$\epsilon_{\rm th}\approx 2.7$ meV. This is significantly above Sgr A*’s
spectral break at $\approx 1$ meV. Consequently, there are substantially fewer
photons available for the pair cascade in Sgr A* than in M87*. Therefore, the
mean free path of pair production $\ell_{\gamma\gamma}\approx 10r_{\rm g}$.
Because $\ell_{\gamma\gamma}$ is on the scale of the system size, there is no
efficient pair cascade after the gap acceleration, $n_{\rm lep}=n_{g}\approx
2.5\times 10^{-3}$ cm-3. Similar to M87*, Broderick & Tchekhovskoy (2015)
infer a total number density of emitters from fitting the 230 GHz source size
and flux density of 3 Jy to be $n_{\rm emit}\approx 10^{6}$ cm-3. As a result,
the positron-to-electron ratio can be inferred:

$f_{\rm pos}\approx 10^{-9}\;\;\;(\mathrm{Sgr\,A^{*}}).$ (14)

Thus, the Broderick & Tchekhovskoy (2015) pair cascade model predicts two
qualitatively distinct values of $f_{\rm e}$ for Sgr A* and M87*. Because the
post-gap cascade in M87* is efficient due to the higher density of seed
photons above the energy threshold, $f_{\rm pos}\approx 1$ and the plasma in
their model is assumed to be pair-dominated. In contrast, for Sgr A* the post-
gap cascade is not efficient and $f_{\rm pos}\approx 0.$

###  3.2 “Pair Drizzle"

Positrons in jets may also originate from annihilating high-energy photons
produced by incoherent structures in the accretion flow, rather than by
photons produced in the coherent gap regions in a cascade (Mościbrodzka et
al., 2011; Wong et al., 2021). In these “pair drizzle" models, high-energy
photons above the pair-production threshold are initially produced in an
accretion flow or jet wall by synchrotron self-Compton. Because the photons
produced by inverse Compton scattering off of typical thermal accretion flow
particles are less energetic than those produced in a pair cascade in a spark-
gap scenario, these models are less efficient at producing pairs (typically,
each photon accelerated by inverse Compton can only produce one pair).

Mościbrodzka et al. (2011) performed the detailed calculation of pair-
production efficiency from a GRMHD simulation tuned to both Sgr A* and M87*.
After determining the simulation accretion rate that produces the correct flux
density at 230 GHz, they determined the background population of high-energy
photons produced by inverse Compton scattering using the Monte-Carlo radiative
transfer code grmonty. The pair production rate per unit volume is then
determined from the distribution of seed photons using the Breit-Wheeler cross
section $\sigma_{\gamma\gamma}$. Recently, Wong et al. (2021) extended the
work of Mościbrodzka et al. (2011) by modeling the pair drizzle scenario using
radiation GRMHD simulations. In these simulations, grmonty is coupled to the
GRMHD code and the photon distribution and number density is solved for self-
consistently along with the plasma variables.

The best-bet “pair-drizzle" models in Mościbrodzka et al. (2011); Wong et al.
(2021) predict maximal pair number densities in the funnel of $n_{\rm
lep}=10^{-8}$ cm-3 ($\approx 10^{-5}$ times the Goldreich-Julian density for
Sgr A*) and $n_{\rm lep}=10$ cm-3 (or $\approx 10^{7}$ times the Goldreich-
Julian density for M87*). These quoted values for the lepton density $n_{\rm
lep}$ are provided above the pole, where the simulation number density is
determined by numerical floors. Hence, we cannot directly estimate a positron-
to-electron ratio directly from these quoted values. Naively, we may assume
the best estimate from these values is $f_{\rm pos}=1$, since the ion density
is only non-zero in the polar region because of simulation floors, so the pair
density from the Mościbrodzka et al. (2011); Wong et al. (2021) calculations
can be seen as representing the only physical matter density along the poles.
However, for the purposes of comparing to semi-analytic models, a more
reasonable approach is to compare the calculated lepton density from the pair
drizzle models to the electron number density in the 230 GHz emitting region.
Neither Mościbrodzka et al. (2011) nor Wong et al. (2021) provide this
information from their models, but we can estimate a reasonable order-of-
magnitude number from the quoted accretion rates Mościbrodzka et al. (2011):
$\dot{M}=2\times 10^{-8}\dot{M}_{\rm Edd}$ for Sgr A* and
$\dot{M}=10^{-6}\dot{M}_{\rm Edd}$ for M87*.444We define the Eddington
accretion rate $\dot{M}_{\rm Edd}=L_{\rm Edd}/0.1c^{2}$, where the Eddington
luminosity is $L_{\rm Edd}=4\pi GMm_{\rm p}c/\sigma_{\rm T}$. At a given
radius $r$, assuming spherical symmetry, we can then estimate:

$n_{\rm emit}\approx\frac{\dot{M}}{4\pi r^{2}v^{r}m_{p}},$ (15)

where for the infall velocity we can assume $v^{r}=c/\sqrt{r/r_{\rm g}}$. We
calculate the density at a radius $r=5r_{\rm g}$, which is a reasonable
estimate of the emission radius for M87* (EHT Collaboration et al., 2019d). We
then find from the quoted simulation accretion rates that $n_{\rm emit}\approx
1.4\times 10^{3}$ cm-3 for Sgr A* and $n_{\rm emit}\approx 2.2\times 10^{4}$
cm-3 for M87*. Using these simple estimates, we can compute positron-to-
electron fractions of:

$f_{\rm pos}\approx 4\times 10^{-3}\;\;\;(\mathrm{M87^{*}}),$ (16)

and

$f_{\rm pos}\approx 10^{-13}\;\;\;(\mathrm{Sgr\,A^{*}}).$ (17)

These estimates are approximation. They depend sensitively on where the 230
GHz emission originates in the accretion flow and what the produced pair
density and background electron density there are. For the pair density, we
have chosen only one value along the pole from Mościbrodzka et al. (2011). For
the background electron density we have chosen an estimate motivated by
spherical symmetry rather than a precise measurement from the emission
location. Furthermore, note that both Mościbrodzka et al. (2011) and Wong et
al. (2021) focus only on one type of accretion flow with weak magnetic flux on
the black hole. The M87* accretion flow, in contrast, is expected to be
strongly magnetized (in the MAD state EHT Collaboration et al. 2021b).
Nonetheless, we take these estimates as motivating values for our choices of
$f_{\rm pos}$ which we explore in the models below.

##  4 Semi-Analytic Models

Our primary objective in this work is to develop representative semi-analytic
models for polarized emission from jet/accretion flow systems that incorporate
the effects of a non-zero positron fraction. we are particularly interested in
predictions from these models for the polarized submillimeter emission
structure that may be resolved by the EHT. The strategy we employ is to modify
existing radiative transfer pipelines to allow for a non-zero positron
content, along with other physical parameters such as the spatial distribution
of particles, the non-thermal particle fraction, and the parameterized energy
distribution function of the emitters.

Here, we discuss the form of the models we use to model emission from M87* and
Sgr A*. While in the next Section, we discuss the implementation of positron
effects we use in our radiative transfer code.

###  4.1 M87*: Semi-Analytic Jet Model

For M87*, we use a self-similar, semi-analytic jet model calibrated to
measurements from a GRMHD simulation. Our semi-analytic jet model is based on
that introduced in Anantua et al. (2020a). However, we extend this model in
significant ways: we add general relativistic ray tracing, we consider non-
thermal distributions with arbitrary power-law indices $p\neq 2$, we map the
particle number density to the total electron/positron pressure rather than
the partial pressure of particles contributing within an octave of the
observed frequency, and we consider overall higher electron-to-magnetic
pressure ratios $\beta_{\rm e}$. Here we summarize the full model description
including these changes.

The semi-analytic jet model assumes self-similarity in the jet parabolic
parameter $\xi=s^{2}/z$, where $(s,z,\phi)$ are cylindrical coordinates. The
model is determined by three fitting functions in $\xi$ we determine from
GRMHD simulation data. These fitting functions are for the magnetic flux
threading the jet cross-section $\Phi_{B}(\xi)$, the field line angular speed
$\Omega_{B}(\xi)$, and the $z$-component of the velocity vector $v_{z}(\xi)$.

We fit to these quantities in the finite simulation region, then extrapolate
to large radii through self-similarity in $\xi$.

The HARM simulation (McKinney et al., 2012) we use as the basis of our semi-
analytic model is a magnetically arrested disk (MAD) accreting on a black hole
with spin $a/M=0.92$, and the fitting forms for plasma variables in the self-
similar model Anantua et al. (2020a) are computed at altitude $z=50M$. The
simulation outputs code units in terms of arbitrary scales for the black hole
mass $M$ and field strength $B$. Physical quantities such as the length scale
$r_{\rm g}\equiv GM/c^{2}$ and time scale $t_{\mathrm{g}}\equiv GM/c^{3}$ are
then determined through specifying the black hole mass $M_{\mathrm{BH}}$ and
the magnetic field scale $B_{0}$ such that we obtain the observed flux of 1.5
Jy for M87* (EHT Collaboration et al., 2019c). 555An alternative procedure for
code unit conversion for the magnetic field can be found in Table 1 in
Appendix A of Anantua et al. (2020a)). We fix the M87* black hole mass
$M_{\mathrm{87}}=6.5\times 10^{9}M_{\odot}$ (EHT Collaboration et al., 2019c),
which gives $r_{\rm g}=9.6\times 10^{14}$cm and $t_{\rm g}\approx 9\mathrm{\
hr}$.

