https://orcid.org/0000-0002-2143-5973
ABSTRACT
The Event Horizon Telescope (EHT) observations of Sgr A* resolved the shadow image and emission ring-like structure, which is associated to the photon ring of the supermassive black hole, at the Galactic Centre, revealing a diameter of |$51.8~\mu \text{as}$|. The ring-like structure is consistent with that of a Kerr black hole. However, the source of the high bright regions in the image and the time variability remain an open question. Besides the plasma properties and emission models, the space-time geometry also holds an important role. We present an image depicting the bright hotspots consistent with Sgr A* observations at a wavelength |$\lambda = 1.3\, {\rm mm}$|. The image is the result of an eclipsing Schwarzschild black hole situated along the line of sight between the Galactic Centre and Earth. The separation from both, primary (Sgr A*) and secondary (eclipsing) black holes is 10 233 AU. The central supermassive black hole located at the centre of the Galaxy has an observational mass of |$4.14\times 10^6\, \, \rm M_{\odot }$| and the secondary eclipsing black hole has an inferred mass of |$1035\, \, \rm M_{\odot }$|.
1 INTRODUCTION
The Galactic Centre, commonly referred as Sagittarius A* (Sgr A*), has been extensively observed using various astronomical instruments across different parts of the electromagnetic spectrum (Hildebrand et al. 1978; Adams et al. 2021; Gallego-Calvente et al. 2021). Notably, Very Long Baseline Interferometry (VLBI) techniques have allowed for a detailed imaging and characterization of this region. Using VLBI, it has been possible to resolve a small region of the Galactic Centre using an extended array of telescopes called the Event Horizon Telescope (EHT) which produced a detailed image of the shadow of a supermassive black hole surrounded by an accretion flow (Event Horizon Telescope Collaboration 2022a). The image clearly shows three bright hotspots around the emission ring-like structure, which is associated to the photon ring of black hole.
In this work we report the shadow produced by the presence of a second Schwarzschild black hole located between Sgr A* and Earth. The secondary black hole with mass of |$1035\, \rm M_\odot$|, produces a lensing effect on the light beams emitted at the accretion disc around the primary supermassive black hole with mass of |$4.14\times 10^6\, \, \rm M_{\odot }$| and generates an eclipsed image of the Sgr A* shadow, similar to the one produced by the Event Horizon Telescope Collaboration (2022b).
The first step in our modelling involves a general relativistic magnetohydrodynamics (GMHD) simulation of a Kerr black hole space-time with dimensionless spin parameter |$a_{\star }=15/16$| and accretion disc with constant specific angular momentum |$\ell =6.76$|, using the code bhac v1.1 (Porth et al. 2017; Olivares et al. 2019). The rotating black hole has a hydrodynamic equilibrium torus surrounding it which is perturbated by a weak single-loop poloidal magnetic field defined by its inner edge |$r_\text{in}=20\, \rm M$| and central radius |$r_\text{c}=40 \,\rm M$| (Fishbone & Moncrief 1976; Font & Daigne 2002; Shiokawa et al. 2012; Rezzolla & Zanotti 2013; Cruz-Osorio, Gimeno-Soler & Font 2020). The plasma is described by a relativistic gas with an ideal equation of state (see e.g. Rezzolla & Zanotti 2013), with adiabatic index |$\Gamma =4/3$|.
