Radiative Models of Sagittarius A* and M87 from Relativistic MHD Simulations

Ongoing millimeter VLBI observations with the Event Horizon Telescope allow unprecedented study of the innermost portion of black hole accretion flows. Interpreting the observations requires relativistic, time-dependent physical modeling. We discuss the comparison of radiative transfer calculations from general relativistic MHD simulations of Sagittarius A* and M87 with current and future mm-VLBI observations. This comparison allows estimates of the viewing geometry and physical conditions of the Sgr A* accretion flow. The viewing geometry for M87 is already constrained from observations of its large-scale jet, but, unlike Sgr A*, there is no consensus for its millimeter emission geometry or electron population. Despite this uncertainty, as long as the emission region is compact, robust predictions for the size of its jet launching region can be made. For both sources, the black hole shadow may be detected with future observations including ALMA and/or the LMT, which would constitute the first direct evidence for a black hole event horizon.

💡 Research Summary

This paper presents a comprehensive framework for interpreting millimeter‑very long baseline interferometry (mm‑VLBI) observations of the supermassive black holes Sagittarius A* (Sgr A*) and M87 by coupling general relativistic magnetohydrodynamic (GRMHD) simulations with physically motivated electron distribution and radiative transfer models. The authors first generate a set of three‑dimensional GRMHD simulations using two independent codes (Cosmos++ and a 3D extension of HARM). These simulations start from a hydrostatic torus threaded by a weak magnetic field in a Kerr spacetime, allowing the magnetorotational instability (MRI) to develop turbulence, transport angular momentum, and drive accretion. The simulated domain extends to ~0.25 M in radius and reaches a quasi‑steady state, but deliberately omits radiation feedback and treats the plasma as a single fluid, so ion and electron temperatures may differ.

To bridge this gap, the authors prescribe electron thermodynamics in two ways. For Sgr A*, they assume the electrons are essentially thermal and adopt a constant ion‑to‑electron temperature ratio (Ti/Te). This simple prescription enables the electron temperature to be derived directly from the simulated pressure and density. For M87, where both a radiatively inefficient accretion flow (disk) and a relativistic jet may contribute to the millimeter emission, they separate the jet region by the criterion b²/ρc² > 1 (magnetically dominated). Jet electrons are modeled as a non‑thermal power‑law population with a density proportional to the magnetic energy density (nnth ∝ b²), while disk electrons remain thermal with the same Ti/Te prescription.

Radiative transfer is performed with the public code grtrans, tracing null geodesics backward from an observer’s camera using the geokerr integrator. Along each geodesic, synchrotron emissivity and absorptivity are computed from the prescribed electron distribution; inverse‑Compton scattering is neglected (though it may be important for high‑energy M87 spectra). The authors generate both time‑averaged images (using the last 2000–4000 M of simulation time) and time‑dependent “movies” by sampling multiple snapshots, thereby capturing intrinsic variability.

For Sgr A*, the model parameters—mass accretion rate (ṁ), Ti/Te, inclination (i), sky orientation (ξ), and the specific simulation—are constrained by fitting 1.3 mm VLBI visibility amplitudes together with the total fluxes at 0.4 mm and 1.3 mm. Using Bayesian inference and χ² minimization, they obtain i ≈ 60° ± 15°, ξ ≈ −70° (+86°/−15°), electron temperature Te ≈ 6 ± 2 × 10¹⁰ K, and ṁ ≈ 3⁺⁷₋₁ × 10⁻⁹ M⊙ yr⁻¹. These values are consistent with earlier radiatively inefficient accretion flow (RIAF) fits and with independent constraints from linear polarization and Faraday rotation measurements. The current VLBI array (SMTO, CARMA, APEX/ALMA) lacks sufficient north‑south baselines to resolve the black‑hole shadow, but the authors predict that adding stations such as ALMA and the Large Millimeter Telescope (LMT) would produce a visibility minimum near 3000 Gλ, revealing the shadow.

For M87, two families of models are explored: a disk‑plus‑jet (DJ1) scenario and a jet‑only (J2) scenario. The large‑scale jet observations provide a prior on the viewing geometry (inclination < 40°, sky angle between −115° and −75°), which is imposed on all models. In the jet‑dominant case, the millimeter emission originates primarily from the counter‑jet, because the forward jet is beamed into the black‑hole shadow and thus largely invisible. Assuming the emission region is compact (r ≲ 10 M) and not completely opaque (optical depth τ ≲ 3), the simulated images predict an apparent size of 33–44 µas (≈ 4–5 Schwarzschild radii), comparable to the measured size of Sgr A*. The visibility curves again show a shallow minimum at ~3000 Gλ on baselines involving LMT‑Chile or Chile‑CARMA, indicating that the shadow could be detected with future north‑south baselines.

The paper emphasizes that relativistic effects—Doppler beaming, light bending, and gravitational lensing—combine to produce crescent‑shaped images with a bright photon‑ring surrounding a central dim region (the shadow). The authors argue that, while current data cannot discriminate between detailed physical models (e.g., different electron heating prescriptions), the inclusion of realistic variability and three‑dimensional dynamics already yields robust predictions for observable quantities such as image size, orientation, and the presence of the shadow.

In conclusion, the study demonstrates that 3‑D GRMHD simulations, when coupled with plausible electron thermodynamics and full relativistic ray‑tracing, can successfully reproduce existing mm‑VLBI observations of Sgr A* and M87, constrain key physical parameters, and forecast the detectability of the black‑hole shadow with modest extensions to the VLBI network. The work thus provides a critical bridge between theoretical models of accretion and jet physics and the emerging observational capability of the Event Horizon Telescope, paving the way for the first direct imaging of event horizons.