The fitting forms for the magnetic flux, field line angular speed, and
$z-$component of the velocity are adapted from Anantua et al. (2020a) (given
in dimensionless code units):

$\displaystyle\Phi_{B}(\xi)$ $\displaystyle=\tanh(0.3\xi),$ (18)
$\displaystyle\Omega_{B}(\xi)$ $\displaystyle=0.15e^{-0.3\xi^{2}},$ (19)
$\displaystyle v_{z}(\xi,z)$
$\displaystyle=\left[-1.2+2.2\mathrm{tanh}(0.85\log
z)\right]e^{-0.001\xi^{4}}.$ (20)

We set $\xi_{\mathrm{max}}=10$ as the boundary of the jet. Given these fitting
forms, the magnetic field in cylindrical coordinates is:

$\displaystyle B_{s}$
$\displaystyle=B_{0}\frac{s\Phi_{B}^{\prime}(\xi)}{2\pi^{2}},$ (21)
$\displaystyle B_{z}$ $\displaystyle=B_{0}\frac{\Phi_{B}^{\prime}(\xi)}{\pi
z},$ (22) $\displaystyle B_{\phi}$
$\displaystyle=B_{0}\frac{-s\Omega_{B}(\xi)\Phi_{B}^{\prime}(\xi)}{\pi z},$
(23)

where $B_{0}$ is the scaling factor between dimensionless code units and
physical units that we measure by the fixing the total flux density of the 230
GHz image. Note that the total magnetic flux in the jet is conserved with
height $z$. If we knew the total magnetic flux in the jet $\Phi_{H}$, we could
instead use it to determine $B_{0}$ via:

$\Phi_{H}=\int_{0}^{s_{\mathrm{max}}(z)}2\pi sB_{z}ds\approx B_{0}r_{g}^{2}.$
(24)

Once we have determined the magnetic field at a point in the jet and converted
it to the cgs units, the magnetic pressure throughout the jet is read as:

$P_{B}=\frac{B^{2}}{8\pi}.$ (25)

The three-velocity vector in this model is:

$\begin{pmatrix}v_{s}\\\ v_{\phi}\\\
v_{z}\end{pmatrix}=\begin{pmatrix}\frac{s}{2z}v_{z}(z,\xi)\\\
s\Omega_{B}\left(1-v_{z}(z,\xi)\right)\\\ v_{z}(z,\xi)\end{pmatrix},$ (26)

where the fitting function for $v_{z0}(z)$ is given by Eq 20.

Considering the model before adding pairs, we assume the emitting electrons
have a power-law distribution function between a minimum Lorentz factor
$\gamma_{\rm min}$ and a maximum Lorentz factor $\gamma_{\rm max}$:

$N_{e^{-}}(\gamma)=\begin{cases}N_{0}\gamma^{-p}&\gamma_{\rm
min}\leq\gamma\leq\gamma_{\rm max}\\\ 0&\mathrm{otherwise}\end{cases},$ (27)

where $N_{0}=n_{e^{-}}(p-1)/\left(\gamma_{\rm min}^{1-p}-\gamma_{\rm
max}^{1-p}\right)$ is the overall normalization for the distribution and
$n_{e^{-}}$ is the total electron number density. In our emission model, we
determine the electron pressure throughout the jet by specifying the electron
pressure to magnetic pressure ratio $\beta_{\mathrm{e}}=P_{e^{-}}/P_{B}$
throughout the emission region. In this work, we assume that the
$\beta_{e}=\beta_{e0}$ is constant. The total electron number density for the
power-law distribution in Equation 27 is then give by:

$n_{e^{-}}=\frac{3}{m_{e}c^{2}}\frac{(p-2)}{(p-1)}\frac{\gamma_{\rm
min}^{1-p}-\gamma_{\rm max}^{1-p}}{\gamma_{\rm min}^{2-p}-\gamma_{\rm
max}^{2-p}}\beta_{e}\,P_{B}.$ (28)

In our emission model, we propose that each emitting positron has the same
distribution function in the $\gamma$ space, but scaled according to a
different total number density. The ratio of positrons to electrons:

$f_{\mathrm{pos}}\equiv\frac{n_{e^{+}}}{n_{e^{-}}},$ (29)

is taken to be constant throughout the emitting region and is a free parameter
in our models. The total positron number density $n_{e^{+}}$ is then given by
$n_{e^{+}}=f_{\mathrm{pos}}n_{e^{-}}$.

###  4.2 Sgr A*: Radiatively Inefficient Accretion Flow Model

For modelling Sgr A* images and spectra, we adapt the radiatively inefficient
accretion flow (RIAF) model first presented inBroderick & Loeb (2006a, b);
Broderick et al. (2009, 2016). This choice is motivated mainly from the most
recent observations of Sgr A* from Issaoun et al. (2019), who showed that disk
dominated models of Sgr A* are better matched with the observations than jet
models. Issaoun et al. (2019) found that jet dominated models are severely
constrained at 86 GHz, and the only jet models in their tested simulations
that are consistent with the measured 1.3 mm and 3 mm image sizes and
asymmetry are viewed nearly face-on at $\lesssim 20^{\circ}$. In a future work
we will consider hybrid disk-jet models for both Sgr A* and M87*.

Before considering pairs, the Broderick & Loeb (2006a) RIAF disk model
features axisymmetric, power-law spatial distributions of both thermal
electrons (with number density $n_{\mathrm{th}}$ and temperature $T_{e}$) and
nonthermal electrons (with number density $n_{\mathrm{nth}}$, power-law index
$p$, minimum Lorentz factor $\gamma_{\rm min}$, and maximum Lorentz factor
$\gamma_{\rm max}$). Following Broderick et al. (2009), we define the electron
temperature, thermal and nonthermal number densities as power-laws:

$n_{\mathrm{th}}=n_{0,\mathrm{th}}\left(\rho/2\right)^{-1.1}e^{\left(-z^{2}/2s^{2}\right)}$
,
---
$T_{e}=T_{0}\left(\rho/2\right)^{-0.84}$ ,
$n_{\mathrm{nth}}=n_{0,\mathrm{nth}}\left(\rho/2\right)^{-2.02}e^{\left(-z^{2}/2s^{2}\right)}$,
(30)

where $\rho$ is the spherical radius while $s=\sqrt{\rho^{2}-z^{2}}$ is the
cylindrical radius. We take both $T_{0}$ and the nonthermal electron fraction
$f_{\rm nth}=n_{0,\mathrm{nth}}/n_{0,\mathrm{th}}$ as free parameters in our
model. We fix $n_{0,\mathrm{th}}$ by requiring that our model produces the
observed 3.5 Jy 230 GHz flux density for Sgr A* (Bower et al., 2015).

The magnetic field strength in the Broderick & Loeb (2006a) model is toroidal
everywhere. The overall field strength is set by a total plasma $\beta$, which
we take as a free parameter:

$\frac{B^{2}}{8\pi}=\frac{1}{\beta}\frac{n_{\mathrm{th}}m_{p}c^{2}}{24r}.$
(31)

The fluid velocity is Keplerian outside of the innermost stable circular orbit
(ISCO) radius. While the fluid is in free-fall inside the ISCO.

Exactly as in the M87 jet model, if a positron population is present, we
assume it has the same properties as the thermal+nonthermal electron
distribution with a different overall density normalization. The ratio of
positron-to-electron number densities $f_{\mathrm{pos}}=n_{e^{+}}/n_{e^{-}}$
is again fixed throughout the plasma and is taken as a free parameter in our
models.

##  5 Implementation and model space

In this Section, we describe the implementation of our semi-analytic jet and
RIAF models in the public GRRT code GRTRANS and we summarized our strategy for
surveying the model space available to each model.

###  5.1 Radiative Transfer with GRTRANS

Polarized radiative transfer through an astrophysical plasma must account for
the effects of differential polarized emissivities, absorption, Faraday
conversion of linear to circular polarization, and Faraday rotation of the
linear polarization EVPA. In a curved spacetime, the emitted light received by
the observer is also modified by the curvature of photon trajectories and the
parallel transport of the polarization basis along geodesics.

Several numerical codes exist that perform polarized radiative transport in
the Kerr spacetime. These include e.g. IPOLE (Mościbrodzka & Gammie, 2017),
BHOSS (Younsi et al., 2020), and GRTRANS (Dexter, 2016), which we use in this
work. GRTRANS first solves for the geodesic trajectories between the pixels on
the observer screen and the emitting region around the black hole. It then
performs radiative transfer along these geodesics to solve for the Stokes
parameters $I,Q,U,V$. It assumes that the radiative transfer on different
geodesics is independent, and so cannot account for scattering processes. In
this work, we consider synchrotron radiation only.

In the local, emitting frame, the full polarized radiative transfer equations
for the Stokes parameters $I,Q,U,V$ take the form:

$\displaystyle\frac{d}{ds}\begin{pmatrix}I^{\prime}\\\ Q^{\prime}\\\
U^{\prime}\\\ V^{\prime}\end{pmatrix}=\begin{pmatrix}j^{\prime}_{I}\\\
j^{\prime}_{Q}\\\ j^{\prime}_{U}\\\
j^{\prime}_{V}\end{pmatrix}-\begin{pmatrix}\chi^{\prime}_{I}&\chi^{\prime}_{Q}&\chi^{\prime}_{U}&\chi^{\prime}_{V}\\\
\chi^{\prime}_{Q}&\chi^{\prime}_{I}&\rho^{\prime}_{V}&\rho^{\prime}_{U}\\\
\chi^{\prime}_{U}&-\rho^{\prime}_{V}&\chi^{\prime}_{I}&\rho^{\prime}_{Q}\\\
\chi^{\prime}_{V}&-\rho^{\prime}_{U}&-\rho^{\prime}_{Q}&\chi^{\prime}_{I}\end{pmatrix}\begin{pmatrix}I^{\prime}\\\
Q^{\prime}\\\ U^{\prime}\\\ V^{\prime}\end{pmatrix},$ (32)

where the $j$s refer to emissivities, the $\chi$ terms are absorption
coefficients, and the $\rho$ describe the "rotativities" that mix linear and
circular polarizations. For instance, $\rho_{V}$ is the Faraday rotation
coefficient that rotates the plane of linear polarization, while $\rho_{Q}$ is
the Faraday conversion coefficient that converts linear into the circular
polarization. GRTRANS simplifies these equations by rotating the emitting
frame so that $j_{U},\chi_{U},\rho_{U}=0$. GRTRANS also accounts for Doppler
boosting of the emitted intensities from the radiation to the lab frame, as
well as the parallel transport of the polarization frame (which can be
specified in terms of a rotation of the Stokes basis in Equation 32, see
Dexter (2016) Section 2.2).

