text:
\section{Introduction}

Merely $d =$ 8.18
kpc away \citep{2019arXiv190405721A}
at the Galactic Center, Sagittarius A* (Sgr A*) is the best known accreting supermassive black hole. With $m_\mathrm{BH}=$ 4.14
million solar masses \citep{2019arXiv190405721A}
(subtending 5.0
$\mu$as at Earth), Sgr A*
is a prime candidate for the next measurement of a black hole shadow by the Event Horizon Telescope (EHT,  \citealt{Doeleman2008})-- after the one in the giant elliptical galaxy M87.
In April 2019, the EHT imaged emission around the central supermassive black hole of
M87, finding a 42 $\mu$as-wide annulus with a Southern excess consistent with relativistic predictions of beamed emission in the Kerr metric of a 6.5 billion solar mass black hole
\citep{2019ApJ...875L...1E}. The distribution of flux density over the EHT M87 image-- along with a conservative lower bound on the jet power of $L_\mathrm{M87}\ge 10^{42}$ erg/s-- precludes models with a non-spinning black hole, supporting the interpretation of the M87 jets as powered by the Blandford-Znajek mechanism \citep{BZ1977}. Image reconstruction of Sgr A* data has presented unique challenges relative to that for M87 due to the dynamical timescales of minutes as opposed to days \citep{2019ApJ...875L...1E}. GRAVITY \citep{Gillessen2010} has recently measured astrometrically the proper motion of several ``hot spots'' (i.e., near infrared flares) orbiting the Galactic Center black hole at $\sim 6$-$10$ gravitational radii ($M\equiv r_g = Gm_\mathrm{BH}/c^2$) \footnote
$M$ for Sgr A* corresponds to $6.08\cdot 10^{9}$ m, 20.25 s and 4.96 $\mu\mathrm{as}$.}, suggesting that the innermost accretion flow may be relatively face on and strongly magnetized
\citep{Gravity2018}. These horizon-scale probes of plasma physics, accretion physics, and strong general relativistic effects provide strong constraints on theoretical models.

Incorporating all these effects into analytic models is difficult, and thus general relativistic magnetohydrodynamic (GRMHD) simulations are the most promising avenues for theoretical modeling \citep{Gammie2003,Sadowski2014,White2016,Porth2017}.  These simulations solve the equations of MHD in the Kerr metric for a rotating black hole in 3+1 dimensions, naturally capturing the magnetorotational instability (MRI), the formation of jets, turbulence, and general relativistic effects.

As an especially low-luminosity  active galactic nucleus (LLAGN)  with an observed Eddington ratio $\sim 10^{-9}$ \citep{Kataoka2018},
the dynamical time for accretion in Sgr A* is much shorter than the electron-ion Coulomb collision time.  This means that the flow can roughly be described as a two-temperature plasma \citep{Mahadevan1997} and that information beyond that provided by the standard GRMHD equations is needed to constrain the electron temperature, a key parameter in emission modeling.  Many approaches have been taken in the Literature for setting the electron temperature.  Simple post-processing prescriptions that ``paint on'' electron temperatures, such as those that set a constant electron-to-proton temperature ratio \citep{Mosci2009} everywhere or that set the electron temperature to be some function of plasma parameters \citep{Shcherbakov2012,Mosci2013,Chan2015}, are useful for rapidly and directly connecting observations to model parameters. More sophisticated treatments directly evolve the electron temperature alongside the single-fluid equations of GRMHD and incorporate knowledge of electron/ion heating gained from particle-in-cell (PIC) simulations of collisionless plasmas \citep{Ressler2015,Ressler2017,Sadowski2017,Chael2018}.  Though these models are more physically motivated and self-consistent, they are more expensive to run which makes it difficult to fully explore the parameter space of electron temperature spanned by our uncertainty in the dominant mechanism of electron/ion heating. Furthermore, conservative GRMHD simulations are unable to properly model the thermodynamics of flows in which the magnetic energy density far exceeds the rest mass energy density (as it often does in the jet).  Thus, two-temperature simulations that rely on an accurate calculation of the total fluid heating rate cannot be trusted in these regions; parametric models that depend only on the more reliable local magnetic energy density may be more appropriate there.\footnote{This may be part of the reason why two-temperature GRMHD simulations of Sgr A* have thus far under-produced the low frequency radio emission of Sgr A* \citep{Ressler2017,Chael2018}, as that emission is often assumed to be powered by the magnetized outflow. Alternatively, a small population of non-thermal particles can also explain the $\sim$ flat low frequency radio spectrum \citep{Ozel2000,Yuan2002}.}

Given this uncertainty, we consider it instructive to further explore the dependence of the spectra and near-horizon-scale images of Sgr A* on different post-processing models of the electron temperature. In doing so, we include not only some of the prescriptions used in past work to model the Galactic Center, but also those attempting to mimic the qualitative behavior seen in recent two-temperature GRMHD simulations and those successfully used to model other systems, (e.g., \citealt{BK1979,Blandford2017,Anantua2018,Anantua2019}). We synthesize our results by categorizing the synthetic images and spectra into a small number of groups according to their emission physics.  This program adds to the foundation of efforts at classifying and understanding jet (or outflow)/accretion disk/black hole (JAB) emission through a unified framework flexible enough to model disk, corona and outflow or jet regions with a small set of parameters-- a methodology we call ``Observing" JAB Simulations.

This work is a small addition to a vast literature on postprocessing simulations to understand JAB systems in the low accretion limit.
For example, \citet{Dexter2009} simulated Sgr A* visibility amplitudes concordant with observed VLBI using $30^\circ$ inclination angle from the black hole spin axis and generated synthetic light curves with 30 min rise times and up to $50\%$ flux modulation.
\cite{Gold2017} compare synthetic Stokes maps of turbulent versus ordered magnetically arrested disk
B-field configurations to EHT polarization data and favor the latter to account for the morphology and degree of linear polarization.
GRMHD simulations in \cite{Mosci2014} explain the Sgr A* radio spectrum and image size primarily by efficient electron heating in the outflow plasma.
\cite{Ryan2018} use general relativistic radiative magnetohydrodynamic (GRRMHD) simulations to model the M87 inner accretion flow and find inverse Compton cooling has a non-negligible backreaction within $10M$. The present work systematically considers simple parametric emission models to directly link microphysical processes to discrete observational signatures in Sgr A* and other AGN.

This paper is structured as follows: Section 2 is a brief synopsis of the observations of Sgr A* used to constrain our models; Section 3 describes the
numerical simulations used in this work; Section 4 presents models for the accretion flow, electron temperature and emission; Section 5 provides results: images, spectra and light curves for our models; Section 6 compares the models in three different simulations to each other and to observations, resulting in a classification distilling various model morphologies, sizes and spectral shapes into four types; and Section 7 concludes.

In what follows, the speed of light, $c$, and the Boltzmann constant, $k_B$, are set to unity. Charge neutrality requires the electron number density, $n_e$, to be $\approx$ the proton number density, $n_p$ $\equiv n_0$. Then, the mass density is $\rho\approx \rho_p=m_p n_0$, where $m_p$ is the mass of a proton. Setting $m_p$ to 1 gives $P_e=\rho T_e$. The total temperature, $T_\mathrm{tot} = T_e + T_p $ is given by the simulation; $T_e$ is modeled.

\section{Observations}

Sgr A* has been observed for many years now in the radio, mm, infrared, and X-rays
\citealp{Fish2011, Neilsen2013,Bower2015, Gravity2018}).  Time variability tends to increase with frequency, with the radio emission being the most stable and the infrared and X-ray emission showing frequent occurrence of large amplitude flares, likely caused by nonthermal particle acceleration.  Since we consider only thermal emission in this work, we treat the 10\% of the quiescent X-ray flux estimated to originate close to the black hole \citep{Neilsen2013} as an upper limit.  For a comprehensive list of the observational data points used in this work as constraints on our models (and plotted alongside our model spectra), see
\citet{Ressler2017}.

We use a 2D Gaussian semi-major axis size constraint of 49-63 $\mu$as (9.9-12.7$M$) incorporating long baseline data  \citep{Johnson2015,Lu2018} in an elliptical ring interpretation of the image to constrain our model image size.
At 3.5 mm (86 GHz), \citet{Issaoun2019} ALMA observations give an intrinsic source axial sizes of $100\pm\ 18\mu$as (16.5-23.8$M$) and $120\pm\ 34\mu$as (17.3-31.0$M$). Recent Sgr A* measurements of non-zero closure phase, e.g., $5.0^{\circ +12.9^\circ}_{\ -4.6^\circ}$ from measurements along the SMT-CARMA-APEX triangle \citep{Lu2018}, rule out a spherically symmetric emission profile.

\section{Principal GRMHD Simulations}

We perform a set of three simulations using the conservative, 3D, ideal GRMHD code {\tt HARM} \citep{Gammie2003}, which we denote as Standard and Normal Evolution (SANE), Magnetically Arrested Disk (MAD) and semi-MAD, all of which are endowed with electron temperature evolution based on \citet{Ressler2015}. The simulations all have dimensionless spin $a=0.5$, start from a \citet{FM1976} torus, and have resolutions of 320 x 256 x 64, uniform in the coordinates $x_1 (r,\theta), x_2 (r,\theta)$, and $x_3\equiv \varphi$, that
are ``cylindrified'' and hyper-exponentiated \citep{SashaMAD} versions of modified Kerr-Schild (MKS) coordinates (\citealt{McKinney2004}; using $h = 0.3$), a process which is described in Appendix B of \citet{Ressler2017}. MKS coordinates focus resolution towards the midplane of the simulation, the ``cylindrification'' process increases the angular width of cells with $r\lesssim 10 M$, while the hyper-exponentiation extends the radial extent of the grid to thousands of $r_g$ by rapidly increasing the radial size of cells at $r > 400 M$. The adiabatic index of the total gas is $\gamma = 5/3$ and the adiabatic index of the electrons is $\gamma_e=4/3$.
Where the three simulations differ is in both the size of the initial torus and the geometry of the initial magnetic field contained within this torus.  The SANE initial torus has an inner boundary of $r_\mathrm{in}=12 M$ and pressure maximum at $r_\mathrm{max}=24 M$, while the semi-MAD and MAD initial tori have inner radii of $r_\mathrm{in} = 15 M$ and pressure maxima at $r_\mathrm{max} = 34.5 M$.   While all simulations start with magnetic field lines that form single loops, the size and shape of the loops varies as determined by the magnetic vector potential, $A_\varphi$.  $A_\varphi$ scales as $A_\varphi\propto \rho$ in the SANE case and $A_\varphi\propto r^{4}\rho^2$ in the semi-MAD case, both of which we normalize such that $\max (P)/\max (P_B)=100$.  The MAD vector potential is more complicated and is computed as described in \citet{SashaMAD} while we normalize it such that $\min (P/P_B) = 100$. The resulting steady-state, time-averaged magnetic flux threading the horizon, $\Phi_{BH}$, for each of the runs are $\ll 1$ $M \sqrt{\dot{m}c}$ (SANE), $\approx 40$ $ M \sqrt{\dot{m}c}$ (semi-MAD), and $\approx 50$ $M \sqrt{\dot{m}c}$ (MAD).

The simulations implement the numerical density floor prescription $u_e\ge 0.01u_g$ from \cite{Ressler2015}.
The simulation grid concentrates 3D spatial resolution in the disk,
allowing for non-axisymmetric turbulence, kink instabilities, etc. in the MHD flow. The code has been parallelized using
message passing interface (MPI).
The physical units of length and time are set by the mass of the black hole.

Our fiducial simulation is the semi-MAD run.  We use a fiducial
simulation time $T=10,000M$ to compare images and spectra.  Time-averaged $\dot{m}$ and $\dot{E}$ are found to be nearly constant in radius for the inner $r< 35M$ for SANE, semi-MAD and MAD simulations alike, indicating that inflow and outflow equilibrium is obtained for the regions of interest in this work. In fact, the MAD simulation is in equilibrium up to $r\lesssim 100M$.

\section{Emission Models}

\subsection{Electron Thermodynamics Models}
\label{sec:models}

\subsubsection{Electron Evolution Model with Turbulent Heating}

All of our simulations also include an electron entropy equation as described in \citet{Ressler2015} (neglecting electron conduction), using the \citet{Howes2010} heating prescription for turbulent heating in collisionless plasmas.
We refer to this as the \say{Electron Evolution Model with Turbulent Heating} (or \say{Electron Evolution Model} for short). Note that the \citet{Howes2010} heating prescription is the result of calculations of the Landau damping of turbulence -- different dissipation mechanisms at the end of a turbulent cascade could produce different results.