To compute the millimetre image of the accretion flow about Sgr A* the radiative-transfer equation along null geodesics, where the electromagnetic radiation propagates (Younsi, Wu & Fuerst 2012; Younsi et al. 2020), is solved. We computed 500 snapshots covering |$\approx 38\ {\rm h}$| of observations given the estimated mass of Sgr A* of |$4.14\times 10^6\, \mathrm{M}_{\odot }$|. The image size is computed adopting a distance to Sgr A* of |$8.127\, {\rm kpc}$| (Event Horizon Telescope Collaboration 2022a, b). The radiative transfer calculations require the use of a non-thermal energy distribution for the modelled electrons (Davelaar et al. 2018; Cruz-Osorio et al. 2022; Fromm et al. 2022; Röder et al. 2023; Zhang et al. 2024), which is a combination of a thermal electron population and of a population with a power-law energy distribution inspired by particle-in-cell simulations (cf. Ball, Sironi & Özel 2018; Meringolo et al. 2023; Imbrogno et al. 2024), with injection radius |$r_{\rm inj}=10$| and the fraction of magnetic energy contributing to the heating of the radiating electrons |$\varepsilon =0.5$|. The electron temperature is computed using the R-|$\beta$| model (Mościbrodzka et al. 2009). To compute the mass accretion rate, we normalize the emission at a resolution of |$600\times 600$| pixels, which corresponds to a field of view of |$150\, \mu {\rm as}$| in order to reproduce the observed flux of Sgr A* at 230 GHz of |$\simeq 2.4\, {\rm Jy}$| (Event Horizon Telescope Collaboration 2022a). The resulting mass accretion rates are |$\dot{M}=[1.41,\, 1.37,\, 1.32, \, 1.29] \times 10^{-8}\, \, \rm M_{\odot }\, \mathrm{yr}^{-1}$|, respectively. The inclination angle of the observation is assumed to be fixed at |$0^{\circ },\, 30^{\circ },\, 45^{\circ },\, 90^{\circ }$|, as deduced from the recent observations of the EHT (Event Horizon Telescope Collaboration 2022b). Under these assumptions, only two of the hotspots in Sgr A* can be reproduced, so far, the numerical GRMHD simulations considering a single black hole have not been capable of reproducing the observed third hotspot.
In this work we construct a model in which a secondary eclipsing black hole with a mass |$\approx 1000\, \textrm {M}_\odot$| at a distance of |$\approx 1200\, \textrm {AU}$| from the primary black hole in Sgr A* is capable of bending light rays. This secondary eclipsing black hole is contained in the line of sight between the Galactic Centre and Earth and as shown in Fig. 1 is capable of bending the original shadow of Sgr A* in such a way as to reproduce the third bright hotspot.
Artistic representation of a secondary Schwarzschild black hole with mass |$M_\text{sec}$| eclipsing the shadow of Sgr A*. Red lines represent light trajectories which are emitted from the single black hole image, lensed by the secondary black hole and arriving to Earth as an eclipsed image. The distances from the secondary black hole to the image of Sgr A* and the one from the secondary black hole to the camera are represented by |$d_{\rm sec}$| and |$d_{\rm obs}$|, respectively.
2 ECLIPSING BLACK HOLES
The eclipsed image was generated by a secondary Schwarzschild black hole using the recently developed numerical code aztekas-shadows, which evolves the geodesic equations using a fourth order Runge–Kutta method (see e.g. Sirca & Horvat 2012), solving the schematic configuration shown in Fig. 1. The algorithm consists of emitting perpendicular light beams from a detector grid with the same size and resolution as that of Sgr A*’s images of |$600\times 600$| pixels, which corresponds to |$150 \,\mu \text{as}$| field of view, located at a distance |$d_{\mathrm{obs}}$| from the secondary black hole. Those beams are deflected by the secondary black hole and some of them would intersect the image of Sgr A* located at a distance |$d_\mathrm{sec}$| from the secondary black hole, in the opposite direction of the grid. The distances |$d_\mathrm{obs}$| and |$d_\text{sec}$| are greater than |$2GM_{_\mathrm{sec}}/c^2\times 10^4$|, which is |$10^4$| times the gravitational radius |$r_\rm{g,\rm{sec}}$| of secondary black hole, where its gravitational influence is essentially negligible (see e.g. Younsi 2014). Therefore, the plane where the image of Sgr A* is located and the detector are parallel to each other. Every light beam which intersects the image of Sgr A* was marked with the same corresponding intensity on the detector grid. The intensity was computed with the original pixels of the image of Sgr A* using a bilinear interpolation. The image obtained in the detector grid is thus the eclipsed shadow. In other words, this final image is the shadow of the shadow of Sgr A*.
