Scientists have shown that the outer boundary of the warm corona surrounding a black hole is defined by a sharp drop in gas temperature where iron ions recombine.

Using a new modelling pipeline called TiNoGyo, which couples the photoionization code TITAN, the Monte Carlo Comptonisation code NOAR, and the relativistic ray‑tracing code GYOTO, the team simulated the spectrum emerging from the corona. The method automatically reproduces both emitted and reflected X‑ray light.

The simulations reveal that, as the distance from the black hole increases, Compton cooling weakens and line cooling by iron takes over. The transition occurs at a specific internal heating rate of about 1.6 × 10⁻²³ erg cm³ s⁻¹, independent of the Eddington ratio. Consequently the size of the corona grows with higher accretion rates: at λ_Edd = 0.05 the edge lies at ~18.2 r_g, at λ_Edd = 0.1 at ~36.6 r_g, and at λ_Edd = 0.3 at ~87.4 r_g.

However, the present model stops at the iron recombination front and does not include further cooling by other elements that could be important at larger radii. Thus the full structure of the disk atmosphere beyond the corona remains unmodelled.

These findings clarify how the temperature structure of the corona and the resulting soft X‑ray excess depend on the accretion rate and help constrain the physics of black‑hole accretion disks.