The \citet{Howes2010} heating function is strongly dependent on plasma $\beta=P_g/P_B$, with a sharp transition between electrons being preferentially heated at $\beta \lesssim 1$ to protons being preferentially heated at $\beta > 1$. A direct consequence is that the relativistically hot electrons are confined to the coronal and jet regions of the simulations while the electrons in the midplane of the disk are non-relativistically cold.

\subsubsection{Critical Beta Electron Temperature Model
}

In an attempt to mimic the behavior of the Electron Evolution Model-- without explicitly including
turbulent heating in an
entropy equation-- we construct a post-processing function for the electron-to-total temperature ratio:
\begin{equation}
\frac{T_e}{T_\mathrm{tot}
} = f e^{-\frac{\beta}{\beta_c}},
	\label{eq:ETempModel}
\end{equation}
where $0<f<1$ is a constant, and $\beta_c$ is the critical value of $\beta$ that approximately sets a maximum $\beta$ contributing to emission.
We call this model the \say{Critical Beta Electron Temperature Model} (or \say{Electron Temperature Model} for short).

This model converges to unique values for the ratio $T_e/T_p$ in the limits $\beta\to 0$ and $\beta\to \infty$
similarly to models in \citet{Mosci2014,Davelaar2018} employed by the EHT for M87 in which the ratio of proton-to-electron temperature is bounded by constants $R_\mathrm{low}$ and $R_\mathrm{high}$.

\subsubsection{Constant Electron Beta Model}

Another viable post-processing prescription for the electron temperature is one in which the electron energy density $u_e$ is some fixed fraction of the magnetic energy density $u_B$.  This may be reasonable if magnetic reconnection is the dominant source of electron heating as it is presumed to be in jet regions.  This \say{Constant Electron Beta Model} (or \say{Constant $\beta_e$ Model}) is described by a single parameter through the relation $\beta_e=P_e/P_B=(\gamma_e-1)u_e/(b^2/2)=\beta_{e0}$ (constant), or:
\begin{equation}
P_e = \beta_{e0}P_B.
	\label{eq:BetaModel}
\end{equation}
where $b^2\equiv b^\mu b_\mu$.

Note that equipartition of particle and electromagnetic energies corresponds to $\beta_{e0} \sim 1$.
Models coupling near-equipartition jets to Sgr A*'s accretion flow have previously been examined in  \cite{Falcke1993} and \cite{Falcke2000}. There, the jets have been invoked to explain the radio emission and,
in some models, the higher frequency spectrum.

\subsubsection{Magnetic Bias Model}

We can generalize the Constant $\beta_e$ Model so that the electron pressure scales as powers of the magnetic pressure
\begin{equation}\label{BiasModel}
P_e=K_nP_B^n \sim b^{2n}
\end{equation}
where
\begin{equation}\label{eq:BiasModel}
K_n=K_1\frac{\langle P_B \rangle}{\langle P_B^n\rangle}=2^{n-1}K_1\frac{\langle b^2 \rangle}{\langle b^{2n}\rangle }
\end{equation}
and $\langle \rangle$ denotes an average over cylindrical radii $2M<R<20M$ as in Appendix \ref{TablesAppendix} Table \ref{OldAvgbToNCylinder}. We call this the \say{Magnetic Bias Model} (or \say{Bias Model} for short).

Note that $n = 1,K_1\equiv\beta_{e0}$ corresponds to the Constant $\beta_e$ Model. By default, for $n\neq 1$, we take $K_1=1$ in this work in the interest of space, although there is a priori no strong motivation for a particular value of $K_n$.

\subsection{Radiative Transport}

We compute (mainly 230 GHz) images using the ray-tracing scheme {\tt IBOTHROS} \citep{Noble2007}, which includes the effects of synchrotron emission and absorption, while we compute spectra using the Monte-Carlo-based {\tt GRMONTY} \citep{Dolence2009}, which includes the effects of synchroton emission, absorption, and inverse Compton scattering.  For the purposes of radiative transport, we exclude regions of the simulation with $\sigma \equiv b^2/\rho>1$ where the GRMHD solution becomes less reliable. The pixel resolution used to produce IBOTHROS images in this work is 1$M$ per pixel in both directions.

Since GRMHD simulations are scale free, for each electron temperature model we choose the physical mass unit such that the flux at 230 GHz matches the 2.4 Jy measurement \citep{Doeleman2008} (to $<2\%$).
Table \ref{MUnit230GHz} lists the resulting mass accretion rate for each of our models. Note that negative accretion rate corresponds to inflow.

Our fiducial viewing angle will be 90$^\circ$ (edge-on disk) and our fiducial observer frequency will be $\nu_\mathrm{Obs}=230$ GHz,
 though we also consider 0$^\circ$ (face on disk), 45$^\circ$ and $\nu_\mathrm{Obs}=140$ THz, the latter for comparison with near infrared observations. In the absence of other labels, all of the images and  spectra shown correspond to fiducial values of viewing angle and frequency and to the semi-MAD simulation.

\begin{table*}
	\centering
	\caption{Mass accretion rate yielding $\approx$ 2.4 Jy 230 GHz flux for our models. Negative values of mass accretion rate correspond to infall. The monochromatic flux is summed over $50M$ x $50M$ image regions. For comparison, the Eddington accretion rate for Sgr A*'s mass is $-0.09 \, M_\odot/\mathrm{yr}$
	}
	\label{MUnit230GHz}
	\begin{tabular}{lcccr}
		\hline
		Model &

        &  Accretion Rate $M_\odot/\mathrm{yr}$ &
        \\
     & SANE

     & semi-MAD
     & MAD
     \\
     		\hline
		Electron Evolution with Turbulent Heating

          & $-2.61\cdot 10^{-8}$
          & $-1.7
          \cdot 10^{-8}$

          & $-1.44\cdot 10^{-11}$
          \\
		\hline
		 Critical Beta Electron Temperature ($f,\beta_c$) &

        & \\
        ($0.1,0.01$)& $-1.11\cdot 10^{-7}$

        &
        $-1.7
        \cdot 10^{-7}

        & $-1.90\cdot 10^{-9}$
        \\
       ($0.1,0.1$)& $-8.10\cdot 10^{-8}$
       &
       $-1.1
       \cdot 10^{-7}

       & $-4.31\cdot 10^{-10}$
       \\
       ($0.1,1.0$)&
       $-6.92

       \cdot 10^{-8}$
       &

       $-9.1
       \cdot 10^{-8}

        & $-2.5
       \cdot 10^{-10}$
       \\

       ($0.5,0.01$)& $-2.57\cdot 10^{-8}$

       &
       $-3.6
       \cdot 10^{-8}

       & $-3.11\cdot 10^{-10}$
       \\
       ($0.5,0.1$)& $-1.52\cdot 10^{-8}$
       &
       $-1.1
       \cdot 10^{-8}

       & $-7.87\cdot 10^{-11}$
       \\
       ($0.5,1.0$) & $-3.9
       \cdot 10^{-9}$
       &
       $-6.0
       \cdot 10^{-9}

       & $-2.1
       \cdot 10^{-11}$
       \\

        \hline
		Constant Electron Beta ($\beta_{e0}$)&  &
		\\
         0.01   & $-4.75\cdot 10^{-9}$
         &
         $-2.9
         \cdot 10^{-9}$
         & $-3.14\cdot 10^{-11}$
         \\
         0.1   &
          $-3.11\cdot 10^{-10}$
           &
         $-6.2
         \cdot 10^{-10}$ & $-4.83\cdot 10^{-12}$
         \\
         1.0   &
         $-3.90\cdot 10^{-11}$

         $-3.8
         \cdot 10^{-10}$
         & $-1.62\cdot 10^{-12}$
         \\
        \hline
		 Magnetic Bias ($n$
         &  &
         \\
		0 & $-4.04\cdot 10^{-9}$
        &
        $-1.4
        \cdot 10^{-9}$
        & $-1.66\cdot 10^{-9}$
        \\
        1 & $-3.90\cdot 10^{-11}$
        &
        $-3.8
        \cdot 10^{-10}$
        & $-1.62\cdot 10^{-12}$
        \\
        2 & $-2.21\cdot 10^{-11} $
        &
        $-4.5
        \cdot 10^{-10}$
        & $-1.95\cdot 10^{-12}$
        \\
        \hline
	\end{tabular}
\end{table*}

\section{Results}

\subsection{Electron Temperature Profiles}
\begin{figure}
\begin{align}\nonumber
& \hspace{1cm}\includegraphics[height=170pt,width=190pt,trim = 6mm 1mm 0mm 1mm]{ElectronEvolModelLog10ThetaE.png}
\end{align}
\begin{align}\nonumber
&  \hspace{1cm}\includegraphics[height=170pt,width=190pt,trim = 6mm 1mm 0mm 1mm]{Log10sigmaPhiAvgRangem5To5Fig.png}
\end{align}
\caption{Slices of electron temperature $r\theta$ profile for the Electron Evolution Model (Top) and slice of $\sigma=b^2/\rho$ (Bottom). We use common log scales to present large dynamical ranges.
}\label{VerticalTeSlicesElectronEvolModelAndSigmaSlice}
\end{figure}

Figure \ref{VerticalTeSlicesElectronEvolModelAndSigmaSlice} shows the azimuthally averaged electron temperature distribution for the Electron Evolution Model and plasma electromagnetic-to-particle flux density $\sigma$ in the $r$-$\theta$ plane. The electron temperature is greatest on the boundary between the high $\sigma$ outflow and lower $\sigma$ outflow and inflow as found in \citet{Ressler2015}. Figure \ref{VerticalTeSlicesElectronTempAndBetaAndBiasModels} shows the azimuthally averaged electron temperature distributions in the $r$-$\theta$ plane resulting in each of our models for a few select parameter choices.
For Critical Beta Electron Temperature Models, a boundary layer on the disk-jet interface (mild outflow+corona) is the region with highest $T_e$. This behavior is consistent throughout the parameter space $f\in\{0.1,0.5\}$ and $\beta_c\in\{0.01,0.1,1\}$.

The electron temperature profiles in various Constant $\beta_e$ Models in the middle panels of Figure \ref{VerticalTeSlicesElectronTempAndBetaAndBiasModels} are hottest for the strong interior outflow, or \say{spine,} characterized by low density and high magnetization.
In Bias Models at the bottom panels of Figure \ref{VerticalTeSlicesElectronTempAndBetaAndBiasModels}, electron temperature is also highest near the coherent, electromagnetically-dominated outflow. Now, however, the  radial profile is strongly dependent on the exponent $n$, which enhances the variation of emission (a function of $b$) with cylindrical radius in the simulation.

\begin{figure}
\begin{align}\nonumber
& \includegraphics[height=100pt,width=110pt,trim = 6mm 1mm 0mm 1mm]{LogElectronTempProfilefEqPt1BetaCEq
.png} & \includegraphics[height=100pt,width=110pt,trim = 6mm 1mm 0mm 1mm]{LogElectronTempProfilefEqPt5BetaCEq
.png}
\end{align}
\begin{align}\nonumber
  & \includegraphics[height=100pt,width=111pt,trim = 6mm 1mm 0mm 1mm]{TeProfileBetae0EqPt0
  .png} & \includegraphics[height=100pt,width=110pt,trim = 6mm 1mm 0mm 1mm]{TeProfileBetae0Eq
  .png} &
\end{align}
\begin{align}\nonumber
  & \includegraphics[height=100pt,width=110pt,trim = 6mm 1mm 0mm 1mm]{LogElectronTempProfilenEq0
  .png} & \includegraphics[height=100pt,width=110pt,trim = 6mm 1mm 0mm 1mm]{LogElectronTempProfilenEq
  .png} &
\end{align}
\caption{Electron temperature $r\theta$ profiles for the Critical Beta Electron Temperature Model with $(f,\beta_c)=(0.1,1)$ (Top Left)
and $(0.5,1)$ (Top Right). Also, electron temperature $r\theta$ profiles for
Constant $\beta_e$ Models with $\beta_{e0}=0.01$ (Middle Left), $\beta_{e0}=0.1$ (Middle Right),
and Bias Models with $n=
$  (Bottom Left) and $n=
$ (Bottom Right). The temperature is expressed in dimensionless form $\theta_e=k_BT_e/m_ec^2$
}\label{VerticalTeSlicesElectronTempAndBetaAndBiasModels}
\end{figure}

\subsection{Electron Evolution Model with Turbulent Heating}

\subsubsection{Electron Evolution Model Images}

The Electron Evolution Model image\footnote
Each individual pixel in the image plane (normal to the observer) has an $I_\nu = d F_\nu / d\Omega$ in cgs units, and $F_\mathrm{tot}\approx2.4\cdot10^{-23}$erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$  = 2.4 Jy is just the sum of all the $dF_\nu$'s. Thus, multiply each cgs intensity colored pixel value by
57.9 to get its flux density in Jy.} in Figure \ref{ElectronEvolutionModel} shows
a ring of lensed emission around the black hole event horizon and a small outflow:
the
whispy
outflow is visible at radii
not exceeding $30M$
and is limb brightened. Asymmetry in the photon ring is apparent, due to Doppler shifts at the edge on viewing angle. The images in this paper are rendered
in observer coordinates left-right and up-down inverted relative to \citet{Ressler2017}.
In our implementation of a new imaging pipeline
90$^\circ$ inclination has the accretion flow approaching on the right and black hole spin pointing down.

