# Accretion Flow Morphology in Numerical Simulations of Black Holes from the ngEHT Model Library: The Impact of Radiation Physics

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## Abstract

**:**

`H-AMR`code, (3) a two-temperature GRMHD simulation from the

`BHAC`code, and (4) a two-temperature radiative GRMHD simulation from the

`KORAL`code. We find that the different models exhibit remarkably similar temporal and spatial properties, except for the electron temperature, since radiative losses substantially cool down electrons near the BH and the jet sheath, signaling the importance of radiative cooling even for slowly accreting BHs such as M87${}^{*}$. We restrict ourselves to standard torus accretion flows, and leave larger explorations of alternate accretion models to future work.

## 1. Introduction

## 2. Numerical Simulations

#### 2.1. RIAF + Hotspot Solutions

#### 2.2. GRMHD Simulations

`H-AMR`code [37], (2) a two-temperature single fluid GRMHD simulation from the

`BHAC`code [38], and (3) a two-temperature radiative GRMHD simulation from the

`KORAL`code [39].

`H-AMR`code, one targeting M87${}^{*}$ and the other Sgr A${}^{*}$. These simulations employ logarithmic Kerr–Schild coordinates and the grid resolutions are ${N}_{r}\times {N}_{\theta}\times {N}_{\phi}=580\times 288\times 512$ for the M87${}^{*}$ simulation and $348\times 192\times 192$ for the Sgr A${}^{*}$ simulation. All the simulations in this work adopt the geometrical unit convention, $G=c=1$ and using ${M}_{\mathrm{BH}}=1$, normalizing the length scale to the gravitational radius ${r}_{\mathrm{g}}=G{M}_{\mathrm{BH}}/{c}^{2}$. The M87${}^{*}$ GRMHD simulation evolves a MAD flow around a black hole with spin $a=0.9375$. The Sgr A${}^{*}$ model also simulates a MAD flow but around a black hole with spin $a=1/2$.

`H-AMR`uses outflowing radial boundary conditions (BCs), transmissive polar BCs, and periodic azimuthal BCs (for more details, see [40]).

`BHOSS`[41] and

`IPOLE`[42] to compute images at 230 GHz and set the compact flux to be approximately 0.5 Jy for M87${}^{*}$ [1] and 2.4 Jy for Sgr A${}^{*}$ [2]. We use the black hole masses and distances of ${M}_{\mathrm{BH}}=6.2\times {10}^{9}{M}_{\odot}$ and ${D}_{\mathrm{BH}}=16.9$ Mpc for M87${}^{*}$ (e.g., [18,43] and references therein) and ${M}_{\mathrm{BH}}=4.14\times {10}^{6}{M}_{\odot}$ and ${D}_{\mathrm{BH}}=8.127$ kpc for Sgr A${}^{*}$ [20,44,45].

#### 2.3. Two-Temperature Physics

`BHOSS`) once the near-horizon flow has reached steady state. The target source is M87${}^{*}$, assuming a black hole mass of ${M}_{\mathrm{BH}}=6.5\times {10}^{9}{M}_{\odot}$ and distance of 16.9 Mpc [1]. The accretion rate is normalized such that the 230 GHz compact flux density is 0.8 Jy. We assume a thermal electron distribution everywhere except in the jet sheath where we adopt a $\kappa -$distribution. More details about the image are provided in Roelofs et al. [4].

#### 2.4. Radiative GRMHD

`KORAL`[39,56,57], which evolves a two-temperature magnetized fluid and treats the radiation field as a second fluid [58]. The conservation equations solved in 2t-GRRMHD are different from that of GRMHD:

## 3. Results

#### 3.1. Temporal Behavior of Horizon Fluxes

#### 3.2. Disk-Averaged Quantities

#### 3.3. Jet Properties

#### 3.4. Axisymmetrized Profiles

#### 3.5. Orbiting Hotspot in a RIAF Model

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | https://www.ngeht.org/ (accessed on 22 November 2022). |

2 | https://challenge.ngeht.org/ (accessed on 22 November 2022). |

3 | Specific enthalpy includes the rest-mass energy contribution in our definition from Section 2.3. |

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**Figure 1.**We show the mass accretion rate $\dot{M}$, dimensionless magnetic flux $\varphi \equiv \Phi /\sqrt{\dot{M}}$, the outflow efficiency ${P}_{\mathrm{out}}/\dot{M}{c}^{2}=1-\dot{E}/\dot{M}{c}^{2}$, and specific radial flux of the angular momentum $\dot{J}/\dot{M}$ over time. Values are calculated at the event horizon.

**Figure 2.**We show equatorial cross-sections of gas density $\rho $ (in arbitrary code units) and electron temperature ${T}_{e}$ (in Kelvin) during a magnetic flux eruption in the 2t-GRMHD simulation of M87${}^{*}$. Magnetic flux eruptions, a characteristic feature of magnetically arrested disks, eject bundles of vertical magnetic fields filled with relativistically hot, low-density plasma. These flux bundles spiral around the black hole and may be responsible for high-energy flares [63]. The time unit M is equivalent to ${r}_{g}/c$.

