# Emission Modeling in the EHT–ngEHT Age

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. “Observing” JAB Systems

- Start with a general relativistic magnetohydrodynamic (GRMHD) simulation or semi-analytic model of a jet (or outflow)/accretion flow/black hole (JAB) system
- Convert GRMHD variables into radiation prescriptions for emission, absorption, polarization, particle acceleration, and/or dissipation to emulate sources, using piecewise models when appropriate to assign parametrizations to each distinct JAB system region
- Add a realistic, synthetic “observer” in postprocessing—which includes all radiating species that significantly contribute to radiative transfer—in order to view sources—specifically, images, spectra, light curves, and Stokes maps.

#### 2.2. GRHMD

`harm`code [14,15], a conservative second-order explicit shock-capturing finite-volume method for solving the equations of ideal GRMHD in arbitrary stationary spacetimes. On a coordinate basis, the governing equations are

`iharm3d`code [18]. The M87 simulations were produced by using

`iharm3d`’s kokkos/GPU-based descendent, kharma.3 Other physically motivated flow geometries are possible. The Bondi spherical accretion flow solution [19] is analytically tractable, but requires an exquisite degree of symmetry and non-magnetized flows to be realistic. Magnetized Bondi flows in numerical calculations display similar properties to those of torus simulations in terms of horizon-scale emission [20,21]. Stellar winds from a small population of $\sim 30$ Wolf–Rayet stars were theorized, e.g., in [22,23,24,25] to source the disk and inflow of Sgr A* in a model that reasonably accounted for the diffuse X-ray emissions. However, the precise knowledge of the position and orbits of this population is currently only possible in our own Galactic Center, while the FM torus model can be generalized to other AGNs. This generalization includes the possibility that the FM torus can be tilted with respect to the black hole spin (a situation that could arise often in a low-luminosity AGN), but we neglect this possibility in our work, which is focused on emission physics.

#### 2.3. Plasma-Heating-Based Emission Models

#### 2.3.1. $R-\beta $ Model

#### 2.3.2. Critical $\beta $ Model

#### 2.4. Sub-Equipartition-Based Models

#### 2.4.1. Constant ${\beta}_{e}$

#### 2.4.2. Magnetic Bias

#### 2.5. Hybrid Models

#### 2.6. Phenomenological Models

#### 2.7. Electron Distribution Functions

#### 2.7.1. Power Law

#### 2.7.2. The Kappa Model

#### 2.8. Emission Modeling in Non-Kerr Spacetimes

## 3. Commencing the Computing: Emission Models in Numerical Codes

#### Positrons’ Effects on Radiative Transfer

## 4. Results: Adding an Observer

#### 4.1. Sgr A*

#### 4.1.1. Parametric Model Comparison

#### 4.1.2. Morphological Classification

- 1
- A thin, compact asymmetric photon ring/crescent with the best fit or flat spectrum (with the spectral energy distribution shown in [9]);
- 2
- Inflow–outflow boundary + thin photon ring with a steep spectrum;
- 3
- Thick photon ring with spectral excesses at high and low frequencies;
- 4
- Extended outflow and a flat low frequency spectrum with excesses at high and low frequencies.

#### 4.2. M87

#### 4.2.1. Parametric Model Comparison

#### 4.2.2. Positron Effects

## 5. Discussion and Conclusions

- The plasma $\beta $ controls the emitting region size in turbulent heating models, where parameter combinations with greater emission contributions from low $\beta $ tend to have more extended outflow/coronal regions, and those with contributions from high $\beta $ are more compact and dominated by near-horizon inflow, as shown in Figure 2, going from top to bottom.
- Inclination has a pronounced effect on the 230 GHz observer plane image morphology due to special relativistic beaming and the focusing properties of gravitational lensing. Thus, we predict a wide variety of image morphologies beyond ring structures that may be uncovered by the ngEHT, as shown in Figure 6.
- SANE and MAD simulations have widely divergent positron effects that are modulated by the larger Faraday depth of SANE plasmas, which are constrained to achieve the same image fluxes that MADs acquire through magnetic fields, with SANEs having EVPAs that are highly sensitive to positron content and MADs having a circular polarization degree that is greatly suppressed by positrons, as shown in Figure 9.

#### 5.1. Limits of Instrumentation

#### 5.2. Universality of Select Measures

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

(ng)EHT | (Next-Generation) Event Horizon Telescope |

GRMHD | General relativistic magnetohydrodynamics |

GRRT | General relativistic radiative transfer |

SED | Spectral energy distribution |

## Appendix A

#### Appendix A.1. List of Emission Models

**Table A1.**JAB Emission Model List. This is a compendium of the phonologically motivated emission models mentioned in this work. The shear stress $S={\gamma}^{2}|d{v}_{z}/ds|$ and dimensional parameters ${L}_{j}$ and ${L}_{S}$ are set by the width of jet system.