\begin{figure}
\begin{align}\nonumber
  & \hspace{1.0cm} \includegraphics[height=150pt,trim = 6mm 1mm 0mm 1mm]{PlotElectronEvolutionModelZoomM_unitgTotalfluxJy1Pt03897e20and2Pt40415T10000v2.png}
  &
\end{align}
\caption{Electron Evolution Model image at 230 GHz.}\label{ElectronEvolutionModel}
\end{figure}

 \subsubsection{Electron Evolution Model Spectra}

The Electron Evolution Model spectrum is shown in Fig. \ref{ElectronEvolSpec}.
The infrared bump is well fit, however the model's low frequency slope steepens under the data around 10 GHz, and the model significantly overpredicts the X-ray emission. Note that this spectrum also appears in \citet{Ressler2017}.

\begin{figure}\nonumber
\begin{align}
\hspace{0.5cm}\includegraphics[height=180pt,width=210pt,trim = 6mm 1mm 0mm 1mm]{ElectronEvolModelSpectrumMUnit1Pt03897e20gRange10e32To10e37wAxesLabe
.png}
\end{align} \caption{
Spectrum generated from the Electron Evolution Model with Turbulent Heating.
}\label{ElectronEvolSpec}
\end{figure}

 \subsection{Critical Beta Electron Temperature Model}

\subsubsection{Electron Temperature Model Images}

The images in Fig. \ref{ElectronTemperatureModelImages}
show the $\beta_c$ variation of the Electron Temperature Model for $f=0.1$ and $f=0.5$, respectively.
For most of the parameter space, the Electron Temperature Model appears to be a fairly uniform projection of the inflow-outflow boundary/coronal region. For the highest values of $f$ and $\beta_c$, the images become more asymmetric and the outflow is limited to smaller radii.
Optical depth effects
are apparent as follows: In the optically thin case, the prescription $T_e=fT_pe^{-\beta/\beta_c}$ at constant $\beta_c$ has line-of-sight intensity proportional to $f$; yet the brightness of intensity maps in Fig. \ref{ElectronTemperatureModelImages} are not simple re-scalings of each other as a function of $f$.
This is most noticeable at the highest value $\beta_c=1$, where the intensity map appears several times more compact and lopsided as we vary $f$ from 0.1 to 0.5.

\begin{figure}\nonumber
\begin{align}
\hspace{0mm} & \includegraphics[height=125pt,width=115pt,trim = 35mm 1mm 0mm 1mm]{PlotfEqPt1BetaTotCritEqPt01HighResZoomMunit1Pt11648e21Flux1Pt85762DX25DY2
.png} & \includegraphics[height=125pt,width=115pt,trim = 10mm 1mm 30mm 1mm]{PlotfEqPt5BetaTotCritEqPt01ZoomHighResMunit2Pt35543e20Flux1Pt99745DX25DY2
.png} &
\end{align}
\begin{align}\nonumber
  \hspace{0mm} & \includegraphics[height=125pt,width=110pt,trim = 35mm 1mm 0mm 1mm]{PlotfEqPt1BetaTotCritEqPt1HighResZoomMunit7Pt58373e20Flux2Pt0889DX25DY2
  .png} & \includegraphics[height=125pt,width=110pt,trim = 10mm 1mm 30mm 1mm]{PlotfEqPt5BetaTotCritEqPt1ZoomHighResMunit7Pt3322e19Flux2Pt32099DX25DY2
  .png} &
\end{align}
\begin{align}
  \hspace{-1.5mm} &  \includegraphics[height=125pt,width=115pt,trim = 35mm 1mm 0mm 1mm]{PlotfEqPt1BetaTotCritEq1HighResZoomMunit5Pt94859e20Flux2Pt20837DX25DY2
  .png} & \includegraphics[height=125pt,width=115pt,trim = 10mm 1mm 30mm  1mm]{PlotfEqPt5BetaTotCritEq1HighResZoomMunit3Pt94188e19Flux2Pt34777DX25DY2
  .png} &
\end{align}
\caption{Critical Beta Electron Temperature Model images for $f=0.1$ (Left Panels) and $f=0.5$ (Right Panels) for $\beta_c=0.01$ (Top Panels), $\beta_c=0.1$ (Middle Panels) and $\beta_c=1$ (Bottom Panels).
}\label{ElectronTemperatureModelImages}
\end{figure}

\subsubsection{Electron Temperature Model Spectra}

Electron Temperature Model spectra generated with parameter values $(f,\beta_c)=(0.1,1)$ and $(0.5,1)$ are shown in Fig. \ref{ElectronEvolAndfAndBeta_cEqPt1And1AndfAndBeta_cEqPt5And1}. Increasing the overall electron temperature prefactor $f$ at constant $\beta_c$ is seen to increase the width of the synchrotron peak-- reflecting a greater range of emitting temperatures. For $(f,\beta_c)=(0.5,1)$, the model fits data points over a
broad frequency range: from microwaves to infrared to X-rays, with possible improvement in the NIR with the addition of nonthermal electrons (and slight reduction in $f,\beta_c$ if this introduces an X-ray excess). From our parsimonious set of two assumptions regarding high and low $\beta$ electron temperature behavior, the simple parametrized Electron Temperature Model performs comparably to the full electron evolution calculation. In fact, for $(f,\beta_c)=(0.5,1)$, the Critical Beta  Electron Temperature Model
is in better agreement with observations
at the high-frequency end.

\begin{figure}\nonumber
\begin{align}
\hspace{1.5cm}\includegraphics[height=150pt,width=180pt,trim = 6mm 1mm 0mm 1mm]{SpectrumfEqPt1BetaTotCritEq1MUnit5Pt94859e20gFlux2Pt40018JyRange10e32To10e37wAxesLabel.png}
\end{align}
\begin{align}\nonumber
  & \hspace{1.5cm}  \includegraphics[height=150pt,width=180pt,trim = 6mm 1mm 0mm 1mm]{SpectrumfEqPt5BetaTotCritEq1MUnit3Pt94188e19gFlux2Pt3596JyRange10e32To10e37wAxesLabe
  }&
\end{align} \caption{
Spectra generated from $(f,\beta_c)=(0.1,1)$ (Top Panel) and $(0.5,1)$ (Bottom Panel) Critical Beta Electron Temperature Models.
}\label{ElectronEvolAndfAndBeta_cEqPt1And1AndfAndBeta_cEqPt5And1}
\end{figure}

\subsection{Constant Electron Beta Model}

\subsubsection{Constant $\beta_{e}$  Model Images}

Constant $\beta_e$ Model images are presented for $\beta_{e0}=0.01,0.1$ and 1 in the top and middle panels of Fig. \ref{EquipartitionInspiredModelImages}.
The images are comprised of a thick torus of lensed disk emission starting at the innermost photon rings and surrounded by dimmer filaments. Decreasing the parameter $\beta_{e0}$ decreases the electron temperature and leads to thinner torii of disk emission
and increasing outflow-to-inflow ratio of filamentary emission. In this limit as well, we find increasing asymmetry as the photon rings emission wane into a narrow crescent on one side.
The increased asymmetry is quantified in Appendix \ref{StatisticalAnalysis:Moments} Table \ref{tab:ImageMoments230GHz}; the centroid of the image for $\beta_{e0}=0.01$ has the furthest lateral displacement for all our models.

\begin{figure}\nonumber
\begin{align}
  \hspace{-1.5mm} & \includegraphics[height=125pt,width=115pt,trim = 35mm 1mm 0mm 1mm]{PlotBetae0EqPt01ModelV2aM_unitgTotalfluxJy1Pt90238e19and2Pt40437T1000
  .png} & \includegraphics[height=125pt,width=115pt,trim = 10mm 1mm 30mm 1mm]{PlotBetae0EqPt1ModelV2aM_unitgTotalfluxJy4Pt08628e182Pt37581T1000
  .png}&
\end{align}
\begin{align}\nonumber
  &\hspace{2.5cm} \includegraphics[height=125pt,width=115pt,trim = 35mm 1mm 0mm 1mm]{PlotBetae0Eq1ModelV2aM_unitgTotalfluxJy2Pt51415e182Pt4266T1000
  .png} &
\end{align}
\begin{align}
  \hspace{-1.5mm} & \includegraphics[height=125pt,width=115pt,trim = 35mm 1mm 0mm 1mm]{PlotnEq0ModelV2aM_unitgTotalfluxJy9Pt18101e18and2Pt43161T1000
  .png} & \includegraphics[height=125pt,width=115pt,trim = 10mm 1mm 30mm 1mm]{PlotnEq2ModelV2aM_unitgTotalfluxJy2Pt95599e182Pt38875T1000
  .png} &
\end{align}
\caption{Constant $\beta_e$ Model image for $\beta{e0}=0.01$ (Top Left) $\beta{e0}=0.1$ (Top Right) and $\beta_{e0}=1$ (Center), along with Bias Model images for $n=0$ (Bottom Left) and $n=2$ (Bottom Right)
}\label{EquipartitionInspiredModelImages
\end{figure}

\subsubsection{Constant $\beta_{e}$ Model Spectra}

Constant $\beta_e$ Model spectra for $\beta_{e0}=0.01$ and 0.1 are shown in Fig. \ref{SpectralComparisonWithObservationsr<30}
excluding regions $r>30M$ that are not in equilibrium. These spectra generally reproduce the low frequency slope well, though flatten near the infrared bump-- particularly for lower values of the constant $\beta_e$.
 The radio spectra can be explained in the context of the Blandford-K\"onigl model \citep{BK1979}, which demonstrates that helical magnetic fields and constant  $\beta_e$ in outflows that are optically thick to synchrotron emission produce radio spectra that are flat in $ L_\nu$, consistent with the low frequency emission in Sgr A*.  The spectral slope rises in the X-ray, as outflow/jet emission overproduces the high-frequency spectrum.

\begin{figure}
\begin{align}\nonumber
  & \hspace{4mm} \includegraphics[height=120pt,width=130pt,trim = 15mm 1mm 0mm 1m
  ]{SpectrumBetaE0EqPt01Munit1Pt95441e19Flux2Pt38785rMax30RangeWAxesLabel
  } & \hspace{0.5mm}\includegraphics[height=120pt,width=130pt,trim = 15mm 1mm 0mm 1m
  ]{SpectrumBetaE0EqPt1Munit5Pt07082e18Flux2Pt36287rMax30WAxesLabel
  } &

\end{align}
\begin{align}\nonumber
  & \hspace{4mm}    \includegraphics[height=120pt,width=130pt,trim = 15mm 1mm 0mm 1mm]{SpectrumNBiasEq0Munit1Pt53302e19Flux2Pt36317rMax30WAxesLabel
  } & \hspace{0.5mm} \includegraphics[height=120pt,width=130pt,trim = 15mm 1mm 0mm 1mm]{SpectrumNBiasEq4Munit4Pt31286e18Flux2Pt42201rMax30WAxesLabel
  } &
\end{align}
\caption{Observed spectra (dots) compared with synthetic model spectra (curves) generated from
$\beta_{e0}=0.01$ Model
(Top Left),
$\beta_{e0}=0.1$ Model (Top Right
,

n=0$ Bias Model (Bottom Left)
and

n=2$ Bias Model
(Bottom Right)
}\label{SpectralComparisonWithObservationsr<30}
\end{figure}

\subsection{Magnetic Bias Model}

\subsubsection{Bias Model Images}

From Eq. \ref{BiasModel}, it is manifest that the simplest ($n=0$) Bias Model has constant relativistic electron gas pressure throughout the simulation, in contrast to its expected decrease along the outflow for higher values of $n$. The image in the bottom left panel of Fig. \ref{EquipartitionInspiredModelImages}
for $n=0$ shows extended emission tracing a funnel shape in the jet/outflow region. For $n=2$,
emission becomes dominated by
a thick photon torus lensed orthogonally to the accretion disk

For Bias Model images, decreasing $n$ leads to more extended outflow contributions to emission.
The Bias and Constant $\beta_e$ Models have the most drastic variation of image intensity and shape over the observer plane for the models considered in this work.

\subsubsection{Bias Model Spectra}

Spectra for
Bias Models with $n=0$ and 2
are shown in Fig. \ref{SpectralComparisonWithObservationsr<30}.
The $n = 2$ model dramatically overproduces the emission at IR-X-rays. Both the $n = 0$ and $n = 2$ models do a reasonable job of explaining the low frequency radio emission, which is not surprising since these models are generalizations of the constant $\beta_e$ model known to explain the radio emission in optically thick AGN jets.