The $j,\chi$ and $\rho$ coefficients we use are all for synchrotron radiation.
They depend on the electron distribution function $N(\gamma)$ as well as the
magnetic field strength and orientation relative to the wavevector in the
fluid rest frame. In this work, we consider both thermal (for the RIAF model)
and non-thermal (for both the jet and RIAF models) lepton distributions. The
coefficients we use in GRTRANS are defined in Appendix of Dexter (2016) for
both the thermal and nonthermal distributions.

In all of our models, we assume a constant positron-to-electron number density
ratio $f_{\mathrm{pos}}$. To include the effects of positrons in the radiative
transfer, we simply modify the emissivity and rotativity coefficients as
follows:

$j_{I,Q,U}\to\left(1+f_{e^{+}}\right)j_{I,Q,U}$,
---
$j_{V}\to\left(1-f_{e^{+}}\right)j_{V}$ ,
$\chi_{I,Q,U}\to\left(1+f_{e^{+}}\right)\chi_{I,Q,U}$,
$\chi_{V}\to\left(1-f_{e^{+}}\right)\chi_{V}$ ,
$\rho_{Q,U}\to\left(1+f_{e^{+}}\right)\rho_{Q,U}$,
$\rho_{V}\to\left(1-f_{e^{+}}\right)\rho_{V}$.
(33)

Equation 33 indicates that e.g. the emissivities for total intensity and
linear polarization increase in magnitude with the addition of emitting
positrons, as their total intensity and linear polarization emission is
indistinguishable from that of electrons. The circular polarization emissivity
is decreased by the addition of positrons as positrons emit circular
polarization in the opposite sense to electrons. However, the Faraday
conversion coefficient which determines how linear polarization is converted
into the circular polarization, $\rho_{Q}$, is _increased_ in magnitude (with
the same sign) in the presence of positrons, while the Faraday rotation
coefficient $\rho_{V}$ _decreases_ ; there is no Faraday rotation in a pure
pair plasma.

###  5.2 M87* Jet Model Parameters

In producing image of our jet model for M87*, we assume a dimensionless black
hole spin of $a/M=0.5$ and a viewing angle of $i=20\deg$ EHT Collaboration et
al. (2019d). We produce resolved images at 230 GHz and 86 GHz, as well as
spectra from $10^{9}-10^{16}$ Hz.

In producing model images and spectra from the M87* jet model, we fix the
electron-to-magnetic pressure ratio $\beta_{e}$ to be constant throughout the
jet (Eq. 28):

$\beta_{e}=\beta_{e0}.$ (34)

This simple emission prescription assumes the relativistic electron pressure
(which we define by _not_ including the positron contribution) is a constant
fraction $\beta_{\mathrm{e0}}$ of the local magnetic pressure everywhere. We
call this model the “Constant Electron Beta Model," following Blandford &
Anantua (2017); Anantua et al. (2018).666 Note that unlike Anantua et al.
(2020a), we drop prefactors $\frac{n-n_{i}}{n}=\frac{n_{e^{-}}+n_{e^{+}}}{n}$
to simplify the changes in the GRTRANS code when incorporating positrons.

We survey a space of constant electron beta values in the range:

$\beta_{\mathrm{e0}}\in\\{10^{-6},10^{-4},10^{-2}\\}$ (35)

We also vary the power-law index $p$ of the emitting distribution:

$p\in\\{2.5,2.8,3.0,3.2,3.5\\}.$ (36)

Finally, we also vary the positron fraction in the range:

$f_{\mathrm{pos}}\in\\{0.0,0.01,0.1,0.5,1.0\\}.$ (37)

This range of positron values spans the range from pure ionic to pure pair
plasmas and allows us to characterize the effect of adding pairs across a
large range of parameter space. However, we focus on small values of the
positron fraction $\lesssim 10\%$, motivated by theoretical estimates for the
positron fraction in M87* and Sgr A* from various pair production mechanisms
(Section 3), (Mościbrodzka et al., 2011; Broderick & Tchekhovskoy, 2015; Wong
et al., 2021). In Anantua et al. (2020b), we found that the constant
$\beta_{e0}$ model could produce low circular polarization fractions $V/I$ of
$\mathcal{O}(10^{-3})$ with low
$\beta_{\mathrm{e0}}=10^{-10},10^{-8},10^{-6}$. For the higher
$\beta_{\mathrm{e0}}$ values which we use in the current work, we see a
different interplay of coupled emissivities, absorption coefficients and
rotativities– resulting in significant Faraday effects and more circular
polarization.

Throughout this work, we vary $\beta_{\mathrm{e0}}$, $p$, and
$f_{\mathrm{pos}}$. We fix all other parameters, such as the minimum and
maximum Lorentz factors: $\gamma_{\mathrm{min}}=10$, and
$\gamma_{\mathrm{max}}=5\times 10^{3}$. We determine the magnetic field scale
$B_{0}$ by finding the value that produces the observed total flux from M87*
at 230 GHz.

####  5.2.1 Sgr A* RIAF model parameters

For modeling Sgr A*, we use the RIAF model from Broderick & Loeb (2006a)
described in Eqs. 30. We fix the fiducial dimensionless black hole spin as
$a=0.1$, inclined $60^{\circ}$ relative to the line of sight, motivated by the
proto-EHT model fitting results from Broderick et al. (2016).

In our implementation, the Broderick & Loeb (2006a) RIAF model is determined
by seven plasma parameters: the normalization of the thermal and nonthermal
number density profiles $n_{0,\mathrm{th}}$ and $n_{0,\mathrm{nth}}$, the
normalization of the temperature profile $T_{0}$, the plasma $\beta$ that sets
the $B$-field strength, the power-law slope of the nonthermal distribution
$p$, and the minimum and maximum nonthermal Lorentz factors
$\gamma_{\mathrm{min}}$ and $\gamma_{\mathrm{max}}$.

Because current EHT polarimetric constraints are stronger for M87* than for
Sgr A*, we explore a smaller parameter space in the Sgr A* RIAF model. Our
goal is not to find a best-fit RIAF model, but to explore the trends in the
model across a few parameters of interest when adding positrons to the
emitting plasma. In particular, we fix $T^{0}_{e}=1.5\times 10^{11}K$. We
determine the non-thermal part of the electron number density through the
ratio $f_{\rm nth}$ by $n_{0,\rm{nth}}=n^{0}_{e}f_{\rm{nth}}$, and we fix the
value $f_{\rm{nth}}=0.01$. We also fix $p=2.8$ and $\gamma_{\rm min}=100$ and
$\gamma_{\rm max}=10^{6}$. Once all other model parameters are set, we fix the
thermal number density $n_{0,\mathrm{th}}$ by requiring the total flux at 230
GHz $F_{230}=3.5\rm{Jy}$.

The two parameters we explore in the RIAF model are $\beta$ and $f_{\rm pos}$.
We explore plasma-beta in the range:

$\beta\in\\{0.01,0.1,1.0,10.0\\},$ (38)

to explore sub-equipartition, equipartition and super-equipartition plasmas.
The positron fractions we explore span the same range as the M87* jet model;
which is:

$f_{\mathrm{pos}}\in\\{0.0,0.01,0.1,0.5,1.0\\}.$ (39)

##  6 Model Exploration and Constraints

After setting up semi-analytic models for M87* and Sgr A* that include
positrons in the emitting lepton population in Section 4 and describing the
model space we explore in Section 5, we scan the parameter space for each
model defined in Section 5, exploring the physics of different emitting
particle distributions, relative magnetic field strengths, and positron
content. For each model parameter set, we produce 230 GHz Stokes maps and
polarized spectra from radio to X-ray frequencies.

In this section, we discuss constraints from observations of M87* and Sgr A*
that we apply on these results from each model. In particular, for both M87*
and Sgr A* we compute reduced-$\chi^{2}$ goodness-of-fit statistics to the
overall SED and disfavor models with the largest $\chi^{2}$s. We also compare
230 GHz images of our M87* jet model to the most recent polarization
constraints from Event Horizon Telescope Collaboration et al. (2021) and
construct a pass/fail table of models according to whether they satisfy the
EHT constraints Finally, we determine best-bet models which best satisfy both
sets of observational constraints. For Sgr A*, on the other hand, we limit our
study to a smaller number of models similar to the ones fit to the proto-EHT
data in Broderick et al. (2016). We add the effects of positrons to the
emission model in this setup and discuss which models are closest to the
observational constraints without determining a “best-bet" fiducial model for
the polarized emission.