We selected the best image that shows two clear and well-defined hotspots, corresponding to a supermassive black hole as described above and an inclinations angle of |$i=30^{\circ }$| between the spin axis and Earth. Then we added a secondary Schwarzschild black hole with masses |$M_\text{sec}~\in \left[10^2\, \rm M_{\odot },\, 10^5\, \rm M_{\odot }\right]$|, placed at various distances |$d_\text{sec}\in \rm [0.2\, AU,2000\, AU]$|. The distance |$d_\text{obs}$| from the secondary black hole to Sgr A* image was always taken with a fixed value of |$r_{g,\text{sec}}\times 10^5$| for every case.
We computed |$\approx 100\,000$| synthetic images of the eclipsing phenomenon, by exploring different masses and locations of the secondary black hole. In particular we explored coplanar circular orbits by varying the radial direction |$r\in \rm [0 \, M, 15\, M]$| and azimuthal position |$\varphi \in [0^\circ ,360^\circ ]$| around the primary black hole. In order to significantly speed up the ray-tracing process to obtain the required images, the program aztekas-shadows required a parallelization procedure in its development. Using |$\approx 16\,000$| GPU units of an NVIDIA card with Compute Unified Device Architecture (CUDA), the computational time of the numerical simulations was reduced by five orders of magnitude.
The distances |$d_\mathrm{sec}$| and |$d_\mathrm{obs}$| on the eclipsed images generated by aztekas-shadows are much greater than the gravitation radius of the secondary black hole (|$d_\mathrm{sec},d_\mathrm{obs}>r_\rm{g,\rm{sec}} \times 10^3$|). While, the mass of the secondary black hole |$M_{_\mathrm{sec}}\lesssim 2.5 M \times 10^{-2}$| is two orders of magnitude smaller than the central supermassive black hole. Since the analysed light beams are detected practically in a perpendicular direction to the detector, their deflection angles are very small. As a consequence, a length scale |$l_i$| on the image’s grid will be modified as |$l_f$| on the image of Sgr A* so that |$l_i-l_f\propto d_\text{sec}q$|. In this way, we can reproduce similar images as long as the length-factor product |$d_\mathrm{sec}\, q$| is constant. In Fig. 2, we present the accretion flow around a single black hole, producing a synthetic image with two distinct brightness regions (top-left). In contrast, the eclipsed image-resulting from the presence of a secondary black hole and representing the best-fitting model-exhibits three hotspots (top-middle), aligning well with the EHT observations (top-right). The bottom panels display the relative errors between the synthetic images and the observational data, highlighting the differences between the models.
The left bottom panel shows the shadow of the primary Kerr black hole in Sgr A* with |$a_{\star }=15/16$|, mass |$M_{\rm BH}=4.14\times 10^{6}\ \, \rm M_{\odot }$| and an inclination angle of |$30^{\circ }$| with respect to the black hole spin axis. The right bottom panel is the shadow of the left bottom panel produced by an eclipsing Schwarzschild black hole. The top panels represent the blurring of their corresponding bottom images using a circular beam with radio of |$20\, \mu\rm as$|. The mass ratio between the supermassive and the secondary black hole is |$q=2.5\times 10^{-4}$|, and the separation distance is |$d_{\rm sec}=1233\, {\rm AU}$|.
3 SUMMARY AND CONCLUSIONS
We explored different configuration positions and masses of the secondary eclipsing black hole following the constraints imposed by Will et al. (2023). The first image of the shadow of Sgr A* was constructed numerically by considering only the supermassive black hole at the Galactic Centre and the emission from the magnetized accretion disc. We assumed that the photons travel in straight paths at the distance of the secondary black hole, so that the light rays are bent when they pass near the secondary black hole. We found that the best suitable candidate to achieve the production of the third hotspot can be done by a secondary Schwarzschild black hole of mass |$M_\text{sec}~= 1035\, \text{M}_\odot$| at a distance of |$d_\text{sec}~= 10233 \, \text{AU}$| from the Galactic Centre as shown in Fig. 3. Smaller secondary black holes can also account for this third hotspot if they follow the simple analytical approximation given by |$d_\text{sec}q \approx 0.308 \, \text{AU}$|, where |$q = M_\text{sec}~/ M$|, |$M_\text{sec}~$| is the mass of the secondary black hole and |$d_\text{sec}~$| the separation distance between the black holes.