\subsection{EHT and GRAVITY}

\citet{Dexter2010} fit Sgr A* thermal synchrotron emission models parameterized by mass accretion rate, orientation angle, spin and electron-to-proton temperature ratio to mm-VLBI and spectral data \citep{Marrone2006SubmillimeterPO}, favoring a wide $50^\circ$ viewing angle. Our view of the Galactic Center has evolved, as
GRAVITY has provided indications from infrared observations that the Galactic Center inner disk (between 6$M$ and 10$M$) appears more face on \citep{Gillessen2017}.

We now compare synthetic EHT-scale (230 GHz) images against GRAVITY-scale ($2.2\ \mu$m, or 1.4 THz) images for
a particular model, $(f,\beta_{c})=(0.5,1)$, in Fig. \ref{EHTvsGravityComparison}, varying the viewing angle from face-on to edge-on (note, we display these on a common log scale to accentuate features). The face-on disk has smoother variation of intensity with radius at 230 GHz, as it slowly goes from spiral edges to circular at $r\sim 10M$ to the innermost stable circular orbit.  At 2.2 $\mu$m, the disk appears spiral throughout, punctuated with distinct bright spots. At 45$^\circ$ inclination, the brightest feature at both frequencies is a circular ring with $r\lesssim 15M$. The edge-on view exhibits greater disk asymmetry due to Doppler brightening at 230 GHz, and has a more prominent outflow with more apparent substructure
at 2.2 $\mu$m.

We see the dynamical behavior of  the
$(f,\beta_c)=(0.5,1)$ model
at IR frequency in Fig. \ref{IRFlares}, where we show face-on images at several different times. The bright spots appear to rotate on time scales between $\Delta T= 100M$ and $1,000M$ where $M=20$s. The image symmetry makes the centroid motion localized to small gyrations centered on the black hole for this model.

\begin{figure}
\begin{align}\nonumber
  & \hspace{-2.5mm} \includegraphics[height=125pt,width=115pt,trim = 15mm 1mm 0mm 1m
  ]{Log10PlotfEqPt5BetaTotCritEq1Zoom_M_unit__g__Total_flux__Jy____3_94188e19_2_3596__T10000_ThetaObsEq0v2.png} & \includegraphics[height=125pt,width=115pt,trim = 6mm 1mm 20mm 1mm]{Log10PlotfEqPt5BetaTotCritEq1Zoom_M_unit__g__Total_flux__Jy____3_94188e19_4_01178em4__T10000_nuEq1Pt4e14Hz_Rangem13Tom6_ThetaObsEq0.png} &
\end{align}
\begin{align}\nonumber
  & \hspace{-2.5mm} \includegraphics[height=125pt,width=115pt,trim = 15mm 1mm 0mm 1mm]{Log10PlotfEqPt5BetaTotCritEq1Zoom_M_unit__g__Total_flux__Jy____3_94188e19_1_93532__T10000_Rangem8Tom3_ThetaObsEq45Deg.png} & \includegraphics[height=125pt,width=115pt,trim = 6mm 1mm 20mm 1mm]{Log10PlotfEqPt5BetaTotCritEq1Zoom_M_unit__g__Total_flux__Jy____3_94188e19_1_83968em4__T10000_nuEq1Pt4e14Hz_Rangem15Tom6_ThetaObsEq45Deg.png} &
\end{align}
\begin{align}\nonumber
  & \hspace{-2.5mm}  \includegraphics[height=125pt,width=115pt,trim = 15mm 1mm 0mm 1mm]{Log10PlotfEqPt5BetaTotCritEq1Zoom_M_unit__g__Total_flux__Jy____3_94188e19_2_3596__T10000.png} & \includegraphics[height=125pt,width=115pt,trim = 6mm 1mm 20mm 1mm]{Log10PlotfEqPt5BetaTotCritEq1Zoom_M_unit__g__Total_flux__Jy____3_94188e19_4_01178em4__T10000_nuEq1Pt4e14Hz_Rangem12Pt9Tom6Pt1.png} &
\end{align}
\caption{EHT scale (230 GHz, Left) versus GRAVITY scale (140 THz, Right) logplot images of the favored
$(f,\beta_{c})=(0.5,1)$ Model. The observer angle varies from $\theta_\mathrm{Obs}\approx0$ to $\frac{\pi}{4}$ to $\frac{\pi}{2}$ from Top to Bottom. These image maps are shown on a log scale in order to accentuate features in the intensity profiles.
}\label{EHTvsGravityComparison}
\end{figure}

\begin{figure}
\begin{align}\nonumber
  & \hspace{1mm} \includegraphics[height=125pt,width=115pt,trim = 15mm 1mm 0mm 1m
  ]{Log10PlotfEqPt5BetaTotCritEq1Zoom_M_unit__g__Total_flux__Jy____3_94188e19_4_01178em4__T10000_nuEq1Pt4e14Hz_Rangem13Tom6_ThetaObsEq0.png} & \includegraphics[height=125pt,width=115pt,trim = 6mm 1mm 20mm 1mm]{Log10PlotfEqPt5BetaTotCritEq1ZoomMunitgTotalfluxJy3Pt94188e19and1Pt25437em4T10010nuEq1Pt4e14HzRangem13Tom6ThetaObsEq0.png} &
\end{align}
\begin{align}\nonumber
  & \hspace{1mm} \includegraphics[height=125pt,width=115pt,trim = 15mm 1mm 0mm 1mm]{Log10PlotfEqPt5BetaTotCritEq1ZoomMunitgTotalfluxJy3Pt94188e19and8Pt35453em5T10100nuEq1Pt4e14HzRangem13Tom6ThetaObsEq0.png} & \includegraphics[height=125pt,width=115pt,trim = 6mm 1mm 20mm 1mm]{Log10PlotfEqPt5BetaTotCritEq1ZoomMunitgTotalfluxJy3Pt94188e19and2Pt98692em5T11000nuEq1Pt4e14HzRangem13Tom6ThetaObsEq0.png} &
\end{align}
\caption{A sequence of face-on $(f,\beta_{c})=(0.5,1)$ Model infrared images at $T=10,000M,10,010M,10,100M$ and 11,000$M$. Variation can be observed after a hundred $M$, but the basic spiral morphology with bright spots is maintained over at least a thousand $M$.
}\label{IRFlares}
\end{figure}

 \subsection{SANE vs. SEMI-MAD vs. MAD Simulations}

 Standard and normal evolution (SANE), magnetically arrested disk (MAD) and semi-MAD simulations represent quite distinct forms of evolution of magnetized accretion flows. The SANE case has the lowest magnetic flux; the MAD case admits the lowest amount of mass accreted by the black hole, as in Table \ref{MUnit230GHz}. Compared to images ray traced from the fiducial semi-MAD simulation, the SANE and MAD images vary in size and asymmetry in a similar manner with changing parameters in our models. For example, Fig. \ref{fEqPt1orPt5ElectronTemperatureModelSANEsemiandMAD} shows that increasing Critical Beta Electron Temperature Model parameter $f$ from 0.1 to 0.5 takes a relatively symmetric photon ring/torus  and extended outflow structure to a compact, asymmetric ring for all three simulations. The same trend holds for increasing $\beta_c$ from 0.01 to 1. For the $f=0.5$ models in all three simulations, a disk feature is visible emanating from the bright spot and extending along the projected equatorial plane. It is also noteworthy that the best $10^{11}$ Hz $<\nu_\mathrm{Obs}<$ $10^{19}$ Hz spectrum across simulations (cf. Fig. \ref{SpectrafEqPt1orPt5ElectronTemperatureModelSANEsemiandMAD}) corresponds to the $(f,\beta_c)=(0.5,1)$ model in the semi-MAD simulation; however, the MAD simulation for this model generates the best low frequency fit. We do add the caveat that with the addition of non-thermal particles, we expect the MAD images to have increased NIR and X-ray contributions.
 As for other models, decreasing Constant Electron Beta Model Parameter $\beta_{e0}$ from 1 to 0.01 changes images from thick photon torii
 to thin rings + outflows. Decreasing Magnetic Bias Model parameter $n$ from 2 to 0 accentuates the outflow in all 3 simulations, with the key difference being greatest collimation in the SANE simulation. For all simulations and models, inclusion of non-thermal particles could readily increase the IR and X-ray emission.

\begin{figure}\nonumber
\begin{align}\nonumber
 \hspace{5mm} & \includegraphics[height=115pt,width=110pt,trim = 35mm 1mm 0mm 1mm]{PlotfEqPt1BetaCritEq1SANEModelV1M_unitgTotalfluxJy4Pt22646e20and2Pt39825T1000
.png} & \includegraphics[height=115pt,width=110pt,trim = 10mm 1mm 30mm 1mm]{PlotfEqPt5BetaCritEq1SANEModelV1M_unitgTotalfluxJy1Pt85173e19and2Pt39114T1000
  .png} &
\end{align}
\begin{align}\nonumber
 \hspace{5mm}  & \includegraphics[height=115pt,width=110pt,trim = 35mm 1mm 0mm 1mm]{PlotfEqPt1BetaTotCritEq1V1fBetaTotCrit0Pt1and1M_unitgTotalfluxJy5Pt94859e20and2Pt40018T1000
  .png} & \includegraphics[height=115pt,width=110pt,trim = 10mm 1mm 30mm 1mm]{PlotfEqPt5BetaTotCritEq1V1fBetaTotCrit0Pt5and1M_unitgTotalfluxJy3Pt94188e19and2Pt3596T1000
  .png} &
\end{align}
\begin{align}
  \hspace{5mm} & \includegraphics[height=115pt,width=110pt,trim = 35mm 1mm 0mm 1mm]{PlotfEqPt1BetaCritEq1MADModelV1M_unitgTotalfluxJy1Pt3886e18and2Pt40796T1000
  .png} & \includegraphics[height=115pt,width=110pt,trim = 10mm 1mm 30mm 1mm]{PlotfEqPt5BetaCritEq1MADModelV1M_unitgTotalfluxJy1Pt06979e17and2Pt43971T1000
  .png} &
\end{align}
\caption{Critical Beta Electron Temperature Model  for $f=0.1$ (Left Panels) and $f=0.5$ images (Right Panels) for $\beta_\mathrm{c}=1$ SANE (Top Panels), semi-MAD (Middle Panels) and MAD simulations (Bottom Panels).}\label{fEqPt1orPt5ElectronTemperatureModelSANEsemiandMAD}
\end{figure}

\begin{figure}\nonumber
\begin{align}\nonumber
  &\hspace{5mm} \includegraphics[height=100pt,width=125pt,trim = 6mm 1mm 0mm 1mm]{SpectrumfEqPt1BetaCEq1Munit4Pt50112e20Flux2Pt39665SANEwAxesLabel
.png} & \includegraphics[height=100pt,width=125pt,trim = 6mm 1mm 0mm 1mm]{SpectrumfEqPt5BetaCEq1Munit2Pt55178e19Flux2Pt42829SANEwAxesLabel
  .png} &
\end{align}
\begin{align}\nonumber
  & \hspace{5mm}\includegraphics[height=100pt,width=125pt,trim = 6mm 1mm 0mm 1mm]{SpectrumfEqPt1BetaTotCritEq1MUnit5Pt94859e20gFlux2Pt40018JyRange10e32To10e37wAxesLabe
  .png} & \includegraphics[height=100pt,width=125pt,trim = 6mm 1mm 0mm 1mm]{SpectrumfEqPt5BetaTotCritEq1MUnit3Pt94188e19gFlux2Pt3596JyRange10e32To10e37wAxesLabe
  .png} &
\end{align}
\begin{align}
  &\hspace{5mm}\includegraphics[height=100pt,width=125pt,trim = 6mm 1mm 0mm 1mm]{SpectrumfEqPt1BetaCEq1Munit1Pt67729e18Flux2Pt39972MADwAxesLabel
  .png} &\includegraphics[height=100pt,width=125pt,trim = 6mm 1mm 0mm 1mm]{SpectrumfEqPt5BetaCEq1Munit1Pt40117e17Flux2Pt42936MADwAxesLabel
  .png} &
\end{align}
\caption{Critical Beta Electron Temperature Model $f=0.1$  (Left Panels) and $f=0.5$ spectra (Right Panels) for $\beta_\mathrm{c}=1$ SANE (Top Panels), semi-MAD (Middle Panels) and MAD simulations (Bottom Panels). }\label{SpectrafEqPt1orPt5ElectronTemperatureModelSANEsemiandMAD}
\end{figure}

\subsection{mm-/NIR
Variability}

In Fig. \ref{ModelLightCurves}, we display mm- and NIR light curves for select models.  The models exhibit different variability timescales, e.g., doubling times, over the 4,000$M$ ($\approx24$h) interval depicted. The $\beta_{e0}=1$ and $n=2$ models exhibit the mildest variation of a few percent within an hour ($\approx 160M$).  These models are immediately precluded if we require at least minute-scale variability. The $n=0$ model is slightly more variable, with slightly larger amplitude local minima and maxima. The $(f,\beta_c)=(0.5,1)$ model is the most rapidly varying, followed by $(f,\beta_c)=(0.1,1)$. The greatest amplitude flare-like feature in the radio is the two-fold peak-to-trough variation for $12,700M\lesssim t_\mathrm{Obs} \lesssim  13,500M$ in the $\beta_{e0}=0.01$ model light curve.