###  6.1 Total Intensity SED $\chi^{2}$ values

As a first test of both the M87* constant beta model and Sgr A* RIAF model, we
construct $\chi^{2}$ factors comparing the predicted SED from each model to
the observational data from each source. We use total intensity SED data from
radio to X-ray frequencies, as described in Section 2. For each model, we use
this test to find a reasonable set of preferred models from the larger
parameter space we explore.

For the M87* jet model, we study a family of 75 models varying
$\left(\beta_{\mathrm{e0}},f_{\mathrm{pos}},p\right)$. Our analysis of the
total intensity SEDs from these models indicate that models with electron
spectral index $p=3.5$ best fit the near-infrared SED data, as we expect from
estimating the power-law index directly from the data in a one-zone power-law
synchrotron emission model. Furthermore, the fit to the total intensity SED
gets progressively better when we increase the lepton pressure by increasing
$\beta_{\mathrm{e0}}$. Table 2 presents the $\chi^{2}$ values for total
intensity SED for all of the M87* jet models we consider. Because the
observational SED is sparse in the interval
$10^{12}\mathrm{Hz}<\nu<10^{13}\mathrm{Hz}$, we report the $\chi^{2}$ values
for the radio and near-infrared parts of the spectrum separately, along with
the overall $\chi^{2}$ from all data points. Recall that we set the magnetic
field scale $B_{0}$ by fitting the model to the 230 GHz total flux of 1.5 Jy
for M87*.

We also study the total intensity SED $\chi^{2}$ for the Sgr A* RIAF model.
Here we fix most parameters from Broderick et al. (2016) and only vary two
parameters: $\beta,f_{\mathrm{pos}}$. The results of the Sgr A* SED fit
analysis are given in Table 3. Recall that for each model we set the thermal
electron number density $n_{0,\mathrm{th}}$ by fitting the model to the 230
GHz total flux of 3.5 Jy for Sgr A*.

We emphasize that in this work, our primary aim is not to determine a single
best fit model in our model space for either Sgr A* or M87* or to claim that
either of the models we put forward for these two sources are the most
complete or physically representative. Our aim is to find a “reasonable" model
for each source, chosen from a well-motivated set of parameters, and to
explore the observational effects of changing the positron content in these
models. In a future work, we will analyze the full parameter space available
to each model more completely and determine a set of best fit parameters and
their uncertainties with an MCMC posterior exploration code.

###  6.2 230 GHz Polarimetric Constraints

From Table 2, it is evident that the M87* jet model $\chi^{2}$ constructed
from the total intensity SED does not show a strong dependence on the positron
fraction. To place constraints on $f_{\rm pos}$ in these models, we need to
turn to polarimetric data constraints. M87* and Sgr A*’s polarized SED’s are
much more poorly constrained than the SED in total intensity. Here, we only
use the most recent observational constraints for the linear and circular
polarization at 230 GHz to distinguish between models. We find, similarly to
the M87* GRMHD analysis in EHT Collaboration et al. (2021b), that these
polarimetric constraints break degeneracies between different models that
equally satisfy the total intensity observations.

####  6.2.1 M87* 230 GHz polarimetric constraints

For M87*, we use the most recent polarimetric constraints at 230 GHz from EHT
Collaboration et al. (2021b). While that work reports the resolved and
unresolved fractional linear polarization, it only provides an upper limit on
the circular polarization from simultaneous ALMA observations during the 2017
EHT campaign (Goddi et al., 2021). In summary, the most recent observational
constraints on the fractional linear and circular polarizations defined in
Eqs. (4-5):

$\displaystyle 1\%\leq|m|_{\mathrm{net}}\leq 3.7\%,$ $\displaystyle
5.7\%\leq\langle|m|\rangle\leq 10.7\%,$ $\displaystyle
0\%\leq|v|_{\mathrm{net}}\leq 0.8\%~{}.$ (40)

The EHT ranges for the polarimetric ratios above are conservative,
incorporating the results from several image reconstruction techniques on the
M87* data. However, these ranges do not incorporate source variability on
timescales longer than $\sim 1$ week, which may be significant. As a
consequence, we score models against the EHT constraints in two ways. First,
we use the direct EHT ranges from EHT Collaboration et al. (2021a) as strict
constraints – none of our models pass these constraints directly. Second, we
interpret the EHT ranges as a central value and $1\sigma$ error region and
consider whether any models pass within $1.5\sigma$ (equivalent to expanding
the allowed range by 50% from the mid-point to account for possible future
source variability). Here we find some of our models do satisfy this looser
constraint. In Table 4 in Appendix C, we compute the aforementioned quantities
in our different models and we give scores of pass (P) or fail (F) to
individual models based on their performance on all three constraints.
Notably, only 4% of models pass all of the constraints (Section 6.3.1) at
1.5$\sigma$.

####  6.2.2 Sgr A* 230 GHz polarimetric constraints

Our exploration of the Sgr A* RIAF model space is more limited than for the
M87* jet model, and we mostly use this model to illustrate physical effects of
adding positrons of a disk-dominate system. However, we do make a preliminary
comparison of Sgr A* to polarimetric data, using observational constraints
from Johnson et al. (2015); Bower et al. (2018b). These constraints are:

$\displaystyle|m|_{\mathrm{net}}\simeq$ $\displaystyle 7\%,$ $\displaystyle
10\%\leq\langle|m|$ $\displaystyle\rangle\leq 30\%,$
$\displaystyle|v|_{\mathrm{net}}\leq$ $\displaystyle 1.0\%~{}.$ (41)

Since our parameter space is more limited and these data constraints will soon
be supplemented with the results from the EHT 2017 observations, in what
follows, we do not attempt to make a pass-fail table for our sample of RIAF
models, and we leave a detailed evaluation of the RIAF models to future
studies. Instead, we take a more qualitative approach and discuss the
implications of these initial constraints on the RIAF models. We provide
values of $|m|_{\rm net}$, $|v|_{\rm net}$ and $\langle|m|\rangle$ for the Sgr
A* RIAF models in Table 5.

###  6.3 Best-Bet Models

Having applied total intensity SED and 230 GHz polarimetric constraints to the
models, here we determine the passing “best-bet" models for the M87* jet. For
the Sgr A* RIAF model, on the contrary, we choose the family of models that
are closer to the current constraints, while we leave a more detailed analysis
for future work.

####  6.3.1 M87* jet model

We choose our best-bet model from the initial set of 75 models for M87* by
selecting models that have a low total $\chi^{2}$ when comparing to the total
intensity SED and that also satisfy the current 230 GHz polarimetric
constraints as described above. In general, models with higher
$\beta_{\mathrm{e0}}$ do a better job fitting the total intensity SED, and
models with small values of $f_{\mathrm{pos}}$ are generally closer to the EHT
polarimetric constraints. We settle on three fiducial best-bet models that we
label A,B,C as follows:

$\displaystyle\mathrm{Model}=$
$\displaystyle[f_{\mathrm{pos}},\beta_{\mathrm{e0}},p],$ $\displaystyle
A\equiv$ $\displaystyle[0.0,10^{-2},3.5],$ $\displaystyle B\equiv$
$\displaystyle[0.01,10^{-2},3.5],$ $\displaystyle C\equiv$
$\displaystyle[0.1,10^{-2},3.5].$ (42)

These models differ only in their positron fraction. From Table 4, it is
evident that some of these only barely satisfy the above polarimetric
constraints, but all perform significantly better than other models in the
set, especially when compared with those with very low values of
$\beta_{\mathrm{e0}}\sim 10^{-6}$ or with $f_{\mathrm{pos}}\gtrsim 0.5$. The
polarimetric constraints are very powerful in breaking the degeneracies
between different models that have similar $\chi^{2}$ values in total
intensity.

####  6.3.2 Sgr A* RIAF model

In considering the Sgr A* RIAF model, we explore a smaller parameter space
focused primarily on varying plasma-$\beta$ and the positron fraction $f_{\rm
pos}$. We fix all other parameters to those fit in Broderick et al. (2016).
Our aim is not to completely explore the parameter space for this model, but
to identify trends in the resulting images and spectra with different values
of plasma $\beta$ and positron fraction $f_{\rm pos}$.

At the level of the total intensity SED $\chi^{2}$, models with different
values of plasma-$\beta$ and positron fraction $f_{\rm pos}$ all fit the data
relatively well. However, their predictions for the 230 GHz linear and
circular polarizations are quite different. Applying the above constraints to
our very limited family of RIAF models, we find that models with lower values
of $\beta$ do a better job. Consequently, we pick the model with
$\beta=10^{-2}$ as our fiducial model. In the next section, we discuss the
effects of varying $f_{\rm pos}$ in this model.

Figure 1:  Stokes maps for the inferred best-bet M87* jet models in Sec.
6.3.1. From the Top to Bottom, we increase the $f_{\mathrm{pos}}$ associated
with models A-C, respectively. Before display, the images were blurred to the
nominal EHT resolution by convolution with a circular Gaussian kernel with
FWHM of 20 $\mu$as. The brightness temperature in each model is displayed in a
linear scale. Ticks overlaid on the Stokes $I$ image denote the EVPA, and are
plotted where the fractional linear polarization is above 0.5 % of its maximum
value.

Figure 2:  Stokes maps for M87* jet model with $f_{\mathrm{pos}}=(0.5,1.0).$
The rest of the model parameters are fixed following the best-bet model. We
plot all panels in the same style as in Fig. 1 to facilitate comparison; as a
result, the polarization maps for $f_{\rm pos}=1$ saturate the color scale.

##  7 Results: Images and Spectra

Having chosen fiducial, “best-bet" model parameters for M87* and Sgr A*, here
we discuss the polarized images and spectra from these models and their
behavior when varying quantities of the interest, particularly the positron
fraction $f_{\rm pos}$. For each family of models, we show several important
examples in the main text and we present a more complete overview of the model
space in Appendix D.

