What sets the edge of a black hole’s warm corona?

A boundary in the X-ray glow

Many active galactic nuclei—the luminous centers of galaxies powered by accreting supermassive black holes—emit more low-energy X-rays than an extrapolation of their harder X-ray continuum would predict. This is called the soft X-ray excess, typically appearing below about 2 kiloelectronvolts.

One explanation involves a warm corona: an optically thick atmosphere above the accretion disk, where photons gain energy by scattering from hot electrons. Another involves X-rays reflected from highly ionized material near the black hole. Both can reproduce aspects of observed spectra, so explaining the excess does not automatically establish which mechanism dominates.

In a November 2026 arXiv preprint, submitted to The Astrophysical Journal, Parikshit Partha Biswas and colleagues investigate a narrower question: if a warm corona exists, what determines its outer edge? Their work is a numerical modeling study, not a new telescope observation or an experimental detection of that boundary.

Building an atmosphere in calculations

The researchers model a standard accretion disk beneath an atmosphere heated both internally and by a compact source of hard X-rays above the black hole. Disk photons enter the atmosphere from below.

They combine three computational tools. TITAN calculates radiation transport, ionization and thermal balance through the atmosphere at individual radii. NOAR follows photon scattering and the associated energy changes. GYOTO then traces light through the black hole’s curved spacetime to predict spectra seen by a distant observer.

The setup uses a nonrotating black hole with 100 million times the Sun’s mass. Its energy budget is fixed: 20% goes to the compact hard-X-ray source, 40% heats the warm atmosphere internally, and 40% is released in the underlying disk. The atmosphere’s optical depth, a measure of its opacity to scattering, is held near five.

These choices allow controlled comparisons, but they are assumptions rather than properties measured for a particular galaxy.

When iron changes the cooling

Moving outward through the modeled disk, the conditions supporting the warm corona weaken. Eventually, cooling through atomic emission lines overtakes Compton cooling, in which electrons transfer energy to photons through scattering.

Iron provides the key transition in these calculations. Nearer the black hole, iron is stripped of almost all its electrons. Around the boundary, it recombines into states with more bound electrons. Collisions can excite those electrons, which subsequently release photons as they return to lower energy levels, carrying energy away from the gas.

For the model with solar iron abundance and an Eddington-scaled accretion rate of 0.1, the boundary begins near 36.6 gravitational radii. A gravitational radius is the distance scale GM/c², set by the black hole’s mass. At this location, temperature falls through part of the atmosphere from roughly 10 million to 4 million kelvin.

The boundary is therefore not a solid rim or an instantaneous radial cutoff. It is a transition region in the atmosphere’s cooling behavior.

More accretion, a larger warm region

For three Eddington-scaled accretion rates—0.05, 0.1 and 0.3—the modeled outer edges lie at about 18.2, 36.6 and 87.4 gravitational radii, respectively. These dimensionless rates express the mass supply relative to the Eddington accretion scale.

At solar composition, the internal-heating parameter at the edge remains close to 1.6 × 10⁻²³ erg cm³ s⁻¹ across those three cases. This is a density-normalized heating coefficient used by the model, not a total luminosity or a universal threshold established observationally.

Higher accretion rates produce stronger soft X-ray emission through both a larger warm emitting region and stronger local contributions. The direction of the trend agrees with earlier observational findings discussed in the paper, but this study does not itself fit new observations.

Iron abundance also matters. At the intermediate accretion rate, increasing iron from solar abundance to ten times solar moves the edge inward, from about 36.6 to 25.6 gravitational radii.

Predictions with important limits

The synthetic warm-corona continuum is smooth, while the iron-line complex comes from reflection. At lower modeled accretion rates, that complex is stronger and more distorted by relativistic effects because line production occurs closer to the black hole.

These are not complete predictions for an observed galaxy’s spectrum. The presented spectra omit the directly observed hard-X-ray continuum and reflection from the disk farther beyond the warm-corona edge. The authors also identify a mismatch between the lower-boundary treatments in TITAN and NOAR that can affect the predicted spectral slope.

Most importantly, the outward growth was calculated at fixed atmospheric optical depth. A thicker atmosphere distributes heating over more material and could shift the edge inward, potentially offsetting that trend. Spin and alternative energy budgets remain unexplored here. The result offers a physical explanation to test, not a settled measurement of where real warm coronae end.