Model Name | Parameters | Functional Form |
---|---|---|

R-Beta | ${R}_{\mathrm{low}},{R}_{\mathrm{high}}$ | $R=\frac{{T}_{i}}{{T}_{e}}=\frac{{\beta}^{2}}{1+{\beta}^{2}}{R}_{\mathrm{high}}+\frac{1}{1+{\beta}^{2}}{R}_{\mathrm{low}}$ |

Critical Beta | $f,{\beta}_{c}$ | $\frac{{T}_{e}}{{T}_{e}+{T}_{i}}=f{e}^{-\beta /{\beta}_{c}}$ |

Const. ${\beta}_{e}$ Jet | ${\beta}_{e0}$ | ${P}_{e}={\beta}_{e0}{P}_{B}$ |

Magnetic Bias Jet | ${\beta}_{e0},n$ | ${P}_{e}={K}_{n}\left({\beta}_{e0}\right){P}_{B}^{n}$ |

R Beta w. Const. ${\beta}_{e}$ Jet | ${R}_{\mathrm{low}},{R}_{\mathrm{high}},{\beta}_{e0}$ | Const. ${\beta}_{e}$ in Jet, $R-\beta $ o.w. |

Critical Beta w. Const. ${\beta}_{e}$ Jet | $f,{\beta}_{c},{\beta}_{e0}$ | Const. ${\beta}_{e}$ in Jet, Crit. $\beta $ o.w |

Current Density | ${L}_{j}$ | ${P}_{e}={\mu}_{0}c{L}_{j}{j}^{\mu}{j}_{\mu}{t}_{\mathrm{cool}}$ |

Jet Alpha | ${\alpha}_{j}$ | ${P}_{e}=\frac{1}{2}\tau S{t}_{\mathrm{cool}}$, $\tau ={\alpha}_{j}\left(\right)open="("\; close=")">\frac{{B}^{\mu}{B}_{\mu}}{2{\mu}_{0}}+\frac{{u}_{g}}{3}$ |

Shear | ${L}_{S}$ | ${P}_{e}=\frac{1}{2}\tau S{t}_{\mathrm{cool}}$, $\tau =\mu S$, $\mu =\frac{{L}_{s}}{3c}\sqrt{\left(\right)open="("\; close=")">\rho {c}^{2}+\frac{{B}^{\mu}{B}_{\mu}}{2{\mu}_{0}}+\left(\right)open="("\; close=")">\frac{{u}_{g}}{3}+\frac{{B}^{\mu}{B}_{\mu}}{2{\mu}_{0}}}$ |

#### Appendix A.2. Resolution Dependence

**Figure A1.**Resolution dependence of the Critical Beta Model. Increasing from 2 pixels/M to 10 pixels/M alters the image morphology such that the overall image is sharper and the brightest features are more localized and extreme in intensity.

## Notes

1 | There have been some notable recent attempts to bridge the gap through hybrid electron distribution functions (edfs), such as the $\kappa $-model smoothly joining thermal electrons to a high-energy power-law tail [10]. |

2 | Here, $\rho $ is the rest mass density, ${u}^{\mu}$ is the four-velocity, and ${T}_{\nu}^{\mu}$ is the stress–energy–momentum tensor. |

3 | This was written using Parthenon; see https://github.com/lanl/parthenon. |

4 | In this treatment, ions can be treated interchangeably with protons. |

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**Figure 1.**Electron-to-proton temperature ratio variation between ${T}_{e}/{T}_{i}=10$ and ∼0 for the R-Beta and Critical Beta Models as a function of $\beta $. The electron-to-ion temperature ratio varies as ${T}_{e}/{T}_{i}=\frac{1}{{(1+{\beta}^{2})}^{-1}{R}_{\mathrm{low}}+{(1+{\beta}^{2})}^{-1}{\beta}^{2}{R}_{\mathrm{high}}}$ for the R-$\beta $ Model and $\frac{f{e}^{-\beta /{\beta}_{c}}}{1-f{e}^{-\beta /{\beta}_{c}}}$ for the Critical Beta Model. For the R-Beta parameters $({R}_{\mathrm{low}},{R}_{\mathrm{high}})=(0.1,100)$ and Critical Beta parameters $(f,{\beta}_{c})=(0.918,0.1)$ (

**Left**), the models nearly coincide. For the R-beta parameters $({R}_{\mathrm{low}},{R}_{\mathrm{high}})=(0.1,100)$ and Critical Beta parameters $(f,{\beta}_{c})=(0.91,1)$ (