###  7.1 M87* constant $\beta_{\mathrm{e}}$ jet model

We start with the constant $\beta_{\mathrm{e}}$ jet model for M87* and present
230 GHz images and polarized spectra for the best-bet models (A-C) defined in
Sec. 6.3.1. Since the best-bet models vary only at the level of the positron
fraction, we also explore the impact of pushing the positron fraction
$f_{\mathrm{pos}}$ to large values out of the favored range from the EHT data
constraints.

####  7.1.1 Impact of $f_{\mathrm{pos}}$ on 230 GHz images

We first consider Stokes $I,Q,U,V$ maps of the 230 GHz emission from our
favored constant $\beta_{\rm e0}$ jet models for M87*. Figure 1 presents
Stokes maps for the fiducial models. Since in each case the magnetic field
scale $B_{0}$ is determined such that the total flux matches observations at
230 GHz, we see few overall changes in the Stokes $I$ image structure and
intensity. In each row, the images are blurred to the nominal EHT resolution
by a convolution with a circular Gaussian kernel with FWHM of 20 $\mu$as.

From the plot, it is apparent that models A and B (with
$f_{\mathrm{pos}}=0.0,0.01$) are visually difficult to distinguish, even in
polarized emission maps. Moving to model C (with $f_{\mathrm{pos}}=0.1$), it
is evident that the polarized intensities $Q,U$ and $V$ increase in magnitude
relative to $I$. In all cases, our best-bet models feature a bilaterally
asymmetric jet emanating from a ring around the central black hole, with
polarization tracing the high intensity regions. The EVPA pattern around the
ring is predominantly azimuthally spiraling as is observed in EHT
Collaboration et al. (2021a). The fractional polarization of the jet emission
is higher than the fractional polarization of the bright ring.

In Fig. 2, we present the case with $f_{\mathrm{pos}}=1.0$. Comparing Fig. 1
with Fig. 2, we infer the following features about the impact of positrons on
230 GHz emission maps of M87:

$\bullet$ Increasing the positron fraction $f_{\mathrm{pos}}$ reduces the
level of the bilateral asymmetry in the polarization maps as Faraday rotation
diminishes. Stokes $Q$ and $U$ increase with increasing positron fraction as
Faraday rotation is suppressed, the polarization pattern becomes more
coherent, and beam depolarization is suppressed (e.g. Jiménez-Rosales &
Dexter, 2018).

$\bullet$ Faraday conversion is the dominant source of circular polarization
in these models. Even in the pair plasma system the value of Stokes $V$ is
nonzero and the Stokes $V$ emission map is qualitatively similar to the
$f_{\rm pos}=0$ case, since the circular polarization is predominantly sourced
by conversion. The linear polarization fraction is thus a better probe of the
positron fraction in this model than the circular polarization fraction.

To justify our interpretation of the impact of Faraday effects in producing
the bilateral asymmetry polarization and sourcing the Stokes V, in Appendix E,
we show several examples in which we have manually turned off Faraday rotation
and conversion effects for comparison with these results.

Figure 3: 86 GHz Stokes maps for the fiducial M87* jet model with
$\beta_{\mathrm{e0}}=10^{-2}$ and $p$ = 3.5. From Top to Bottom, we vary the
fraction of positrons as $\beta_{\rm e0}=0.0,0.01,0.1,1.0$. All images are
shown in a linear scale and are blurred with a circular Gaussian of FWHM 20
$\mu$as.  Figure 4:  Total intensity spectral energy distribution (SED).
(Left) for our best-bet M87* jet models with $f_{\mathrm{pos}}=(0.0,0.01,0.1)$
and for the best-bet model modified with a maximum pair fraction
$f_{\mathrm{pos}}=1.0$. Spectrum of the unresolved fractional linear
polarization $|m|_{\rm net}$ (Center), compared with the EHT 2021 allowed
range of 1.0-3.7% (EHT Collaboration et al., 2021b). Spectrum of the
unresolved fractional circular polarization $|v|_{\rm net}$ (Right). Note that
none of our models satisfy the EHT constraints within $1\sigma$, but the
fractional linear and circular polarization for our best-bet models are within
$1.5\sigma$ of the EHT constraints. In contrast, both linear and circular
polarization fractions are significantly above the current observational
constraints for the full pair plasma case $f_{\rm pos}=1$.

###  7.2 Impact of $f_{\mathrm{pos}}$ on 86 GHz images

So far, we have only presented Stokes maps at a single frequency of 230 GHz.
In Figure 3, we present Stokes maps for the M87* jet model at the frequency of
$\nu=86$ GHz for different values of the positron fraction. We notice several
trends in the 86 GHz images w/r/t our 230 GHz images:

$\bullet$ The intensity distribution at 86 GHz is more extended (as the plasma
becomes optically thick) and the EVPA patterns become more ordered and radial
than at 230 GHz.

$\bullet$ The linear polarization no longer peaks at the position of the black
hole (as emission is optically thick there), but rather peaks farther along
the jet. We may see a similar effect in M87* Hada et al. (2016).

$\bullet$ The simple quadrupolar pattern in linear polarization azimuthal
variation observed at 230 GHz is no longer present, and we see a more complex
pattern in the most polarized region downstream from the black hole

$\bullet$ The 86 GHz circular polarization patterns remain ordered and
bilaterally anti-symmetric in the non-pair plasma case. In the pair plasma
case, the circular polarization peaks at the position of the black hole,
whereas for the ionic-dominated cases it peaks further down the jet.

The 86 GHz jet linear and circular polarization maps are distorted relative to
the 230 GHz cases such as Fig. 1 further from the core, as the $\tau=1$
surface of unit optical depth expands outward (note the absorption
coefficients for intensity and linear polarization scale as
$\nu^{-\frac{p}{2}-2}$ and for circular polarization as
$\nu^{-\frac{p}{2}-\frac{5}{2}}$).

####  7.2.1 Impact of $f_{\mathrm{pos}}$ on polarized spectra

Next, we focus on the total intensity SED and the spectra of fractional linear
polarizations $|m|_{\rm net}$ and $|v|_{\rm net}$ for our best-bet models
exploring the impact of $f_{\mathrm{pos}}$ in the polarized spectra. In
producing spectra, we have limited the field of view (FOV) to 120 $r_{g}$.
Consequently, these spectra do not contain extended emission on large scale
and cannot account for the low-frequency slope in total intensity below
$\nu=10^{10}$ Hz.

Fig. 7.2.1 presents the total intensity spectral energy density (SED), the
spectrum of the unresolved fractional linear polarization and the spectrum of
the unresolved fractional circular polarization for our best-bet model. We
also present spectra for the case with $f_{\mathrm{pos}}=1$ for comparison.
Included in each plot are the different observational constraints for the
total intensity SED and for the linear and circular polarization discussed in
Section 2. We see that while the best-bet models are relatively close to the
current polarimetric observational constraints, models with
$f_{\mathrm{pos}}=1$ sit well above the current EHT constraints are thus fully
ruled out.

###  7.3 RIAF Model Images and Spectra

Here we turn our attention to the RIAF model images and spectra for Sgr A*. As
we discussed above, both because current EHT polarimetric constrains for Sgr
A* are less robust than for M87* and because these models have been well-fit
to available data in Broderick et al. (e.g. 2016), we fix most parameters and
only vary plasma $\beta$ and $f_{\rm pos}$ in the present work.

####  7.3.1 RIAF Model: Impact of $f_{\mathrm{pos}}$ on 230 GHz images

First, we consider the Stokes maps for our fiducial RIAF model. The parameters
we use are $\beta=10^{-2}$, $f_{\mathrm{nth}}=0.01$, $T_{e}=1.5\times
10^{11}K$, $\gamma_{\mathrm{min}}=10$, $\gamma_{\mathrm{max}}=10^{6}$, and $p$
= 2.8. We fix the inclination at $i=60^{\circ}$ and orient the image at a
position angle of $\xi=156^{\circ}$ East of North. The BH spin is kept fixed
at $a$ = 0.1 (Broderick et al., 2016). We blur all image with a circular
Gaussian beam to a resolution of 15$\mu$as.

Fig. 6 shows the effects of increasing positron content in the RIAF models
with all other parameters kept fixed. All images show a polarized crescent
with a radial EVPA distribution. The linear polarization images $Q$ and $U$
exhibit azimuthal variation in two peak-valley cycles – these features can be
naturally interpreted as arising from the toroidal field geometry we fix in
the model. Note that special relativistic beaming causes the approaching half
of the disk to have a pronounced brightness enhancement. The addition of
positrons result in a more ordered, more radial EVPA pattern. As in our
exploration of M87* jet models, this increases in linear polarization with
increasing positron fraction occurs because Faraday rotation (and thus beam
depolarization) is suppressed (e.g. Jiménez-Rosales & Dexter, 2018)

In reality, a pair plasma RIAF accretion flow model is un-physical, as some
ions are required to maintain the disk virial temperature. Furthermore, both
pair-production scenarios we recap in Section 3 predict a polarization
fraction $f_{\rm pos}\approx 0$ for Sgr A*. Our results indicate that the
addition of a relatively small population of positrons will not significantly
modulate the polarized emission at 230 GHz.

Figure 5: Stokes maps for the Sgr A* RIAF models in Section 7.3. Before
display, the images were blurred with a circular Gaussian of 15$\mu$as FWHM.
EVPA ticks in the left panel are shown above 5% of the maximum linear
polarization fraction. The models shown have with $\beta=10$,
$f_{\mathrm{nth}}=0.01$, $T_{e}=1.5\times 10^{11}K$ and $p$ = 2.8. From Top to
Bottom, we vary the positron fraction as $f_{\rm pos}=0,0.01,0.1,1.0$.