![The top left panel shows the synthetic image a single Kerr black hole. The top middle panel is the shadow of the top left panel produced by an eclipsing Schwarzschild black hole. The top right is the observed image of Sgr A* of the EHT collaboration (Event Horizon Telescope Collaboration. 2022a), with green contours of the top middle image superimposed. The synthetic images were blurred using a circular beam of radio $20\, \mu\rm as$ and the white lines represent constant flux contours with values of [2.56,3.62,5.12,7.24] $\times 10^{-5}\mathrm{Jy}$. The mass of the Schwarzschild black hole is $M_{\rm BH}= 1035\ \, \rm M_{\odot }$ so that the ratio between the primary to the secondary black hole is $q\approx 2.5\times 10^{-4}$, and the separation distance between them is $d_{\rm sec}=1233\, {\rm AU}$. The bottom panels show the relative errors between the observation of Sgr A* and the synthetic images, $\Delta S= S_{\rm obs}-S_{\rm syn}$.](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/mnrasl/540/1/10.1093_mnrasl_slaf033/1/m_slaf033fig3.jpeg?Expires=1748459621&Signature=ehaXX5ViBJ-2F0giWj0mqyNPbETh0Wki3iIZyXl1mFRTmWoqfJRQV1rjfD5KMLiJ7w~UCeme85dq6ornm2BB9hH-ph8aVwdvljmjSgVgBdPOAckP79AMSS~u1krv~NQfc31nbFd0pMkP6J9JKgPGpfMHInL2fQaEueYn-AmwZYnhiUC6c2dVYMGPnzm-yUccVRSXb7BBTZSUXZoIZ7KeAlnih0HNFC4QrxFmYyUb4jXWXPn-d~jqtPMjRnYQ1relhVxSu6HYRdNithbokPQqnm30j5yi6hp4z58Wjl7EPiB56j3W9YHH9ibcmvwb9QYYagAqOIsAJ81IL2Bmo7~97Q__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
The top left panel shows the synthetic image a single Kerr black hole. The top middle panel is the shadow of the top left panel produced by an eclipsing Schwarzschild black hole. The top right is the observed image of Sgr A* of the EHT collaboration (Event Horizon Telescope Collaboration. 2022a), with green contours of the top middle image superimposed. The synthetic images were blurred using a circular beam of radio |$20\, \mu\rm as$| and the white lines represent constant flux contours with values of [2.56,3.62,5.12,7.24] |$\times 10^{-5}\mathrm{Jy}$|. The mass of the Schwarzschild black hole is |$M_{\rm BH}= 1035\ \, \rm M_{\odot }$| so that the ratio between the primary to the secondary black hole is |$q\approx 2.5\times 10^{-4}$|, and the separation distance between them is |$d_{\rm sec}=1233\, {\rm AU}$|. The bottom panels show the relative errors between the observation of Sgr A* and the synthetic images, |$\Delta S= S_{\rm obs}-S_{\rm syn}$|.
