First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way

What can we learn from these images and their variability properties? Focusing first on the astrophysics of the accretion process and jet launching, we can explore which physical scenarios are most consistent with our results, under the assumption that Sgr A* is a Kerr black hole with mass and distance accurately determined from stellar orbits. Our constraints on the properties of the black hole and potential deviations from GR are explored in Section VII.

We assume that the accretion structure around Sgr A∗ is approximately governed by ideal GRMHD, as was done for M87∗, which is common in the literature for modeling SMBHs [49, see, e.g.,]. Decades of observations and semi-analytical modeling (see for details, references, and caveats) constrain the average plasma properties close to the event horizon of Sgr A∗, allowing us to make several additional simplifying approximations. In particular, for Sgr A∗ we can assume that radiative cooling does not strongly affect the dynamics, and that the electrons and ions are weakly coupled by Coulomb collisions, so that ions and electrons can have distinct temperatures in some parts of the flow.

Because we model the plasma as a fluid with a single temperature, one of our main sources of uncertainty is the treatment of the electrons, whose presence is not explicitly accounted for in the simulation evolution equations. We explore several parameterized models to assign the electron distribution function (eDF), assuming that the electron temperature is proportional to the proton temperature, with a proportionality that depends on the gas-to-magnetic pressure ratio. The eDFs include thermal and variations of non-thermal, the latter of which were not explored in the M87∗ 2017 papers. Our fiducial thermal models employ the same eDF prescription as for the M87∗ papers, using only one free parameter, Rhigh, to specify the proton-to-electron temperature ratio in regions where gas pressure dominates the magnetic pressure. This ratio is typically larger in the disk midplane than in the jet/outflow. Since the radiation is produced by electrons, increasing Rhigh effectively increases the brightness of the jets/outflow region relative to the disk, and changes the resulting images/spectra. Compared to M87∗, we also allow the inclination angle to vary.

We employ five different ideal GRMHD codes to explore a large swath of overlapping parameter space, in some cases with very similar set-ups allowing for consistency checks. In other cases, with a more exploratory sampling of parameter space, we also allow differences in, for example, adiabatic index, resolution, and size of the tori and/or computational domain. Most models are initialized with an orbiting torus of plasma with characteristic radius ∼20GM/c². The torus is seeded with a weak, poloidal magnetic field, and can be either prograde or retrograde with respect to the black hole spin (a free parameter). We also consider a limited set of exploratory models such as those with ‘tilt’ where the black hole spin axis is misaligned from the rotation axis of the torus, as well as a model initiated on a very large grid using a more realistic set-up for the outer boundary conditions in which the accretion flow is directly fed by winds from orbiting stars. Both of these are more realistic physical scenarios but also allow a much larger range of parameter space than we could fully explore.

We classify the fiducial models as being in either the magnetically arrested [88, MAD;] or standard and normal evolution [90, SANE;] modes. In MAD models the ordered magnetic fields significantly affect the dynamics of the flow, episodically halting accretion onto the black hole, while SANE models have weaker, more turbulent magnetic fields. Because the dynamical timescale in Sgr A∗ is short compared to a night of observations, it is important to run each model for enough time to capture the range of spectral and structural variations. The simulations are typically run for 30,000tg, while some are run for more than 100,000tg in order to sample a broader distribution of behavior.

For each time-dependent GRMHD simulation, with an eDF prescription and inclination with respect to the line-of-sight, we calculate a sequence of model images (movies) using ray tracing and including synchrotron emission and absorption. We also calculate spectra including synchrotron emission and absorption, bremsstrahlung emission and, using Monte Carlo methods, Compton scattering. These synthetic data sets are then used to generate simulated EHT images as well as multi-wavelength light curves and spectral energy distributions (SEDs) for comparison with the Sgr A∗ data described in. We scale all images to a benchmark average flux density of 2.4 Jy at 230 GHz to match the average synchrotron flux density of Sgr A∗ (see and Wielgus et al. 120).

We evaluate the simulations against three types of observational constraints: EHT interferometric measurements, emission at other wavelengths, and variability. The EHT constraints include 1) a measure of the image size, 2) salient features from the visibility amplitudes, such as the location of the first deep minimum, and 3) the diameter, asymmetry, and width of simplified ring models fitted to well-sampled portions of the April 7 visibility data. The constraints from other wavelengths include the flux densities at 86 GHz, 2.2μm in the NIR, and X-ray, and the major axis source size at 86 GHz, constrained from observations with the Global mm-VLBI Array. Finally, the variability constraints are 1) the fractional 230 GHz variability on 3 h timescales, derived from more than a decade of measurements, and 2) the structural variability of the source, calculated at a baseline length of 4Gλ after fitting a parameterized model to the visibility amplitude variation versus baseline length. See for the full ranges of tests and pass/fail conditions.