Figure 6: Stokes maps for the Sgr A* RIAF models in Section 7.3. Before
display, the images were blurred with a circular Gaussian of 15$\mu$as FWHM.
EVPA ticks in the Left panel are shown above 5% of the maximum linear
polarization fraction. The models shown have with $\beta=10$,
$f_{\mathrm{nth}}=0.01$, $T_{e}=1.5\times 10^{11}K$ and $p$ = 2.8. From the
Top to Bottom, we vary the positron fraction as $f_{\rm pos}=0,0.01,0.1,1.0$.

####  7.3.2 RIAF Model: impact of $f_{\mathrm{pos}}$ on polarized spectra

Having presented the Stokes maps for RIAF models, below we discuss their total
intensity SEDs and fractional polarization spectra. Figure 7 presents the
total intensity SED (left), the unresolved fractional linear (center) and
circular (right) polarizations for our fiducial models with $\beta=10^{-2}$
and with $f_{\mathrm{pos}}=(0.0,0.01,0.1)$. Added to each panel, we also
present the case with $f_{\mathrm{pos}}=1.0$. In each panel, we have also
added the current observational constraints as they were mentioned in Sec.
2.3.

$\bullet$ First considering the total intensity SED, it is evident that
different models are degenerate in the radio band (dominated by thermal self-
absorption) and in the IR (dominated by power-law emission). Changing the
plasma-$\beta$ (and thus the magnetic field strength), shifts the position of
the synchrotron peak in the intermediate range $5\times
10^{11}\leq\nu/\mathrm{Hz}\leq 5\times 10^{13}$. As we saw in the M87* jet
model, smaller values of $\beta$ (larger field strengths) peak at higher
frequencies. Models at the same $\beta$ show only small differences in the
transition region from radio to IR frequencies with different values of
$f_{\rm pos}.$

$\bullet$ For the unresolved fractional linear polarization, at low
frequencies, $\nu\leq 10^{11}$ Hz, models with higher positron fractions are
more polarized. The enhancement in $|m|_{\rm net}$ with large $f_{\rm pos}$ is
again due to the suppression of Faraday rotation in models with more positron
content. As in M87* models, internal Faraday rotation is the primary source of
depolarization in such models, as the intrinsic polarization pattern is
rotated by different amounts on different lines-of-sight. While Faraday
rotation peaked in the submm in our M87* jet models, however, here it is most
significant at frequencies $<10^{11}$Hz.

$\bullet$ Finally, we see the same pattern in the unresolved fractional
circular polarization as we observed in the M87* jet model. The circular
polarization spectrum drops significantly in the pair plasma toward higher
frequencies, owing to the inefficiency of the Faraday conversion in this limit
(note $\rho_{V}\propto\nu^{-2}$) and the suppression of all intrinsic Stokes
$V$ emission in a pair plasma.

Figure 7:  Total intensity spectral energy distribution (SED) (Left) for our
best-bet Sgr A* RIAF models with $f_{\mathrm{pos}}=(0.0,0.01,0.1)$ and for the
best-bet model modified with a maximum pair fraction $f_{\mathrm{pos}}=1.0$.
Spectrum of the unresolved fractional linear polarization $|m|_{\rm net}$
(Center), compared with the Johnson et al. (2015). Spectrum of the unresolved
fractional circular polarization $|v|_{\rm net}$ (Right). We fix the fraction
of nonthermal-to-thermal particles as $f_{\mathrm{nth}}=10^{-2}$ and the
nonthermal power-law slope $p=2.8$.

##  8 Conclusions and Future Directions

###  8.1 Conclusions

In this paper, we have considered the effects of adding a non-zero positron
fraction to models of M87* and Sgr A*. Positrons may be generated by the
Breit-Wheeler process in LLAGN, either in a high-efficiency spark-gap electron
acceleration and inverse Compton cascade scenario (Broderick & Tchekhovskoy,
2015), or in a low-efficiency "pair drizzle" scenario (Mościbrodzka et al.,
2011; Wong et al., 2021). Both scenarios can give non-negligible positron-to-
electron fractions in the range $f_{\rm pos}\sim 10^{-3}-1$ in M87*; however,
both predict essentially zero positron fraction in Sgr A*. While we take these
results from the literature as motivating values for our choice of positron
fractions explored in this work, we extend the explored range of values in
both the Sgr A* and M87* models we consider to examine the observational
effects of a nonzero pair fraction in different source models in a way which
is agnostic to the production mechanism.

The first model we considered was for the M87* jet. We abstracted a GRMHD
simulation into a semi-analytic model of a Poynting flux jet populated by non-
thermal electrons and positrons. Fixing the flux at 230 GHz to its observed
values from the recent observations of M87*, we found the magnetic field scale
$B_{0}$ for each model, varying the distribution function slope $p$, the
electron-to-magnetic pressure ratio $\beta_{\rm e0}$, and the positron
fraction $f_{\rm pos}$. Analyzing the reduced $\chi^{2}$ of the resulting
total intensity SEDs and the 230 GHz constraints on the polarized structure,
we selected a reasonable fiducial model. We investigated the impact of
changing $\beta_{\rm e0}$ and $f_{\rm pos}$ on the Stokes maps and the
spectra. We found that increasing the positron content in the constant
$\beta_{\rm e0}$ model increases both the 230 GHz linear polarization fraction
(from decreased Faraday rotation and beam depolarization) and the circular
polarization fraction (from increased Faraday conversion). As a consequence,
the pair content in the M87* 230 GHz emitting region is unlikely to exceed
$f_{\rm pos}\approx 10^{-2}$. At 86 GHz, our jet models are most polarized
downstream from the core, with a distinct polarization pattern from the 230
GHz images.

We next considered a RIAF model of Sgr A*’s accretion flow (Broderick et al.,
2009). This geometric model describes a torus with power-law distributions of
temperature and number density height and a coherent, toroidal $B$-field
(poloidal and toroidal). We put $1\%$ of electrons and positrons in a power
law component, but thermal electrons and positrons supply most of the
emission. Throughout out analysis, we fixed the best-fit parameters from
(Broderick et al., 2016) and investigated the impact of varying plasma-$\beta$
and the positron fraction $f_{\mathrm{pos}}$ on the Stokes maps at 230 GHz and
on the polarized spectra. We show that, we can account for the total intensity
and total flux spectra with $1\%$ of the number density in power law electrons
and/or positrons. Changing $\beta$ does not alter the total intensity SED
significantly. The key discriminant among degenerate compositions is in the
polarization, where decreased Faraday rotation from higher pair fractions
result in higher linear polarization fractions at low frequencies. On the
other hand, the increased Faraday rotation from lower pair fractions at low
frequencies leads to somewhat smaller linear polarization at low frequencies.

In both models, the overall changes in both the linear and circular
polarization with low values of $f_{\rm pos}$ are small. The analysis in this
work suggests that large positron fractions $f_{\rm pos}\gtrsim 0.5$ have a
significant effect on the polarized SEDs and 230 GHz images from both model
classes, and the application of EHT constraints from (EHT Collaboration et
al., 2021b) suggests that the 230 GHz M87* emission region is predominantly
ionic. However, these results also suggest that current constraints are
unlikely to rule out or distinguish between pair fractions $\ll 50$%.

###  8.2 Future Directions

The analysis in this paper is an initial foray into examining the effects on
non-zero positron content on realistic physical models of the 230 GHz images
and broadband spectra from jet/accretion flow models of Sgr A* and M87*. We
plan to extend this analysis in several directions in a future work:

  1. 1.

A major limitation of the present analysis is the assumption of a constant
positron-to-electron fraction throughout the source. Pair-drizzle models, in
contrast, predict azimuthal structure in $n_{e^{+}}$, with the pair fraction
peaking at the poles in the jet region (Mościbrodzka et al., 2011; Wong et
al., 2021). In a forthcoming work, we will extend our models to include
spatially varying parameterizations for $f_{\rm pos}.$

  2. 2.

The current work has focused on introducing the models we consider and
presenting a few best-bet models and qualitative trends in the models when
increasing $f_{\rm pos}$. In a forthcoming work, we will narrow this on the
preferred parameter space determined here and provide a more complete fit to
the observational (total intensity and polarimeteric) multi-frequency data,
determining fitted values for model parameters like $p$, $\beta_{\rm e0}$ and
critically $f_{\rm pos}$ with associated uncertainties from an MCMC posterior
sampler.

  3. 3.

We have chosen a particular semi-analytic jet model for M87* derived from a
GRMHD simulation (Anantua et al., 2020a) for our M87* jet investigation. The
field structure in this model is fixed and not varied in our analysis.
Similarly, in our main RIAF model from Broderick & Loeb (2006b), we fix the
field orientation. The fixed field orientations determine our underlying EVPA
and Stokes $V$ patterns in both models.

  4. 4.

In this work, we consider only a pure jet model for M87* and a pure disk model
for Sgr A*. We chose these models to bracket the parameter space and explore
the effects of positrons on images of two well-known source morphologies.
However, a face-on jet model could explain both the observed Sgr A* SED and
230 GHz image size (Issaoun et al., 2019), and M87*’s 230 GHz emission may
originate in the equatorial plane from a MAD disk (EHT Collaboration et al.,
2019d; Chael et al., 2021). In realistic near-horizon accretion flow
simulations, both disk and jet emissions are present and can contribute to the
230 GHz image. A hybrid jet+accretion flow model may be a better choice to
model both sources than the limiting cases which we explore here.