The NIR luminosity differs by orders of magnitude among the models in Fig. \ref{ModelLightCurves}.  The models with the greatest overall flux are for $n=2$ and  $\beta_{e0}=1$. The lowest flux model, $(f,\beta_c)=(0.5,1)$, exhibits the fastest peak-to-trough variability and the largest percentage swings.  But despite these large relative excursions from the mean flux, the low absolute amplitude NIR luminosity suggests that this model requires a non-thermal particle contribution.

\subsubsection{Time Averaging}\label{TimeAveraging}
We have primarily compared models at a particular time $T=10,000M$ for which the simulation flow is statistically in steady state.  However, given the $20$s light crossing time of Sgr A*'s gravitational radius, variability challenges EHT's temporal imaging capacity for Sgr A*-- and all but the most massive black holes.
Fig. \ref{TimeAveragedImage} shows an image averaged over $1000 \, M$ (6 hr for Sgr A*) for our $(f,\beta_c) = (0.5,1)$ model.  The result is very similar to the instantaneous image in Fig. \ref{ElectronTemperatureModelImages}.   Examining time evolution of the image in detail, we find that there is image variability on roughly hourly timescales, with some bright features moving around.   The overall image morphology is, however, relatively stable.

\begin{figure}\nonumber
\begin{align}
 \hspace{1.3cm} & \includegraphics[height=170pt,width=200pt,trim = 35mm 1mm 0mm 1mm]{mmLightCurvesT10000To13960.png}
\end{align}
\begin{align}\nonumber
 \hspace{0.3cm} &
\includegraphics[height=170pt,width=200pt,trim = 10mm 1mm 30mm 1mm]{NIRLightCurvesT10000To13960.png}
\end{align}
\caption{Light curves at 230 GHz (Top)  and 140 THz (Bottom) for parametric models with $(f,\beta_c)=(0.1,1)$, $(f,\beta_c)=(0.5,1)$,  $\beta_{e0}=0.01$, $\beta_{e0}=1$, $n=0$  and $n=2$.
}\label{ModelLightCurves}
\end{figure}

\section{Comparison of Models}

We now synthesize our results to assess which models are favored by observations, starting with our fiducial semi-MAD simulation.
A summary of how the different models fare against observational constraints is given in Table \ref{PassFail}.

\subsection{Comparison of Images}

In the images for the Electron Evolution Model with Turbulent Heating, the emission is concentrated in an asymmetric photon ring, with some some contribution from outflow at small radii. The Critical Beta  Electron Temperature Model images have emission smeared out broadly over inflow and outflow
at low $f$ or $\beta_c$, and approach a compact, asymmetric photon ring for $(f,\beta_c)\to(0.5,1)$, with even less outflow contribution relative to the Electron Evolution Model. The Constant $\beta_e$ and Bias Models have drastically varying morphology over the parameter space scanned-- from long, extended outflow filaments for the low $\beta_{e0}$ and low $n$ models, to thick photon torii
in the high $\beta_{e0}$ and high $n$ limits.

The relative asymmetry of model images can be compared using image moments in Appendix \ref{StatisticalAnalysis:Moments} Table \ref{tab:ImageMoments230GHz}, where models in which most of the flux density emanates from compact regions regions such as $\beta_{e0}=0.01$ or $(f,\beta_c)=(0.5,1)$ are shown to have centroids shifted furthest to the right in our fiducial simulation. We make
further quantitative comparisons of the images
by ascribing emitting region sizes to our intensity maps.

\subsubsection{Emitting Region}

\begin{table*
	\centering
	\caption{Elliptical Gaussian semi-axes $\theta_\mathrm{min}$ and $\theta_\mathrm{maj}$ generated from second order central image moments and image covariance matrices for our models for SANE, MAD and semi-MAD simulations at the fiducial 230 GHz frequency and 90$^\circ$ (edge-on) viewing angle.  The anisotropic observational size constraints  are $5.44M<\theta_\mathrm{min}<10.68M$ and  $9.87M<\theta_\mathrm{maj}<12.69M$ \citep{Johnson_2018}

    }
	\label{tab:Gtr10AndGtr20PctAreas}
	\begin{tabular}{lccccccr}
		\hline
		Model &
        &&

        $\frac{\theta_\mathrm{min\ }}{M}=\sqrt{(8\ln{2})\lambda_\mathrm{min}}$ &&
        &

        $\frac{\theta_\mathrm{maj\ }}{M}=\sqrt{(8\ln{2})\lambda_\mathrm{max}}$ &  \\
        &&&&&&\\
     &
     &SANE&  semi-MAD  &MAD&
     SANE& semi-MAD & MAD  \\
     		\hline
	 Electron Evolution with Turbulent Heating
          &

          \textbf{8.2
          &

          \textbf{9.0

          11.

          18.
          &
          15.

          15.
          \\
		\hline
		Critical Beta Electron Temperature ($f,\beta_c$) &    & & & \\
        ($0.1,0.01$)&

        \textbf{9.4
        &

        24.

        38.
        &

        28.
        &
        48.

        73.
        \\
       ($0.1,0.1$)&

         \textbf{8.8
       &
       21.

       18.
       &

       24.
       &
       39.

       34.
       \\
       ($0.1,1.0$)&
       &
       \textbf{8.8
       &
       20.
       &
       15.
       &
       22.
       &
       33.

       23.
       \\
       ($0.5,0.01$) &

       \textbf{9.4
       &
       33.
       &
       25.
       &

       29.
       &
       43.

       55.
       \\
       ($0.5,0.1$)&

       \textbf{8.7
       &
       16.

       19.
       &

       22.
       &
       24.

       34.
       \\
       ($0.5,1.0$)&

       \textbf{9.3
       &
       15.

       12.
       &

       \textbf{11.7
       &
       18.

       14.
       \\
        \hline
		Constant Electron Beta ($\beta_{e0}$)&  &  & & &\\
         0.01   &

         14.
         &
         22.

         22.
         &

         45.
         &
         44.

         44.
         \\
         0.1   &

         19.
         &
         45.

         47.
         &

         26.
         &
         46.

         53.
         \\
         1.0   &

         37.
         &
         45.

         57.
         &

         39.
         &
         56.

         58.
         \\
        \hline
		 Magnetic Bias ($n$
         &  &  & & & \\
		0 &

        43.
        &
        58.

        30.
        &

        75.
        &
        66.

        75.
        \\
        1 &

        37.
         &
         45.

         57.
         &

         39.
         &
         56.

         34.
        \\
        2 &

        30.
        &
        43.

        60.
        &

        47.
        &
        52.

        62.
        \\
        \hline
	\end{tabular}\label{AnisotropicGaussianSize}
\end{table*}

We adopt the image covariance approach that \citet{Johnson_2018} used to derive the $9.87M<\theta_\mathrm{maj,Obs
<12.69M$ elliptical Gaussian size constraint for 230 GHz Sgr A* EHT observations to formulate characteristic emitting region sizes for our synthetic images. The intensity can be expressed as a moment generating function

\addtocounter{equation}{7}

\begin{equation}
\tilde{I}(\mathbf{u})=\int d^2\mathbf{x} I(\mathbf{x})e^{-2\pi i \mathbf{u}\cdot\mathbf{x}}
\end{equation}
The image covariance matrix is
   \begin{align}\label{ImageCovariance}
M =   \begin{pmatrix} \frac{M_{20}}{M_{00}}-\left(\frac{M_{10}}{M_{00}}\right)^2 & \frac{M_{11}}{M_{00}}-\frac{M_{10}M_{01}}{M_{00}^2} \\
  \frac{M_{11}}{M_{00}}-\frac{M_{10}M_{01}}{M_{00}^2} & \frac{M_{02}}{M_{00}}-\left(\frac{M_{01}}{M_{00}}\right)^2  \\
  \end{pmatrix}.
\end{align}
where
\begin{equation}
M_{n_1,n_2}=\sum_{i,j}x_i^{n_1}y_j^{n_2}I(x_i,y_j)
\end{equation}
The equivalent Gaussian FWHM for images in Observer Plane coordinates is
\begin{align}\nonumber
(\theta_\mathrm{maj})_\mathrm{Obs} = \sqrt{-\frac{2\ln 2}{\pi^2\int d^2\mathbf{x} I(\mathbf{x})}}\times \hspace{4.5cm}\\ \nonumber
\sqrt{\left((\hat{u}_\mathrm{maj})_1^2\frac{\partial^2\tilde{I}}{\partial u_1^2}+2(\hat{u}_\mathrm{maj})_1(\hat{u}_\mathrm{maj})_2\frac{\partial^2\tilde{I}}{\partial u_1 \partial u_2}+ (\hat{u}_\mathrm{maj})_2^2\frac{\partial^2\tilde{I}}{\partial u_2^2}\right)}
\end{align}

\begin{equation}
=2\sqrt{2\ln 2\frac{(\hat{u}_\mathrm{maj})_1^2M_{20}+2(\hat{u}_\mathrm{maj})_1(\hat{u}_\mathrm{maj})_2M_{11}+(\hat{u}_\mathrm{maj})_2^2M_{02}}{M_{00}}}
\end{equation}
where $\hat{u}_\mathrm{maj}$ is the eigenvector of the covariance matrix corresponding to greatest eigenvalue $\lambda_\mathrm{maj}$ and $\hat{u}_\mathrm{min}$ is defined analogously. We may conceptualize $(\theta_\mathrm{maj})_\mathrm{Obs}$ as a normalized second directional derivative along the image major axis. We take our characteristic image size to be the centroid-adjusted equivalent Gaussian FWHM
\begin{equation}
\theta_\mathrm{maj} = 2\sqrt{(2\ln{2})\lambda_\mathrm{maj}}
\end{equation}

In Table \ref{AnisotropicGaussianSiz
} using the elliptical Gaussian size constraint, we have the following trends in image size:
\begin{itemize}
    \item Image size decreases with $f$ or $\beta_c$
    \item Image size increases with $\beta_{e0}$ (with the exception of the SANE simulation at low $\beta_{e0}$
    \item Image size decreases with $n$ (exceptions: high $n$ SANE and MAD)
\end{itemize}
The preferred models with smallest size occur for $(0.5,1)$, particularly in the SANE simulation, which satisfies the EHT size constraint for $\theta_\mathrm{maj}$.

Note that observational effects such as scattering, the point spread function and instrument-specific cadence tend to wash out image features. While we do not include a model for scattering in this work, we have time-averaged our preferred model in
Fig. \ref{TimeAveragedImage} to reflect uncertainties due to temporal resolution.
The time averaged image for the $\sim 6$h interval $10,000M<T<11,000M$ is similar to the instant one in Fig. \ref{ElectronTemperatureModelImages} (Lower Right), and the spectral shape does not vary appreciably even over the $\sim 12$h interval $9,000M<T<11,000M$, as discussed in Section \ref{TimeAveraging}

  \begin{figure}
\begin{align}\nonumber
  & \hspace{1.0cm} \includegraphics[height=175pt,width=200pt,trim = 6mm 1mm 0mm 1mm]{PlotfEqPt5BetaTotCritEq1MUnit3Pt94188e19gT10000MTo11000MAveragedZoom.png}
  &
\end{align}
\caption{Preferred model ($(f,\beta_c)=(0.5,1)$) averaged over simulation times $10,000M<T<11,000M$ at 230 GHz.}\label{TimeAveragedImage}
\end{figure}

\subsubsection{Image Size Frequency and Viewing Angle Dependence}

In Table \ref{tab:86GHzVs230GHzAnd90DegVs0Deg}, image sizes are compared at angles $0^\circ$ (face on) vs. $90^\circ$ (edge on) and frequencies 86 GHz (Global mm VLBI Array (GMVA)) vs. 230 GHz (EHT). For both frequencies, the edge-on image sizes tend to decrease with each of $f$ and $\beta_c$, increase in $\beta_{e0}$ and decrease in $n$; the face-on images do not exhibit monotonic behavior. Moreover, at a given frequency, the edge-on images tend to be larger than their face-on parametric counterparts for Critical $\beta$ Models (except at high $(f,\beta_c)$); face-on images are larger than edge-on images for the Constant $\beta_e$ Model (except at low $\beta_{e0}$) and Magnetic Bias Model (except at higher frequency and low $n$). Face on and at low frequency, intensity profiles appear broad and point symmetric due to the projection of co-axial outflows and disk onto the observer plane-- particularly for low $n$ and low $(f,\beta_c)$. The smallest of our flux-normalized images tend to occur for the most asymmetric models, e.g., $(f,\beta_c)=(0.5,1)$.

