# Spins of Supermassive Black Holes M87* and SgrA* Revealed from the Size of Dark Spots in Event Horizon Telescope Images

## Abstract

**:**

## 1. Introduction

## 2. Event Horizon Silhouette ≡ Lensed Image of the Black Hole Event Horizon

## 3. Forms and Sizes of Dark Spots in the Black Hole Images

## 4. Spin Values of SMBHs SgrA* and M87*

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EHT | Event Horizon Telescope |

SMBH | Supermassive Black Hole |

## Appendix A. Geodesics in Kerr Metric

**Figure A1.**Classical black hole shadows, projected at the celestial sphere, for different spin parameters a in the Kerr metric and possible inclinations of SMBH rotation axes with respect to the polar angle ${\theta}_{0}$ of a distant observer: SgrA* in the left panel (${\theta}_{0}={90}^{\circ}$, $a=1$ (red), $a=0.65$ (purple), $a=0$ (magenta)) and M87* in the right panel (${\theta}_{0}={163}^{\circ}$, $a=1$ (red), $a=0.75$ (purple), $a=0$ (magenta)).

## References

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**Figure 1.**Reconstruction of the Shwarzschild black hole event horizon silhouette using $3D$ trajectories of photons (multicolored curves), which start very near the black hole event horizon and are registered by a distant observer (by a distant telescope). The event horizon silhouette always projects at the celestial sphere within the classical black hole shadow with a radius $3\sqrt{3}$. Meanwhile, the corresponding radius of event horizon silhouette is ${r}_{\mathrm{h}}\simeq 4.457$ [50,51].

**Figure 2.**The face profiles of the event horizon silhouettes in the Shwarzschild case (right panel, $a=0$) and extreme Kerr black hole (left panel, $a=1$). It is supposed that a distant observer is placed slightly above the black hole equatorial plane.

**Figure 3.**A $3D$ picture of the SMBH SgrA* with the spin parameter $a=1$. It is supposed that a distant observer is placed slightly above the black hole equatorial plane at the polar angle ${\theta}_{0}={84}^{\circ}$. The inner part of the dark spot is a lensed image of the north hemisphere of the event horizon globe. The outline of the dark spot is a lensed image of the event horizon equator. The multicolored curve is a numerically calculated trajectory of photon, starting from the radius $r=1.01{r}_{h}$ at the black hole equatorial plane and coming to a distant observer.

**Figure 4.**Examples of dark spots for the case of SMBH SgrA* with different values of spin parameter a, obtained by numerical calculation of possible photon trajectories in the model of geometrically thin accretion disk: left panel, $a=1$, middle panel, $a=0.65$, right panel, $a=0$. It is supposed that a distant observer is placed slightly above the black hole equatorial plane at the polar angle ${\theta}_{0}={84}^{\circ}$.

**Figure 5.**Superposition of the modeled dark spot with the EHT image of SgrA*: left panel, $a=1$, right panel, $a=0.65$. The dark spots, which are the lensed images of event horizons, are always projected at the celestial sphere inside the awaited positions of the classical black hole shadows, which are invisible in both cases of SgrA* and M87*.

**Figure 6.**Examples of dark spots for the case of SMBH M87* with a distant observer at the polar angle ${\theta}_{0}={84}^{\circ}$: left panel, $a=1$; middle panel, $a=0.75$; right panel, $a=0$.

**Figure 7.**Superposition of the modeled dark spot with the EHT image of M87*: Left panel, $a=1$; right panel, $a=0.75$. A ${17}^{\circ}$ inclination angle of the black hole rotation axis at the celestial sphere is supposed.

**Figure 8.**A $3D$ picture of the SMBH M87* with spin parameter $a=1$ surrounded by thin accretion disk. A ${17}^{\circ}$ inclination angle of M87* rotation axis with respect to a distant observer is supposed.

**Figure 9.**Dependence of the dark spot size, ${r}_{\mathrm{h}}\left(a\right)$, on the black hole spin parameter a, respectively, for SgrA* (

**left**panel) and M87* (

**right**panel). The light green regions mark the possible range of the black hole spin parameters: $0.65\le a\le 0.9$ for SgrA* and $0.75\le a\le 1$ for M87*.

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**MDPI and ACS Style**

Dokuchaev, V.I.
Spins of Supermassive Black Holes M87* and SgrA* Revealed from the Size of Dark Spots in Event Horizon Telescope Images. *Astronomy* **2023**, *2*, 141-152.
https://doi.org/10.3390/astronomy2030010

**AMA Style**

Dokuchaev VI.
Spins of Supermassive Black Holes M87* and SgrA* Revealed from the Size of Dark Spots in Event Horizon Telescope Images. *Astronomy*. 2023; 2(3):141-152.
https://doi.org/10.3390/astronomy2030010

**Chicago/Turabian Style**

Dokuchaev, Vyacheslav Ivanovich.
2023. "Spins of Supermassive Black Holes M87* and SgrA* Revealed from the Size of Dark Spots in Event Horizon Telescope Images" *Astronomy* 2, no. 3: 141-152.
https://doi.org/10.3390/astronomy2030010