  5. 5.

Both our Sgr A* and M87* models are static with smooth plasma and field
structure, but we know both sources are turbulent and dynamical and feature
fine-scale structure. In M87* in particular, turbulent magnetic fields can
help depolarize the net emission, which may bring our 230 GHz results more
comfortably in line with observations (though Faraday depolarization is the
dominant effect, see Jiménez-Rosales & Dexter 2018). We plan to extend our
models by adding a post-processing parameterization for turbulent structure.

  6. 6.

Finally, we will incorporate models for non-zero positron fractions (with
spatial dependence) into radiative post-processing of full GRMHD simulations.
Adding positrons to GRMHD images will naturally allow us to explore the
effects of positrons on different emitting regions (jet vs disk) as well as
the effects of fine-scale turbulence and source variability in our models. By
adding a prescription for nonzero $f_{\rm pos}$ to the production of ray-
traced images from GRMHD simulations such as presented in (EHT Collaboration
et al., 2019d, 2021b), we can also constrain the positron fraction directly
from the simulation images following the procedures in (EHT Collaboration et
al., 2021b) for model evaluation.

## Acknowledgements

It is a great pleasure to thank Charles Alcock, Charles Gammie, Michael
Johnson, Geoff Bower, Lars Hernquist, Angelo Ricarte, David Spergel and Craig
Walker for the very insightful conversations. R.E. and R.A. acknowledges the
support by the Institute for Theory and Computation at the Center for
Astrophysics. This work was supported in part by the Black Hole Initiative at
Harvard University, which is funded by grants from the John Templeton
Foundation and the Gordon and Betty Moore Foundation. The computations in this
paper were run on the FASRC Cannon cluster supported by the FAS Division of
Science Research Computing Group at Harvard University. RA acknowledges the
Center for Computational Astrophysics funded by the Simons Foundation. Support
for this work was provided by NASA through the NASA Hubble Fellowship grant
HST-HF2-51431 awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS5-26555.

Software: GRTRANS (Dexter, 2016), matplotlib (Hunter, 2007), numpy (van der
Walt et al., 2011), scipy (Oliphant, 2007).

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##  Appendix A Estimate of $\beta_{\rm e}$ for M87* in a BZ Jet Interpretation

In this section, we provide estimates of $\beta_{\rm e0}$ in the submm
emission region of M87* that we use to motivate our choices for the jet model
explored in Section 5. We make use of the the partial electron pressure,
(Blandford & Anantua, 2017; Anantua et al., 2020a):

$\tilde{P}_{\rm
e}=\frac{1}{3}\gamma^{{}^{\prime}2}N_{e^{-}}(\gamma^{\prime})m_{\rm e}c^{2},$
(A1)

which is the contribution to the total pressure from relativistic electrons
emitting within an octave of the observed frequency $\nu$ (where primes denote
the fluid co-moving frame, so $\gamma^{\prime}$ is the local electron Lorentz
factor that produces the observed emission at $\nu$, and $N_{\rm e^{-}}$ is
the electron distribution function per unit Lorentz factor). In the context of
mm-emitting electrons, we can relate the partial pressure to the observed
luminosity $L_{\nu}$ and local cooling time:

$\tilde{P}_{\rm
e}=\frac{1}{3}\frac{L_{\nu}t_{\mathrm{cool}}}{V_{\mathrm{emit}}}$ (A2)

where $V_{\mathrm{emit}}$ is the volume of the emitting region and
$t_{\mathrm{cool}}$ is the synchrotron cooling time:

$t_{\mathrm{cool}}=\frac{10}{\gamma}\left(\frac{10^{-4}}{B^{2}}\right)^{2}\mathrm{yr}$
(A3)

If we neglect the effects of red/blue-shift, the emitting Lorentz factor
$\gamma^{\prime}(\nu)$ is:

$\gamma^{\prime 2}=\frac{m_{\rm e}}{3\pi eB}\nu.$ (A4)

Note we have modeled the emission in terms of the total pressure in the main
text. Here, we seek an independent estimate of characteristic $\beta_{\rm e0}$
using only the partial pressure contribution from emitters that contribute to
the observed frequency.

To estimate $\beta_{\rm e0}$ in different emission regions along the M87* jet,
we use the jet geometry given by Hada et al. (2013): the cylindrical radius
varies with height as $s\propto z^{0.56}$ out to a de-projected distance
$z\approx 200r_{\rm g}$. The synchrotron luminosity $L_{\nu}$ can be deduced
from the spectrum from Prieto et al. (2016). Then, we can estimate
$\tilde{P}_{\rm e}$ and $\beta_{\mathrm{e}}=\tilde{P}_{\rm e}/P_{B}$ if we
have an estimation for the B-field along with the jet. For this, we use the
same jet model in Eq. 21, originally set in Anantua et al. (2020a) based on
the MB09D simulation from McKinney et al. (2012). Although in the main text we
set the scaling factor $B_{0}$ based on which value gives the desired 230 GHz
flux, here we make estimates using a fixed field strength set by using the jet
magnetic flux (Eq. 24) $\Phi_{H,\mathrm{M87}}\approx B_{0}r^{2}_{\rm g}\approx
10^{34}\ \mathrm{Mx}$ (Anantua et al., 2020a) (conversions between the cgs
scaling factors used in this paper and the mks unit scaling factors in Anantua
et al. (2020a) are given in Table 1). Note this jet flux of $10^{34}$ Mx
produces a jet much more powerful than the EHT lower limit of
$L_{\mathrm{jet}}\gtrsim 10^{42}$ erg s-1 used in EHT Collaboration et al.
(2019d).

Table 1: Code-to-physical unit conversions used in Anantua et al. (2020a). The mapping from these to the magnetic field conversion factors we use here to conserve energy between synthetic and observed M87* intensity maps with target flux 1.5 Jy is multiplication the magnetic field scale $B_{0}$, as given in Table 2. In Anantua et al. (2020a), there are prefactors $\Phi_{26}10^{26}$ and $r_{13}10^{13}$ where $\Phi_{26}\approx 1\approx r_{13}$ that link code units to physical units (in cgs) that normalize the horizon flux to 1034 Mx (by comparison, our flux normalization $B_{0}$ in this paper gives a total intensity of 1.5 Jy). Including both of these scaling methods allows the reader to go back and forth between the two papers.  Code Unit | mks | cgs |
---|---|---|---
Flux threading (one hemisphere |  $\Phi_{26}10^{26}$ Wb |  $\Phi_{34}10^{34}$ Mx |
of) the horizon ($\Phi_{B}$) |  |  |
Magnetic Field Near Horizon ($B_{\mathrm{scale}}$) |  |  |
$\frac{\Phi_{B}}{(\mathrm{Length\ Unit})^{2}}$ | $\frac{10^{26}\Phi_{26}\mathrm{Wb}}{(r_{13}10^{13}\mathrm{m})^{2}}=\frac{\Phi_{26}}{r_{13}^{2}}\mathrm{T}$ | $\frac{10^{34}\Phi_{34}\mathrm{Mx}}{(r_{15}10^{15}\mathrm{cm})^{2}}=\frac{\Phi_{34}}{r_{15}^{2}}10^{4}\mathrm{Gs}$ |
Pressure Near Horizon |  |  |
$\left(\frac{\Phi_{B}}{(\mathrm{Length\ Unit})^{2}}\right)^{2}$ |  $\frac{1}{2\mu_{0}}\left(\frac{\Phi_{26}}{r_{13}^{2}}\mathrm{T}\right)^{2}=\frac{1}{8\pi\cdot 10^{-7}}\frac{\Phi_{26}^{2}}{r_{13}^{4}}$ Pa |  $\frac{1}{8\pi}\left(\frac{\Phi_{34}}{r_{15}^{2}10^{4}}\mathrm{Gs}\right)^{2}=\frac{10^{8}}{8\pi}\frac{\Phi_{34}^{2}}{r_{15}^{4}}$ Ba |

Using the values of emitting region height and size from Hada et al. (2013)
and the powerful $B$-field estimates from the $10^{34}$ Mx jet model (Anantua
et al., 2020a), we estimate values of the synchrotron cooling time (at the
boundary of the Hada et al. (2013) optically thick core) as $\sim 2.6\times
10^{4}$s, $4.3\times 10^{5}$s, $6.6\times 10^{6}$s, and $3.9\times 10^{8}$s
for $\nu=230$ GHz, 86 GHz, 43 GHz, and 22 GHz respectively. Then we estimate
$\beta_{\rm e0}$ in these emission regions as $\beta_{\rm e0}\approx\\{(230\
\mathrm{GHz},1.8\times 10^{-5}),(86\ \mathrm{GHz},9.2\times 10^{-6}),(43\
\mathrm{GHz},2.0\times 10^{-4}),(23\ \mathrm{GHz},2.2\times 10^{-1})\\}$.

The observational $\tilde{P}_{\mathrm{e}}$ estimation give us a ballpark
$10^{-6}\lesssim\beta_{e0}\lesssim 10^{-2}$ which helps motivate our choice of
parameter space for the constant $\beta_{\rm e0}$ jet model in in Section 5.

##  Appendix B $\chi^{2}$ Tables

In this section, we present tables of reduced-$\chi^{2}$ values we compute by
comparing model Stokes $I$ SED against the observed M87* or Sgr A* SEDs to
select models. We present $\chi^{2}$ values considering only points in the
radio-submm band ($\chi^{2}_{\rm radio}$, in the IR ($\chi^{2}_{\rm IR}$) and
from the full SED data-sets ($\chi^{2}$).