The synthetic image obtained from a lensed shadow image due to the presence of a secondary intermediate black hole in the Galactic Centre has an overall good agreement with the reconstructed average image from the EHT observations of Sgr A*. We found that the binary system with separation of |$10\,233\, {\rm AU}$| and a secondary black hole with mass of |$1035\, \, \rm M_{\odot }$| produces a lensed image with three hotspots. Fig. 3 shows a direct image comparison between the observed image of Sag A* with the numerical constructed images with and without a secondary black hole, showing that the image with a secondary black hole is a better model. Our results are also in agreement with the recent theoretical constraints of the distance and mass of the secondary compact object. The study performed by Will et al. (2023) using the Newtonian three-body problem considering the binary black hole system and the S2 star orbit found two constriction regimes for the S2 orbit not to be disrupted. The first one is when the secondary black hole is farther away than the S2 star orbit at |$\sim 1020\, {\rm AU}$| finding that distances |$d_\text{sec}~$| in the range of |$d_\mathrm{sec}\in [1000\, \mathrm{AU},4000\, \mathrm{AU}]$| and black hole masses |$M_{_\mathrm{sec}}\in [10^3\, \mathrm{M}_{\odot }, 10^5\, \mathrm{M}_{\odot }]$|. The second region is inside the S2 orbit and excludes the masses for the secondary black hole in the same range, |$M_{_\mathrm{sec}}\in [10^3\, \mathrm{M}_{\odot },10^5\, \mathrm{M}_{\odot }]$|. These limits ensure that the orbit of S2 is not affected by quadrupolar perturbations and the amplitude of gravitational waves generated by the binary black holes is negligible. This means that in the search parameter space for the simulations (|$d_\text{sec}~> 1020, {\rm AU}$|), a smaller value of |$d_\mathrm{sec}$| requires also a smaller value of |$M_{_\mathrm{sec}}$|. Our lensed synthetic image from the binary system satisfies these constraints.
Since the angular diameter |$d_{\rm sh}$| of Sgr A* is given by Event Horizon Telescope Collaboration (2022a): |$d_{\rm sh} \approx 10\, \rm GM / c^2 D \approx 48.7 \,\mu \rm \text{as}$| and the angular diameter of the eclipsing secondary black hole turns out to be |$0.013\, \mu \text{as}$|, the ratio between the secondary black hole and Sgr A* is |$\approx 2.7 \times 10^{-4}$|. This result clearly shows that the eclipse is partial and localized in order to get the required third hotspot.
Finally, in our approach, the eclipsing binary black hole could be a fly-by object or an orbiting one about the Galactic Centre. In the latter case for a Keplerian circular orbit, with a radius |$\approx 10^4\, {\rm AU}$| around the Galactic Centre, it would have an orbiting period of |$\approx 26\, \text{yr}$|. In general terms, we expect the third hotspot to be absent in the next release of Sgr A* observations if the secondary black hole is in orbit about the supermassive black hole, Sgr A*.
The results presented here highlight the need for further research and enhancements, particularly in modelling that incorporates the self-consistent dynamics of binary black holes surrounded by an accretion flow, and the analysis of the time variability of the light curve of Sgr A*.
ACKNOWLEDGEMENTS
This work was supported by the Programa de Apoyo a Proyectos de Investigacion e Inovacion Tecnologica, Direccion General de Asuntos del Personal Academico, Universidad Nacional Autonoma de Mexico (PAPIIT DGAPA-UNAM )projects (IN110522, IN118325, IA103725). ACO, GMG, SM, MJSA acknowledge economic support from ‘Secretaría de Ciencia, Humanidades, Tecnología e Innovación’ (Secihti, México), numbers: 257435, 378460, 26344, 751147. GMG and MJSA acknowledge economic support from the Coordinacion General de Estudios de Posgrado, Universidad Nacional Autonoma de Mexico (CGEP-UNAM) for a research visit at Goethe Universität, Frankfurt. ACO gratefully acknowledges ‘Ciencia Básica y de Frontera 2023-2024’ (Sechti) Grant No. CBF2023-2024-1102 and the European Horizon Europe staff exchange (SE) programme HORIZON-MSCA2021-SE-01 Grant No. NewFunFiCO-101086251.
DATA AVAILABILITY
The data that support the plots within this paper and other findings of this study are available on: https://archive.org/details/sagAbinaryblackhole. The public released version of the GRMHD code bhac can be found at https://bhac.science. The public version or the code aztekas-shadows will be released soon.