The images satisfying the 86 GHz size constraint occur for the Critical Beta Electron Temperature Models $(f,\beta_c)=(0.1,0.01),(0.1,0.1),(0.1,1),(0.5,0.01),(0.5,0.1)$ and $(0.5,1)$ for the SANE and MAD simulations and MAD $\beta_{e}=0.01,0.1$ and 1 Constant $\beta_{e0}$ and $n=0,1$ and 2 Bias Models viewed at $0^\circ$; and the $(f,\beta_c)=(0.5,1)$ Model for the SANE, semi-MAD and MAD simulations at
$90^\circ.$
The unique model satisfying the 230 GHz size constrain is the $(f,\beta_c)=(0.5,1)$ Model in the SANE simulation viewed edge on.

\begin{table*}
	\centering
	\caption{Elliptical Gaussian semi-axes $\theta_\mathrm{min}$ and $\theta_\mathrm{maj}$ generated from second order central image moments and image covariance matrices for our semi-MAD models at varying viewing angles (0$^\circ$ vs. 90$^\circ$) and frequencies (86 GHz vs. 230 GHz). The anisotropic size constraints are $5.44M<\theta_\mathrm{min}<10.68M$ and  $9.87M<\theta_\mathrm{maj}<12.69M$ at 230 GHz and $16.5<\theta_\mathrm{min}<23.8M$ and $17.3M<\theta_\mathrm{maj}<31.0M$ at 86 GHz \citep{Issaoun2019}

    }
	\label{tab:86GHzVs230GHzAnd90DegVs0Deg}
	\begin{tabular}{lccccccr}
		\hline
		Model &
        && $\theta_\mathrm{min}(M)$

        &&
        & $\theta_\mathrm{maj}(M)$

        &  \\
        &&&&&&\\
     &
     & 86 GHz, 0$^\circ$ &  86 GHz, 90$^\circ$  &  230 GHz, 0$^\circ$ & 86 GHz, 0$^\circ$ & 86 GHz, 90$^\circ$  & 230 GHz, 0$^\circ$   \\
     		\hline

	 Electron Evolution

          &
          &
          37.6
          &

           25.7
          &
          26.8
          &
          39.7
          &
          47.8
          &
          28.5
          \\
		\hline

		Electron Temperature ($f,\beta_c$) &    & & & \\
        ($0.1,0.01$)&
        &
        43.7
        &

         30.4
        &
        33.8
        &

        54.7
        &
        69.7
        &
        39.1
        \\
       ($0.1,0.1$)&
       &
       39.8
       &
       29.6
       &
       28.9
       &

       48.7
       &
       62.5
       &
       32.2
       \\
       ($0.1,1.0$)&
       &

       50.6
       &
       29.6
       &
       25.9
       &
       45.5
       &
       56.4
       &
       28.6
       \\
       ($0.5,0.01$) &

       &
       42.5
       &
       28.4
       &
       34.9
       &
       47.6
       &
       59.5
       &
       39.5
       \\
       ($0.5,0.1$)&

       &
       36.6
       &
       28.7
       &
       25.7
       &
       38.4
       &
       38.6
       &
       27.2
       \\
       ($0.5,1.0$)&
       &
       33.9
       &
       27.6
       &
       24.3
       &

        34.8
       &
       \textbf{29.4}
       &
       25.4
       \\
        \hline
		Constant Electron Beta ($\beta_{e0}$)&  &  & & &\\
         0.01   &
         &
         44.1
         &
         32.9
         &
         40.1
         &

        48.2
         &
         49.9
         &
         42.1
         \\
         0.1   &
         &
         56.6
         &
         45.8
         &
         54.0
         &

         59.5
         &
         50.3
         &
         57.6
         \\
         1.0   &

         61.5
         &
         45.7
         &
         61.5
         &

        62.8
         &
         57.0
         &
         62.9
         \\
        \hline
		 Magnetic Bias ($n$
         &  &  & & & \\
		0 &
        &
        64.5
        &
        58.8
        &
        63.5
        &
        66.9
        &
        62.0
        &
        66.2
        \\
        1 & &
        61.5
         &
         45.7
         &
         61.5
         &

           62.8
         &
         57.0
         &
         62.9
        \\
        2 &
        &
        60.7
        &
        44.4
        &
        58.9
        &
        62.6
        &
        55.0
        &
        61.1
        \\
        \hline
	\end{tabular}
\end{table*}\label{AnisotropicGaussianSizeChangingThetaAndNu}

\subsection{Comparison of Spectra}

The
Electron Evolution Model with Turbulent Heating fits most of the data near the sub-mm bump. Spectra in models inspired by equipartition (Constant $\beta_e$ and Bias) are dominated by near-horizon outflow emission (this becomes more apparent in images at lower frequencies, as the lensed disk becomes less prominent). These spectra significantly overproduce frequencies above infrared.
For the Bias Model, lower $n$ reduces the falloff of $u_e$ with radius, accounting for flatter radio and X-ray spectra.  Spectra in the Critical Beta Electron Temperature Model tend to be dominated by outflow or coronal emission, especially at lower $f$ or $\beta_c$,
and tend to be more peaked than the data.

The following summarizes trends appearing upon comparing spectra in different models:
\begin{itemize}
\item{The Critical Beta Electron Temperature Model reproduces low- and high-energy spectral amplitudes over the $10^{11}$-$10^{19}$ Hz frequency domain better than the Constant $\beta_e$ and Bias Models.}
\item{The radio-IR spectrum is flatter for lower $\beta_{e0}$ in the Constant $\beta_e$ Model.}
\item{The radio-IR spectrum is flatter for lower $n$ in the Bias Model (until a sharp low frequency turnover at $\lesssim10^{11}$ Hz, where lower $n$ models steepen).}

\item{Spectra in the
Constant $\beta_e$ and Bias Models appear flatter than spectra in the Critical Beta Electron Temperature Model.
}

\end{itemize}
Some of these trends
are a consequence of varying
mass accretion rate and synchrotron absorption
in different models, as discussed below.

\subsubsection{Trends With Mass Accretion Rate}

From Table \ref{MUnit230GHz},
model families with flatter spectra (Constant $\beta_e$ and Bias Models) tend to have lower (magnitude) mass accretion rates than the family with steeper spectra (the Critical Beta Electron Temperature Model). This can be explained from the simulation as follows: As $|\dot{m}|$ decreases at fixed flux at 230 GHz,
temperature increases and plasma in regions that have not previously been emitting synchrotron radiation begin to emit, broadening the range of emitting temperatures and, in turn, the spectra. This also predicts an
anticorrelation between the magnitude of the mass accretion rate and the size of the emitting region in images in optically thin synchrotron models, but optically thick models
deviate markedly from this trend.

\subsection{Phenomenological Classification}

For our fiducial semi-MAD simulation, upon exploration of
parameter space, $(f,\beta_c)$ models tend to outperform the jet-inspired Constant $\beta_e$ and Bias $(n)$ Models with respect to
spectral observations, and often outperform the detailed Electron Evolution Model with Turbulent Heating as well.

With respect to image morphology, observations favor compactness.

The images best satisfying the  86 GHz GMVA and 230 GHz EHT elliptical Gaussian size constraints for $\theta_\mathrm{maj}$ are from the $(f,\beta_c)=(0.5,1)$ model. It is noteworthy that this highest $(f,\beta_c)$ pair producing the best spatial fit for the Critical Beta Electron Temperature Model also provides the best overall spectral fit, giving us indication that the image and spectral properties are correlated. Moreover, image compactness is related to image shape in our models, as the lowest $\beta_{e0}=0.01$ providing the best (smallest) emitting region size for equipartition-inspired models has image morphology resembling the asymmetric crescent from the $(f ,\beta_c )=(0.5,1)$ Model.

Upon a scan of model parameter space for our fiducial  simulation, we find that images tend to aggregate into (at least) 4 broad categories, which are closely tied to
spectra
and related to variability as well:
\renewcommand{\theenumi}{\Roman{enumii}}
\begin{enumerate}
    \item I.) Thin, Compact, Asymmetric Photon Ring/Crescent\newline
  This image morphology is exhibited for the Electron Evolution Model with Turbulent Heating and in parameterized models with high $(f,\beta_c)$, e.g., $\{(0.5,0.1),(0.5,1)\}$ or low $\beta_{e0}$, e.g.,  0.01. The concomitant spectra are the best fit across our models, with the largest deviation coming from the $\beta_{e0}=0.01$ Model's very flat spectrum in the IR band and mild X-ray excess. The mm-variability is characterized by light curves with moderate or high amplitude, oscillating rapidly on intra-hour scales-- with the greater variability in the Constant $\beta_{e0}$ branch breaking image degeneracy with $(f,\beta_c)$ models.
  \item II.) Inflow-Outflow Boundary + Thin Photon Ring\newline
  This image morphology is exhibited in models with $(f,\beta_c)\in \{(0.1,0.01),(0.1,0.1),(0.1,1),(0.5,0.01)\}$, and is accompanied by the steepest spectra, sharply peaking near the IR bump. The radio variability is moderate amplitude and fast.
  \item III.) Thick Photon Toru
  \newline
  This image morphology is exhibited in the Constant $\beta_e$ Model with $\beta_{e0}\in \{0.1,1\}$, and the Bias Model with
  n=2$, and is accompanied by spectra with large X-ray excesses (and a flat overall spectrum for $\beta_{e0}=0.1$). These models have low amplitude, slowly oscillating radio light curves.
  \item IV.) Extended Outflow\newline
  Occurring in the
  n=0$ Bias Model, this image morphology is linked to a flat overall spectrum with X-ray excess. This class is characterized by low amplitude, nearly monotonic radio variability within a day.
  \end{enumerate}

It is noteworthy that dominant image features are closely linked with dominant spectral features and tied to variability as well, and that these associations form the basis for distinct classes into which models governed by related physics-- albeit different parameterizations thereof-- can be identified.

\subsubsection{SANE vs. SEMI-MAD vs. MAD Simulation Comparison}

As we consider simulations outside of the fiducial simulation, new image morphologies emerge. For example, for the SANE simulation, the $(f,\beta_c)=(0.1,1)$ Model has a distinctly helical outflow intensity map and a remarkably good spectral fit for all but the lowest frequencies (cf.
Figs. \ref{fEqPt1orPt5ElectronTemperatureModelSANEsemiandMAD} and \ref{SpectrafEqPt1orPt5ElectronTemperatureModelSANEsemiandMAD}).
Furthermore, it is notable that some of our models become similar to our favored $(f,\beta_c)=(0.5,1)$ Model in appropriate limits. In particular, as $\beta_{e0}$ goes from 1 to 0.01 in the Constant $\beta_e$ Mode
, intensity maps become more confined to small cylindrical radius and asymmetric for SANE, semi-MAD and MAD cases alike; and, for the MAD simulation, closer spectral fits are produced for the microwave, IR and (especially) X-ray bands.
The theoretical advancement of a single model unifying emission characteristics of accretion flow, corona and outflow using a single simulation and small set of parameters would enable us to directly identify observed features with plasma and emission physics of different AGN components.

\section{Conclusions and Future Directions}

 We have used two simple classes of parametric emission prescriptions-- the
turbulent-heating-based Critical Beta  Electron Temperature
Model $T_e=fT_pe^{-\beta/\beta_c}$ and the equipartition-based Constant Electron Beta/Magnetic Bias Models
$P_e=K_nP_B^n,\ (\beta_{e0}\equiv K_1\ \mathrm{for}\ n=1)$-- to
explore a wide range of possible models for the images and spectra
in the inner tens of gravitational radii around the supermassive black hole at the Galactic Center. One of our models, $(f,\beta_c)=(0.5,1)$,
is observationally favored for Sgr A* due to its agreement with respect to the spectrum, emitting region compactness and asymmetry. We stress that both observationally preferred images and spectra are associated with emission concentrated in a bright, crescent-shaped portion of the photon ring near the horizon, and that forthcoming EHT and GRAVITY data will further constrain our models.

It is worth noting that our intuitive one- and two-parameter models span new
electron physics beyond what has been previously explored, and promise to bear upon AGN beyond Sgr A*. By surveying synthetic images and spectra in other parts of model parameter space, we may isolate the emission physics underlying
particular observational phenomena. We summarize these results as follows.