The $\chi^{2}$ values for Constant $\beta_{\rm e0}$ jet models for M87* models
are given in Table 2. The $\chi^{2}$ values for the Sgr A* RIAF model can be
found in Table 3. Note that in the RIAF model we consider a smaller parameter
space than in the M87* jet model by only varying $\beta$ and $f_{\rm pos}$ and
fixing other parameters to the fitted values from Broderick et al. (2016).

Table 2: Stokes I SED reduced $\chi^{2}$ values for different constant
electron $\beta_{\rm e0}$ M87* jet models. In each section of five rows, we
fix the value of the positron fraction $f_{\rm pos}$ and vary the electron
distribution spectral slope $p\in\\{2.5,2.8,3.0,3.2,3.5\\}.$ The three
sections of five columns each contain data for different values of $\beta_{\rm
e0}\in\\{10^{-6},10^{-4},10^{-2}\\}$. The best $\chi^{2}$ over
$f_{\mathrm{pos}},p$ and $\beta_{e0}$ for each segment of the spectrum are
given in bold.

$f_{\mathrm{pos}}$ $p$ $\beta_{\mathrm{e0}}$ $B_{0}$($10^{3}$)
$\chi_{\mathrm{radio}}^{2}$ $\chi_{\mathrm{IR}}^{2}$ $\chi^{2}_{\mathrm{tot}}$
$\beta_{\mathrm{e0}}$ $B_{0}$($10^{3}$) $\chi_{\mathrm{radio}}^{2}$
$\chi_{\mathrm{IR}}^{2}$ $\chi^{2}_{\mathrm{tot}}$ $\beta_{\mathrm{e0}}$
$B_{0}$($10^{3}$) $\chi_{\mathrm{radio}}^{2}$ $\chi_{\mathrm{IR}}^{2}$
$\chi^{2}_{\mathrm{tot}}$ 0.00 2.5 $10^{-6}$ 11.7 11.0 $>10^{4}$ $>10^{4}$
$10^{-4}$ 3.1 7.78 $>10^{3}$ $>10^{3}$ $10^{-2}$ 0.88 6.1 $\gtrsim 10^{3}$
$\sim 10^{3}$ 0.00 2.8 $10^{-6}$ 11.3 9.6 $\gtrsim 10^{3}$ $\sim 10^{3}$
$10^{-4}$ 3.2 7.4 610.2 319.2 $10^{-2}$ 0.93 5.5 93.0 50.7 0.00 3.0 $10^{-6}$
11.7 10.3 426.2 225.4 $10^{-4}$ 3.3 7.1 92.3 51.1 $10^{-2}$ 0.98 5.2 7.00 6.1
0.00 3.2 $10^{-6}$ 12.1 10.4 82.4 47.6 $10^{-4}$ 3.5 7.3 13.1 12.3 $10^{-2}$
1.07 5.6 1.2 3.3 0.00 3.5 $10^{-6}$ 12.5 9.8 1.9 5.7 $10^{-4}$ 3.9 8.0 3.1 5.5
$10^{-2}$ 1.2 5.7 7.7 6.8 0.01 2.5 $10^{-6}$ 11.6 10.6 $>10^{4}$ $>10^{4}$
$10^{-4}$ 3.1 7.88 $>10^{3}$ $>10^{3}$ $10^{-2}$ 0.88 6.15 $\gtrsim 10^{3}$
$\sim 700$ 0.01 2.8 $10^{-6}$ 11.6 10.7 $\gtrsim 10^{3}$ $\sim 10^{3}$
$10^{-4}$ 3.2 7.3 591 309.5 $10^{-2}$ 0.92 5.4 80.7 44.3 0.01 3.0 $10^{-6}$
11.6 9.96 394.4 208.8 $10^{-4}$ 3.4 7.4 107.7 59.3 $10^{-2}$ 0.99 5.39 10.0
7.78 0.01 3.2 $10^{-6}$ 12.2 10.92 92.96 53.35 $10^{-4}$ 3.5 7.29 13.18 10.34
$10^{-2}$ 1.07 5.51 1.19 3.27 0.01 3.5 $10^{-6}$ 12.5 9.91 2.0 5.82 $10^{-4}$
3.97 8.63 2.56 5.49 $10^{-2}$ 1.2 5.75 7.65 6.7 0.1 2.5 $10^{-6}$ 11.3 10.45
$>10^{4}$ $>10^{4}$ $10^{-4}$ 3.1 7.82 $>10^{3}$ $>10^{3}$ $10^{-2}$ 0.84 5.79
$\gtrsim 10^{3}$ 583 0.1 2.8 $10^{-6}$ 10.99 9.35 $\gtrsim 10^{3}$ $\sim 870$
$10^{-4}$ 3.2 7.93 720 376.2 $10^{-2}$ 0.92 5.61 102.9 55.9 0.1 3.0 $10^{-6}$
11.3 9.72 373.2 197.7 $10^{-4}$ 3.3 7.34 104.4 57.53 $10^{-2}$ 0.95 5.2 6.57
5.9 0.1 3.2 $10^{-6}$ 11.6 9.54 64.78 38.11 $10^{-4}$ 3.5 7.81 17.99 13.08
$10^{-2}$ 1.03 5.36 1.31 3.26 0.1 3.5 $10^{-6}$ 12.2 9.7 1.93 5.68 $10^{-4}$
3.8 7.88 3.17 5.44 $10^{-2}$ 1.18 5.54 8.18 6.91 0.5 2.5 $10^{-6}$ 10.2 9.6
$>10^{4}$ $\sim 10^{4}$ $10^{-4}$ 2.8 8.0 $>10^{3}$ $>10^{3}$ $10^{-2}$ 0.78
5.9 $\gtrsim 10^{3}$ 573 0.5 2.8 $10^{-6}$ 10.2 9.39 $\gtrsim 10^{3}$ $\sim
10^{3}$ $10^{-4}$ 2.9 7.6 624.6 326.7 $10^{-2}$ 0.83 5.4 70.5 39.0 0.5 3.0
$10^{-6}$ 10.5 10.1 403.5 213.6 $10^{-4}$ 3.0 7.2 96.2 53.3 $10^{-2}$ 0.88 5.1
4.8 5.0 0.5 3.2 $10^{-6}$ 10.9 10.2 78.8 45.7 $10^{-4}$ 3.2 7.5 14.8 11.3
$10^{-2}$ 0.98 5.5 1.1 3.2 0.5 3.5 $10^{-6}$ 11.3 9.5 1.9 5.6 $10^{-4}$ 3.5
7.5 3.5 5.4 $10^{-2}$ 1.07 5.2 9.3 7.3 1.0 2.5 $10^{-6}$ 9.4 6.5 $>10^{4}$
$>10^{3}$ $10^{-4}$ 2.5 7.4 $>10^{3}$ $>10^{3}$ $10^{-2}$ 0.71 5.9 $\sim
10^{3}$ 405.7 1.0 2.8 $10^{-6}$ 9.4 9.3 $\gtrsim 10^{3}$ $\sim 10^{3}$
$10^{-4}$ 2.6 7.0 437.9 229.9 $10^{-2}$ 0.78 5.7 73.3 40.7 1.0 3.0 $10^{-6}$
9.8 9.7 379.3 201.0 $10^{-4}$ 2.7 6.8 63.7 36.2 $10^{-2}$ 0.83 5.5 5.5 5.5 1.0
3.2 $10^{-6}$ 10.2 10.0 74.5 43.4 $10^{-4}$ 2.9 7.0 9.4 8.2 $10^{-2}$ 0.9 5.6
1.15 3.3 1.0 3.5 $10^{-6}$ 10.4 8.8 1.9 5.2 $10^{-4}$ 3.3 7.7 3.2 5.3
$10^{-2}$ 1.0 6.0 8.6 7.2

Table 3: Stokes $I$ SED reduced $\chi^{2}$ values for different Sgr A* RIAF
models. In each block of 4 rows/5 columns, we fix the the positron fraction
$f_{\mathrm{pos}}\in\\{0.0,0.01,0.1\\}$ and vary the
plasma-$\beta\in\\{10^{-2},10^{-1},1,10\\}$. All other parameters are fixed to
their values from Broderick et al. (2016).

$\beta$ $f_{\mathrm{pos}}$ $n_{\rm th0}$($10^{3}$) $\chi_{\mathrm{radio}}^{2}$
$\chi_{\mathrm{IR}}^{2}$ $\chi^{2}_{\mathrm{tot}}$ $f_{\mathrm{pos}}$ $n_{\rm
th0}$($10^{3}$) $\chi_{\mathrm{radio}}^{2}$ $\chi_{\mathrm{IR}}^{2}$
$\chi^{2}_{\mathrm{tot}}$ $f_{\mathrm{pos}}$ $n_{\rm th0}$($10^{3}$)
$\chi_{\mathrm{radio}}^{2}$ $\chi_{\mathrm{IR}}^{2}$ $\chi^{2}_{\mathrm{tot}}$
$10^{-2}$ 0.0 314 1.61 4.51 1.89 0.01 313.6 1.6 4.48 1.88 0.1 283.1 1.78 4.8
2.53 $10^{-1}$ 0.0 893 2.02 4.95 2.30 0.01 893.4 2.0 4.93 2.28 0.1 832.3 2.14
5.06 2.42 $10^{0}$ 0.0 278 2.28 4.88 2.53 0.01 2785.3 2.25 4.86 2.50 0.1
2663.2 2.28 4.86 2.53 $10^{1}$ 0.0 8888 2.46 4.68 2.67 0.01 8888.2 2.44 4.66
2.65 0.1 8888.2 2.24 4.42 2.45 $10^{-2}$ 0.5 253 1.71 4.57 1.98 1.0 207 1.95