**Figure 3.**We show the radial profiles of gas density $\rho $, plasma$-\beta $, proton temperature ${T}_{\mathrm{p}}$, and electron temperature ${T}_{\mathrm{e}}$. Quantities are disk-averaged and time-averaged over the raytracing period.

**Figure 4.**We show the radial profiles of disk scale height $h/r$, radial velocity $|{v}_{r}|$, angular velocity $\Omega $, and specific angular momentum ${u}_{\phi}$. Quantities are disk-averaged and time-averaged over the raytracing period.

**Figure 5.**We show the jet radius ${R}_{\mathrm{jet}}$ and the jet Lorentz factor $\gamma $ from the M87${}^{*}$ GRMHD and 2t-GRRMHD models, and the Sgr A${}^{*}$ GRMHD model. The gray circles indicates the deprojected jet radius of the M87 jet assuming a BH mass of $6.2\times {10}^{9}{M}_{\odot}$ and a source inclination of ${14}^{\circ}$ [70]. The data points are a compilation of various papers [59,60,71,77,78,79].

**Figure 6.**We show t- and $\phi $-averaged data: electron number density ${n}_{\mathrm{e}}$ (top row) and temperature ${T}_{\mathrm{e}}$ (bottom row). We also denote the jet boundary with $\sigma =1$ (black lines). The time-averaging is performed over the $5000\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{g}}/c$ for each model. RIAF plots are for Sgr A${}^{*}$ while the rest are for M87. The Sgr A${}^{*}$ GRMHD model produces similar plots of ${n}_{\mathrm{e}}$ and ${T}_{\mathrm{e}}$ as the M87${}^{*}$ model, and, hence, we do not show it here.

**Figure 7.**We show the $\phi -$averaged hotspot electron number density as a function of radius and time. The hotspot falls into the BH and becomes sheared over time.

**Table 1.**GRMHD simulations considered in this work. Simulation grid resolution, simulation time period over which raytracing was performed, modulation index (MI) of the mass accretion rate $\dot{M}$, dimensionless magnetic flux $\varphi $, outflow efficiency ${P}_{\mathrm{out}}/\dot{M}{c}^{2}$, and the specific angular momentum flux $\dot{J}/\dot{M}$ for each GRMHD model. The MI is calculated over the final $5000\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{g}}/c$ in runtime and at the event horizon (see Figure 1).

Model | Grid Resolution | Sim. Time | MI($\dot{\mathit{M}}$) | MI($\mathit{\varphi}$) | MI | MI |
---|---|---|---|---|---|---|

Name | (${\mathit{N}}_{\mathit{r}}\times {\mathit{N}}_{\mathit{\theta}}\times {\mathit{N}}_{\mathit{\phi}}$) | ($\times {10}^{3}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathit{r}}_{\mathbf{g}}/\mathit{c}$) | (${\mathit{P}}_{\mathbf{out}}/\dot{\mathit{M}}{\mathit{c}}^{2}$) | ($\dot{\mathit{J}}/\dot{\mathit{M}}$) | ||

M87${}^{*}$ GRMHD | $580\times 288\times 512$ | 5.6–10.6 | 0.27 | 0.15 | 0.26 | 0.33 |

M87${}^{*}$ 2t-GRMHD | $348\times 192\times 192$ | 10–15 | 0.29 | 0.14 | 0.25 | 0.31 |

M87${}^{*}$ 2t-GRRMHD | $288\times 224\times 128$ | 11–16 | 0.28 | 0.14 | 0.14 | 0.31 |

Sgr A${}^{*}$ GRMHD | $348\times 192\times 192$ | 30–35 | 0.23 | 0.21 | 0.39 | 0.57 |

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**MDPI and ACS Style**

Chatterjee, K.; Chael, A.; Tiede, P.; Mizuno, Y.; Emami, R.; Fromm, C.; Ricarte, A.; Blackburn, L.; Roelofs, F.; Johnson, M.D.;
et al. Accretion Flow Morphology in Numerical Simulations of Black Holes from the ngEHT Model Library: The Impact of Radiation Physics. *Galaxies* **2023**, *11*, 38.
https://doi.org/10.3390/galaxies11020038

**AMA Style**

Chatterjee K, Chael A, Tiede P, Mizuno Y, Emami R, Fromm C, Ricarte A, Blackburn L, Roelofs F, Johnson MD,
et al. Accretion Flow Morphology in Numerical Simulations of Black Holes from the ngEHT Model Library: The Impact of Radiation Physics. *Galaxies*. 2023; 11(2):38.
https://doi.org/10.3390/galaxies11020038

**Chicago/Turabian Style**

Chatterjee, Koushik, Andrew Chael, Paul Tiede, Yosuke Mizuno, Razieh Emami, Christian Fromm, Angelo Ricarte, Lindy Blackburn, Freek Roelofs, Michael D. Johnson,
and et al. 2023. "Accretion Flow Morphology in Numerical Simulations of Black Holes from the ngEHT Model Library: The Impact of Radiation Physics" *Galaxies* 11, no. 2: 38.
https://doi.org/10.3390/galaxies11020038