\subsection{Images Summary}

Synthetic 230 GHz intensity maps on the scale of tens of gravitational radii appear:

\begin{itemize}
\item{Dominated by the inflow/outflow interface for the Critical Beta Electron Temperature Model }
\item{More compact and asymmetric with increasing $f$ or $\beta_c$ }
\item{Mixed with outflow/near-horizon photon ring emission for
the Constant $\beta_e$ and Bias Models}
\item{More concentrated around
horizon-circulating lensed disk emission} for increasing $\beta_{e0}$ and increasing $n$
\end{itemize}

\subsection{Spectra Summary}

In comparison to the data, our parameterized synthetic $10^{11}$ Hz $<\nu<10^{19}$ Hz spectra have:
\begin{itemize}
\item{More peaked slopes fairly consistent with the sub-mm bump but underproducing the low frequency tail for the Critical Beta Electron Temperature Model}
\item{Fairly consistent low frequency slope but flatter peak and overproduction of low and (especially) high frequency emission for the Constant $\beta_e$ Model}
\item{Consistent low frequency slope but broader peak and (in some cases vast) overproduction of high frequency emission for the Bias Model}

\end{itemize}

The steep radio spectra seen in the Electron Temperature Model is characteristic of emission from an adiabatically expanding coronal outflow, in which temperature rapidly declines in radius. The flatter radio spectra from the Constant $\beta_e$ and Bias prescriptions are due to contributions from the highly magnetized outflow and the dependence of the latter emission functions exclusively on $P_B$. Although these latter models can explain many features of jets/outflows (including Sgr A*) such as Doppler beaming, knots and other magnetic field substructure, we have shown that they do not accurately describe the inner regions of discs/jets around Sgr A* (and perhaps, by analogy, other systems).

\subsection{Future Directions}

Sgr A* viewed at 1.3 mm has exhibited intrahour variability \citep{Johnson2015} in the inner  $\sim 6r_g$ around a black hole whose Schwarszchild-radius-light-crossing-time is 40 s.
In the future, it would be valuable to produce models with light curves in the simulations closely replicating the observational cadence and understand which range of models is most consistent with the observed variability. We may also add polarization maps to our pipeline to test whether electromagnetically dominated emission models in simulations with ordered magnetic field substructure, e.g., helical Blandford-K\"onigl outflows, can
help explain observations of a high degree of linear polarization in AGN cores.

The \say{observing} simulations methodology is a key link between ever-advancing simulations and observations of the central engines of JAB systems. The EHT serves as a timely testbed for our
emission models that aim to unite simulations and observations. The next works planned in this ```Observing' JAB Simulations" series are applications to the prominent jets in the giant elliptical galaxy M87 and the highly variable quasar 3C 279. Other EHT target sources to be observed-- and possibly, \say{observed}-- in the future include: Cen A, NGC 1052 and OJ 287.

\section*{Acknowledgements}

This work was supported in part by NSF grants AST 13-33612 and AST 1715054, Chandra theory grant TM7-18006X from the Smithsonian Institution, and a Simons Investigator award from the Simons Foundation. This work was made possible by computing time granted by UC Berkeley on the Savio cluster. RA carried out part of this work supported by the California Alliance and Simons Foundation and the remainder of this work at the Black Hole Initiative at Harvard University, which is supported by a grant from the John Templeton Foundation. RA is also supported by the NSF MSIP Grant AST-1440254. SR was supported in part by the NASA Earth and Space Science Fellowship and the Gordon and Betty Moore Foundation
through Grant GBMF7392 during part of the duration of this work. We all thank the scientific editor's thorough and incisive comments, which greatly improved the overall quality of this work.

\bibliographystyle{mnras}

\section{Introduction}

The journal \textit{Monthly Notices of the Royal Astronomical Society} (MNRAS) encourages authors to prepare their papers using \LaTeX.
The style file \verb'mnras.cls' can be used to approximate the final appearance of the journal, and provides numerous features to simplify the preparation of papers.
This document, \verb'mnras_guide.tex', provides guidance on using that style file and the features it enables.

This is not a general guide on how to use \LaTeX, of which many excellent examples already exist.
We particularly recommend \textit{Wikibooks \LaTeX}\footnote{\url{https://en.wikibooks.org/wiki/LaTeX}}, a collaborative online textbook which is of use to both beginners and experts.
Alternatively there are several other online resources, and most academic libraries also hold suitable beginner's guides.

For guidance on the contents of papers, journal style, and how to submit a paper, see the MNRAS Instructions to Authors\footnote{\label{foot:itas}\url{http://www.oxfordjournals.org/our_journals/mnras/for_authors/}}.
Only technical issues with the \LaTeX\ class are considered here.

\section{Obtaining and installing the MNRAS package}
Some \LaTeX\ distributions come with the MNRAS package by default.
If yours does not, you can either install it using your distribution's package manager, or download it from the Comprehensive \TeX\ Archive Network\footnote{\url{http://www.ctan.org/tex-archive/macros/latex/contrib/mnras}} (CTAN).

The files can either be installed permanently by placing them in the appropriate directory (consult the documentation for your \LaTeX\ distribution), or used temporarily by placing them in the working directory for your paper.

To use the MNRAS package, simply specify \verb'mnras' as the document class at the start of a \verb'.tex' file:

\begin{verbatim}
\documentclass{mnras}
\end{verbatim}
Then compile \LaTeX\ (and if necessary \bibtex) in the usual way.

\section{Preparing and submitting a paper}
We recommend that you start with a copy of the \texttt{mnras\_template.tex} file.
Rename the file, update the information on the title page, and then work on the text of your paper.
Guidelines for content, style etc. are given in the instructions to authors on the journal's website$^{\ref{foot:itas}}$.
Note that this document does not follow all the aspects of MNRAS journal style (e.g. it has a table of contents).

If a paper is accepted, it is professionally typeset and copyedited by the publishers.
It is therefore likely that minor changes to presentation will occur.
For this reason, we ask authors to ignore minor details such as slightly long lines, extra blank spaces, or misplaced figures, because these details will be dealt with during the production process.

Papers must be submitted electronically via the online submission system; paper submissions are not permitted.
For full guidance on how to submit a paper, see the instructions to authors.

\section{Class options}
\label{sec:options}
There are several options which can be added to the document class line like this:

\begin{verbatim}
\documentclass[option1,option2]{mnras}
\end{verbatim}
The available options are:
\begin{itemize}
\item \verb'letters' -- used for papers in the journal's Letters section.
\item \verb'onecolumn' -- single column, instead of the default two columns. This should be used {\it only} if necessary for the display of numerous very long equations.
\item \verb'doublespacing' -- text has double line spacing. Please don't submit papers in this format.
\item \verb'referee' -- \textit{(deprecated)} single column, double spaced, larger text, bigger margins. Please don't submit papers in this format.
\item \verb'galley' -- \textit{(deprecated)} no running headers, no attempt to align the bottom of columns.
\item \verb'landscape' -- \textit{(deprecated)} sets the whole document on landscape paper.
\item \verb"usenatbib" -- \textit{(all papers should use this)} this uses Patrick Daly's \verb"natbib.sty" package for citations.
\item \verb"usegraphicx" -- \textit{(most papers will need this)} includes the \verb'graphicx' package, for inclusion of figures and images.
\item \verb'useAMS' -- adds support for upright Greek characters \verb'\upi', \verb'\umu' and \verb'\upartial' ($\upi$, $\umu$ and $\upartial$). Only these three are included, if you require other symbols you will need to include the \verb'amsmath' or \verb'amsymb' packages (see section~\ref{sec:packages}).
\item \verb"usedcolumn" -- includes the package \verb"dcolumn", which includes two new types of column alignment for use in tables.
\end{itemize}

Some of these options are deprecated and retained for backwards compatibility only.
Others are used in almost all papers, but again are retained as options to ensure that papers written decades ago will continue to compile without problems.
If you want to include any other packages, see section~\ref{sec:packages}.

\section{Title page}

If you are using \texttt{mnras\_template.tex} the necessary code for generating the title page, headers and footers is already present.
Simply edit the title, author list, institutions, abstract and keywords as described below.

\subsection{Title}
There are two forms of the title: the full version used on the first page, and a short version which is used in the header of other odd-numbered pages (the `running head').
Enter them with \verb'\title[]{}' like this:
\begin{verbatim}
\title[Running head]{Full title of the paper}
\end{verbatim}
The full title can be multiple lines (use \verb'\\' to start a new line) and may be as long as necessary, although we encourage authors to use concise titles. The running head must be $\le~45$ characters on a single line.

See appendix~\ref{sec:advanced} for more complicated examples.

\subsection{Authors and institutions}

Like the title, there are two forms of author list: the full version which appears on the title page, and a short form which appears in the header of the even-numbered pages. Enter them using the \verb'\author[]{}' command.

If the author list is more than one line long, start a new line using \verb'\newauthor'. Use \verb'\\' to start the institution list. Affiliations for each author should be indicated with a superscript number, and correspond to the list of institutions below the author list.

For example, if I were to write a paper with two coauthors at another institution, one of whom also works at a third location:
\begin{verbatim}
\author[K. T. Smith et al.]{
Keith T. Smith,$^{1}$
A. N. Other,$^{2}$
and Third Author$^{2,3}$
\\
$^{1}$Affiliation 1\\
$^{2}$Affiliation 2\\
$^{3}$Affiliation 3}
\end{verbatim}
Affiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.

Email addresses can be inserted with the \verb'\thanks{}' command which adds a title page footnote.
If you want to list more than one email, put them all in the same \verb'\thanks' and use \verb'\footnotemark[]' to refer to the same footnote multiple times.
Present addresses (if different to those where the work was performed) can also be added with a \verb'\thanks' command.

\subsection{Abstract and keywords}

The abstract is entered in an \verb'abstract' environment:
\begin{verbatim}
\begin{abstract}
The abstract of the paper.
\end{abstract}
\end{verbatim}
\noindent Note that there is a word limit on the length of abstracts.
For the current word limit, see the journal instructions to authors$^{\ref{foot:itas}}$.

Immediately following the abstract, a set of keywords is entered in a \verb'keywords' environment:
\begin{verbatim}
\begin{keywords}
keyword 1 -- keyword 2 -- keyword 3
\end{keywords}
\end{verbatim}
\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.
Do \emph{not} make up new keywords!
For the current list of allowed keywords, see the journal's instructions to authors$^{\ref{foot:itas}}$.

\section{Sections and lists}

Sections and lists are generally the same as in the standard \LaTeX\ classes.

\subsection{Sections}
\label{sec:sections}
Sections are entered in the usual way, using \verb'\section{}' and its variants. It is possible to nest up to four section levels:
\begin{verbatim}
\section{Main section}
 \subsection{Subsection}
  \subsubsection{Subsubsection}
   \paragraph{Lowest level section}
\end{verbatim}
\noindent The other \LaTeX\ sectioning commands \verb'\part', \verb'\chapter' and \verb'\subparagraph{}' are deprecated and should not be used.

Some sections are not numbered as part of journal style (e.g. the Acknowledgements).
To insert an unnumbered section use the `starred' version of the command: \verb'\section*{}'.

See appendix~\ref{sec:advanced} for more complicated examples.

\subsection{Lists}

Two forms of lists can be used in MNRAS -- numbered and unnumbered.

For a numbered list, use the \verb'enumerate' environment:
\begin{verbatim}
\begin{enumerate}
 \item First item
 \item Second item
 \item etc.
\end{enumerate}
\end{verbatim}
\noindent which produces
\begin{enumerate}
 \item First item
 \item Second item
 \item etc.
\end{enumerate}
Note that the list uses lowercase Roman numerals, rather than the \LaTeX\ default Arabic numerals.

For an unnumbered list, use the \verb'description' environment without the optional argument:
\begin{verbatim}
\begin{description}
 \item First item
 \item Second item
 \item etc.
\end{description}
\end{verbatim}
\noindent which produces
\begin{description}
 \item First item
 \item Second item
 \item etc.
\end{description}

Bulleted lists using the \verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.

\section{Mathematics and symbols}

The MNRAS class mostly adopts standard \LaTeX\ handling of mathematics, which is briefly summarised here.
See also section~\ref{sec:packages} for packages that support more advanced mathematics.

Mathematics can be inserted into the running text using the syntax \verb'$1+1=2$', which produces $1+1=2$.
Use this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.

\subsection{Equations}
Equations should be entered using the \verb'equation' environment, which automatically numbers them:

\begin{verbatim}
\begin{equation}
 a^2=b^2+c^2
\end{equation}
\end{verbatim}
\noindent which produces
\begin{equation}
 a^2=b^2+c^2
\end{equation}

By default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \verb'\numberwithin{equation}{section}' to the preamble.

It is also possible to produce un-numbered equations by using the \LaTeX\ built-in \verb'\['\textellipsis\verb'\]' and \verb'$$'\textellipsis\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.