Compared to, we explore a larger range of models and model parameter space, and we also include some additional observational constraints. These include the degree of intrinsic variability as well as the broadband spectral constraints given above. Accordingly, we find that all our models fail at least one of the observational constraints. These results indicate the power of combining interferometric data with other observational constraints to narrow down the viable physical parameter space. We now summarize our main results and their implications for our understanding of Sgr A∗’s accretion state and geometry.

We primarily focus on a set of “fiducial” simulations, which use aligned (prograde or retrograde) accretion flows and thermal eDFs defined via the Rhigh prescription. We declare a model to fully pass a set of constraints only when all GRMHD simulations with those parameters pass. This approach helps to ensure that our selection of favored models is resilient to small variations in the GRMHD simulation choices and software. Figure 4 summarizes these results.

All edge-on (high inclination) models fail the combined set of EHT-only constraints for at least one simulation, and almost all retrograde models (a∗<0) fail. There are two interesting groupings of models that pass all EHT constraints for all simulations: both have positive/prograde spin (a∗=0.5, 0.94) with lower (≤ 50°) inclination, but some are MAD (10) and some are SANE (8). With only ∼10% pass rate for all models, it is clear that EHT imaging data alone are capable of strongly down-selecting the potential model space. The more heterogeneous non-EHT constraints prefer a rather different set of models. However even with all these constraints, 11/200 models pass for all simulations; all of these are MADs with all but one having Rhigh=160. On the other hand, they cover a wider range of spin compared to EHT-only constraints, with a slight preference for a∗≤0 and higher inclinations, and including retrograde and edge-on models.

None of the 200 fiducial models pass all 11 constraints in combination, the most strict of which are the 86 GHz size, the ring diameter and the light curve variability on 3-hour timescales. Of these, the light curve variability turns out to be the most stringent constraint, passing only 4% of fiducial models. SANE models, which are less variable than MADs, are preferred, while all the other EHT/non-EHT constraints generally favor MADs. If we consider the set of models that pass 10/11 constraints, it is notable that for both MADs and SANEs, there is still a marked preference for models with prograde spin, a lower inclination and Rhigh>40 with several clustered at the maximum of Rhigh=160.

The “exploratory” models cover less of a range in parameter space but still indicate important trends. In particular, including non-thermal electrons (which are especially important for modeling flares) tends to push the limits of allowed NIR flux and, to some extent, also the X-ray flux. Because the non-thermal particles also enhance the 230 GHz flux density, rescaling to a fixed flux density results in a smaller accretion rate. The smaller resulting opacity then affects the image properties, such as producing a somewhat narrower ring morphology at 230 GHz and a larger image at 86 GHz. Nevertheless, the addition of non-thermal models does not drastically change the preferred parameter space, and passing models also favor prograde spin and lower inclination. Increased tilt of the accretion flow tends to increase variability and the NIR flux, and thus leads to model failures. Similarly, neither of the two wind-fed models passed, but this is not surprising as they only model a single spin (a∗=0) and two instances of the thermal Rhigh eDF. With such sparse sampling of parameter space, these classes of models require a more focused study to draw conclusions.

Overall, very few of our models are as quiet as the data. Although this wasn’t investigated in, subsequent work suggests a similar result for M87∗ simulations. In general SANE models are less variable than MADs, and face-on models are somewhat less variable than edge-on. However, there are limitations of the modeling that may affect the variability. For instance, collisionless effects, radiative cooling, and improved electron heating models could all potentially reduce variability. Thus, our variability constraints may be too strict.

There are only two models that pass everything but the variability constraints. The models have similar parameters: both are MAD, both are prograde spin (one has a∗=0.5, and the other has a∗=0.94), and both have i=30° and Rhigh=160. Figure 5 shows a representative snapshot from one of these simulations and its corresponding image reconstruction. If we take these two models as indicative, there is a preference for accretion rates at the lower end of the range found by other constraints, 10⁻⁹−10⁻⁸M⊙/yr, and an outflow power of ∼10³⁸ erg/s, which is comparable to the bolometric luminosity of stellar-mass black holes in X-ray binaries. The combination of high Rhigh and relatively low inclination satisfies the spectral constraints without their associated jet structure producing too much asymmetry to satisfy the EHT constraints. A low inclination is also consistent with the independent estimate based on tracking the motions of NIR flaring structures using the VLTI. Our results highlight the value of continued EHT plus multiwavelength monitoring of Sgr A∗, and there is clearly much more exploration to conduct, including improved theoretical models and numerical simulations, full SED modeling, and polarimetric imaging, all of which will yield deeper physical insights [37, see, e.g.,].