**Right**), the Critical Beta model has a softer transition.

**Figure 2.**A parameter scan of the Critical Beta Model with (

**Left**) $f=0.1$, (

**Right**) $f=0.5$, ${\beta}_{c}=0.01$ (

**Top**), ${\beta}_{c}=0.1$ (

**Middle**), and ${\beta}_{c}=1$ (

**Bottom**) from an edge-on view. For Sgr A* models, the cgs conversion into Jy is found by multiplying each cgs-intensity-colored pixel value by 57.9 to get its flux density in Jy [9]. The scale is $M\equiv G{M}_{\mathrm{BH}}/{c}^{2}$ when used as a length and $\equiv G{M}_{\mathrm{BH}}/{c}^{3}$ when used as time. The notation 1e2 is a compact form of the scientific notation $1\times {10}^{2}$.

**Figure 3.**A parameter scan of the Constant Electron Beta Model (

**Left**) with $f=0.1$, (

**Right**) $f=0.5$, ${\beta}_{e0}=0.01$ (

**Top**), ${\beta}_{e0}=0.01=0.1$ (

**Middle**), and ${\beta}_{e0}=1$ (

**Bottom**). Magnetic Bias Model with ${\beta}_{e0}=1$ and $N=0$ (

**Bottom Right**).

**Figure 4.**A parameter scan of the face-on Critical Beta Model with (

**Left**) $f=0.1$, (

**Right**) $f=0.5$, ${\beta}_{c}=0.01$ (

**Top**), ${\beta}_{c}=0.1$ (

**Middle**), and ${\beta}_{c}=1$ (

**Bottom**).

**Figure 5.**Semi-MAD simulation [9] models that were ray traced at 230 GHz at T = 10,000 M: (

**Top Left**) best-fit Critical Beta model (

**Top Right**) R Beta with $(f,{\beta}_{c})=(0.5,1)$; (

**Bottom Left**) Constant Electron Beta model with ${\beta}_{e0}=1$; (

**Bottom Right**) Magnetic Bias with $N=0$ jet.

**Figure 7.**Synthetic intensity with the electric vector polarization angle (EVPA) and circular polarization maps for models that include positron effects and piecewise modeling. For the $a=-0.5$ SANE at 230 GHz and at T = 25,000 M: (

**Top Left**) R-Beta with a ${\beta}_{e0}=0.01$ jet model; (

**Top Right**) R-Beta with a ${\beta}_{e0}=0.01$ jet; (

**Bottom Left**) Critical Beta Model; (

**Bottom Right**) Critical Beta with a ${\beta}_{e0}=0.01$ jet. For each case, the intensity is overplotted with the electric vector polarization angle on the left panel, and the circular polarization degree is mapped on the right panel.

**Figure 8.**For the $a=-0.5$ MAD at 230 GHz and at T = 25,000 M: (

**Top Left**) R-Beta Model; (

**Top Right**) R-Beta with a ${\beta}_{e0}=0.01$ jet; (

**Bottom Left**) Critical Beta Model; (

**Bottom Right**) Critical Beta with a ${\beta}_{e0}=0.01$ jet. For each case, the intensity is overplotted with the electric vector polarization angle on the left panel, and the circular polarization degree is mapped on the right panel.

**Figure 9.**For $a=-0.5$ at 230 GHz and at T = 25,000 M: (

**Top**) Critical Beta Model without positrons; (

**Bottom**) Critical Beta with ${n}_{\mathrm{pairs}}/{n}_{0}=1$. We also compare MAD (

**Left**) and SANE (

**Right**).

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## Share and Cite

**MDPI and ACS Style**

Anantua, R.; Dúran, J.; Ngata, N.; Oramas, L.; Röder, J.; Emami, R.; Ricarte, A.; Curd, B.; Broderick, A.E.; Wayland, J.;
et al. Emission Modeling in the EHT–ngEHT Age. *Galaxies* **2023**, *11*, 4.
https://doi.org/10.3390/galaxies11010004

**AMA Style**

Anantua R, Dúran J, Ngata N, Oramas L, Röder J, Emami R, Ricarte A, Curd B, Broderick AE, Wayland J,
et al. Emission Modeling in the EHT–ngEHT Age. *Galaxies*. 2023; 11(1):4.
https://doi.org/10.3390/galaxies11010004

**Chicago/Turabian Style**

Anantua, Richard, Joaquín Dúran, Nathan Ngata, Lani Oramas, Jan Röder, Razieh Emami, Angelo Ricarte, Brandon Curd, Avery E. Broderick, Jeremy Wayland,
and et al. 2023. "Emission Modeling in the EHT–ngEHT Age" *Galaxies* 11, no. 1: 4.
https://doi.org/10.3390/galaxies11010004