\subsection{Special symbols}

\begin{table}
 \caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}
 \label{tab:anysymbols}
 \begin{tabular}{lll}
  \hline
  Command & Output & Meaning\\
  \hline
  \verb'\sun' & \sun & Sun, solar\\[2pt]
  \verb'\earth' & \earth & Earth, terrestrial\\[2pt]
  \verb'\micron' & \micron & microns\\[2pt]
  \verb'\degr' & \degr & degrees\\[2pt]
  \verb'\arcmin' & \arcmin & arcminutes\\[2pt]
  \verb'\arcsec' & \arcsec & arcseconds\\[2pt]
  \verb'\fdg' & \fdg & fraction of a degree\\[2pt]
  \verb'\farcm' & \farcm & fraction of an arcminute\\[2pt]
  \verb'\farcs' & \farcs & fraction of an arcsecond\\[2pt]
  \verb'\fd' & \fd & fraction of a day\\[2pt]
  \verb'\fh' & \fh & fraction of an hour\\[2pt]
  \verb'\fm' & \fm & fraction of a minute\\[2pt]
  \verb'\fs' & \fs & fraction of a second\\[2pt]
  \verb'\fp' & \fp & fraction of a period\\[2pt]
  \verb'\diameter' & \diameter & diameter\\[2pt]
  \verb'\sq' & \sq & square, Q.E.D.\\[2pt]
  \hline
 \end{tabular}
\end{table}

\begin{table}
 \caption{Additional commands for mathematical symbols. These can only be used in maths mode.}
 \label{tab:mathssymbols}
 \begin{tabular}{lll}
  \hline
  Command & Output & Meaning\\
  \hline
  \verb'\upi' & $\upi$ & upright pi\\[2pt]
  \verb'\umu' & $\umu$ & upright mu\\[2pt]
  \verb'\upartial' & $\upartial$ & upright partial derivative\\[2pt]
  \verb'\lid' & $\lid$ & less than or equal to\\[2pt]
  \verb'\gid' & $\gid$ & greater than or equal to\\[2pt]
  \verb'\la' & $\la$ & less than of order\\[2pt]
  \verb'\ga' & $\ga$ & greater than of order\\[2pt]
  \verb'\loa' & $\loa$ & less than approximately\\[2pt]
  \verb'\goa' & $\goa$ & greater than approximately\\[2pt]
  \verb'\cor' & $\cor$ & corresponds to\\[2pt]
  \verb'\sol' & $\sol$ & similar to or less than\\[2pt]
  \verb'\sog' & $\sog$ & similar to or greater than\\[2pt]
  \verb'\lse' & $\lse$ & less than or homotopic to \\[2pt]
  \verb'\gse' & $\gse$ & greater than or homotopic to\\[2pt]
  \verb'\getsto' & $\getsto$ & from over to\\[2pt]
  \verb'\grole' & $\grole$ & greater over less\\[2pt]
  \verb'\leogr' & $\leogr$ & less over greater\\
  \hline
 \end{tabular}
\end{table}

Some additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.

Many other mathematical symbols are also available, either built into \LaTeX\ or via additional packages. If you want to insert a specific symbol but don't know the \LaTeX\ command, we recommend using the Detexify website\footnote{\url{http://detexify.kirelabs.org}}.

Sometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.

To produce bold symbols in mathematics, use \verb'\bmath' for simple variables, and the \verb'bm' package for more complex symbols (see section~\ref{sec:packages}). Vectors are set in bold italic, using \verb'\mathbfit{}'.

For matrices, use \verb'\mathbfss{}' to produce a bold sans-serif font e.g. \mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\nabla$ (del, used in gradients, divergence etc.) use \verb'$\nabla$'.

\subsection{Ions}

A new \verb'\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.
For example, to typeset singly ionised calcium use \verb'\ion{Ca}{ii}', which produces \ion{Ca}{ii}.

\section{Figures and tables}
\label{sec:fig_table}
Figures and tables (collectively called `floats') are mostly the same as built into \LaTeX.

\subsection{Basic examples}
\begin{figure}
 \includegraphics[width=\columnwidth]{example}
 \caption{An example figure.}
 \label{fig:example}
\end{figure}
Figures are inserted in the usual way using a \verb'figure' environment and \verb'\includegraphics'. The example Figure~\ref{fig:example} was generated using the code:
\begin{verbatim}
\begin{figure}
 \includegraphics[width=\columnwidth]{example}
 \caption{An example figure.}
 \label{fig:example}
\end{figure}
\end{verbatim}

\begin{table}
 \caption{An example table.}
 \label{tab:example}
 \begin{tabular}{lcc}
  \hline
  Star & Mass & Luminosity\\
   & $M_{\sun}$ & $L_{\sun}$\\
  \hline
  Sun & 1.00 & 1.00\\
  $\alpha$~Cen~A & 1.10 & 1.52\\
  $\epsilon$~Eri & 0.82 & 0.34\\
  \hline
 \end{tabular}
\end{table}
The example Table~\ref{tab:example} was generated using the code:
\begin{verbatim}
\begin{table}
 \caption{An example table.}
 \label{tab:example}
 \begin{tabular}{lcc}
  \hline
  Star & Mass & Luminosity\\
   & $M_{\sun}$ & $L_{\sun}$\\
  \hline
  Sun & 1.00 & 1.00\\
  $\alpha$~Cen~A & 1.10 & 1.52\\
  $\epsilon$~Eri & 0.82 & 0.34\\
  \hline
 \end{tabular}
\end{table}
\end{verbatim}

\subsection{Captions and placement}
Captions go \emph{above} tables but \emph{below} figures, as in the examples above.

The \LaTeX\ float placement commands \verb'[htbp]' are intentionally disabled.
Layout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.
Simply place the \LaTeX\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.

By default a figure or table will occupy one column of the page.
To produce a wider version which covers both columns, use the \verb'figure*' or \verb'table*'  environment.

If a figure or table is too long to fit on a single page it can be split it into several parts.
Create an additional figure or table which uses \verb'\contcaption{}' instead of \verb'\caption{}'.
This will automatically correct the numbering and add `\emph{continued}' at the start of the caption.
\begin{table}
 \contcaption{A table continued from the previous one.}
 \label{tab:continued}
 \begin{tabular}{lcc}
  \hline
  Star & Mass & Luminosity\\
   & $M_{\sun}$ & $L_{\sun}$\\
  \hline
  $\tau$~Cet & 0.78 & 0.52\\
  $\delta$~Pav & 0.99 & 1.22\\
  $\sigma$~Dra & 0.87 & 0.43\\
  \hline
 \end{tabular}
\end{table}
Table~\ref{tab:continued} was generated using the code:

\begin{verbatim}
\begin{table}
 \contcaption{A table continued from the previous one.}
 \label{tab:continued}
 \begin{tabular}{lcc}
  \hline
  Star & Mass & Luminosity\\
   & $M_{\sun}$ & $L_{\sun}$\\
  \hline
  $\tau$~Cet & 0.78 & 0.52\\
  $\delta$~Pav & 0.99 & 1.22\\
  $\sigma$~Dra & 0.87 & 0.43\\
  \hline
 \end{tabular}
\end{table}
\end{verbatim}

To produce a landscape figure or table, use the \verb'pdflscape' package and the \verb'landscape' environment.
The landscape Table~\ref{tab:landscape} was produced using the code:
\begin{verbatim}
\begin{landscape}
 \begin{table}
  \caption{An example landscape table.}
  \label{tab:landscape}
  \begin{tabular}{cccccccccc}
    \hline
    Header & Header & ...\\
    Unit & Unit & ...\\
    \hline
    Data & Data & ...\\
    Data & Data & ...\\
    ...\\
    \hline
  \end{tabular}
 \end{table}
\end{landscape}
\end{verbatim}
Unfortunately this method will force a page break before the table appears.
More complicated solutions are possible, but authors shouldn't worry about this.

\begin{landscape}
 \begin{table}
  \caption{An example landscape table.}
  \label{tab:landscape}
  \begin{tabular}{cccccccccc}
    \hline
    Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\
    Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\
    \hline
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    \hline
  \end{tabular}
 \end{table}
\end{landscape}

\section{References and citations}

\subsection{Cross-referencing}

The usual \LaTeX\ commands \verb'\label{}' and \verb'\ref{}' can be used for cross-referencing within the same paper.
We recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.
This ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).

It is best to give each section, figure and table a logical label.
For example, Table~\ref{tab:mathssymbols} has the label \verb'tab:mathssymbols', whilst section~\ref{sec:packages} has the label \verb'sec:packages'.
Add the label \emph{after} the section or caption command, as in the examples in sections~\ref{sec:sections} and \ref{sec:fig_table}.
Enter the cross-reference with a non-breaking space between the type of object and the number, like this: \verb'see Figure~\ref{fig:example}'.

The \verb'\autoref{}' command can be used to automatically fill out the type of object, saving on typing.
It also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.
For example, \verb'\autoref{tab:journal_abbr}' produces \autoref{tab:journal_abbr}.

\subsection{Citations}
\label{sec:cite}

MNRAS uses the Harvard -- author (year) -- citation style, e.g. \citet{author2013}.
This is implemented in \LaTeX\ via the \verb'natbib' package, which in turn is included via the \verb'usenatbib' package option (see section~\ref{sec:options}), which should be used in all papers.

Each entry in the reference list has a `key' (see section~\ref{sec:ref_list}) which is used to generate citations.
There are two basic \verb'natbib' commands:
\begin{description}
 \item \verb'\citet{key}' produces an in-text citation: \citet{author2013}
 \item \verb'\citep{key}' produces a bracketed (parenthetical) citation: \citep{author2013}
\end{description}
Citations will include clickable links to the relevant entry in the reference list, if supported by your \LaTeX\ compiler.

\defcitealias{smith2014}{Paper~I}
\begin{table*}
 \caption{Common citation commands, provided by the \texttt{natbib} package.}
 \label{tab:natbib}
 \begin{tabular}{lll}
  \hline
  Command & Ouput & Note\\
  \hline
  \verb'\citet{key}' & \citet{smith2014} & \\
  \verb'\citep{key}' & \citep{smith2014} & \\
  \verb'\citep{key,key2}' & \citep{smith2014,jones2015} & Multiple papers\\
  \verb'\citet[table 4]{key}' & \citet[table 4]{smith2014} & \\
  \verb'\citep[see][figure 7]{key}' & \citep[see][figure 7]{smith2014} & \\
  \verb'\citealt{key}' & \citealt{smith2014} & For use with manual brackets\\
  \verb'\citeauthor{key}' & \citeauthor{smith2014} & If already cited in close proximity\\
  \verb'\defcitealias{key}{Paper~I}' &  & Define an alias (doesn't work in floats)\\
  \verb'\citetalias{key}' & \citetalias{smith2014} & \\
  \verb'\citepalias{key}' & \citepalias{smith2014} & \\
  \hline
 \end{tabular}
\end{table*}

There are a number of other \verb'natbib' commands which can be used for more complicated citations.
The most commonly used ones are listed in Table~\ref{tab:natbib}.
For full guidance on their use, consult the \verb'natbib' documentation\footnote{\url{http://www.ctan.org/pkg/natbib}}.

If a reference has several authors, \verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \bibtex\ (see section~\ref{sec:ref_list}) then this is handled automatically. If not, the \verb'\citet*{}' and \verb'\citep*{}' commands can be used at the first citation to include all of the authors.

\subsection{The list of references}
\label{sec:ref_list}

It is possible to enter references manually using the usual \LaTeX\ commands, but we strongly encourage authors to use \bibtex\ instead.
\bibtex\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.
An MNRAS \bibtex\ style file, \verb'mnras.bst', is distributed as part of this package.
The rest of this section will assume you are using \bibtex.

References are entered into a separate \verb'.bib' file in standard \bibtex\ formatting.
This can be done manually, or there are several software packages which make editing the \verb'.bib' file much easier.
We particularly recommend \textsc{JabRef}\footnote{\url{http://jabref.sourceforge.net/}}, which works on all major operating systems.
\bibtex\ entries can be obtained from the NASA Astrophysics Data System\footnote{\label{foot:ads}\url{http://adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.
Simply copy this into your \verb'.bib' file or into the `BibTeX source' tab in \textsc{JabRef}.

Each entry in the \verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.
Simply cite it in the usual way, as described in section~\ref{sec:cite}, using the specified key.
Compile the paper as usual, but add an extra step to run the \texttt{bibtex} command.
Consult the documentation for your compiler or latex distribution.

Correct formatting of the reference list will be handled by \bibtex\ in almost all cases, provided that the correct information was entered into the \verb'.bib' file.
Note that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.
If in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\ref{foot:itas}}$ for the current guidelines on how to format the list of references.

\section{Appendices and online material}

To start an appendix, simply place the \verb'

redpajama_set_name: RedPajamaArXiv