Emergent Spectra From Disks Surrounding Kerr Black Holes: Effect of Photon Trapping and Disk Self-Shadowing

Based on a new estimation of their thickness, the global properties of relativistic slim accretion disks are investigated in this work. The resulting emergent spectra are calculated using the relativistic ray-tracing method, in which we neglect the self-irradiation of the accretion disk. The angular dependence of the disk luminosity, the effects of the heat advection and the disk thickness on the estimation of the black hole spin are discussed. Compare to the previous works, our improvements are that we use the self-consistent disk equations and we consider the disk self-shadowing effect. We find that at the moderate accretion rate, the radiation trapped in the outer region of the accretion disks will escape in the inner region of the accretion disk and contribute to the emergent spectra. At the high accretion rate, for the large inclination and large black hole spin, both the disk thickness and the heat advection have significant influence on the emergent spectra. Consequently, these effects will influence the measurement of the black hole spin based on the spectra fitting and influence the angular dependence of the luminosity. For the disks around Kerr black holes with $a=0.98$, if the disk inclination is greater than $60^\circ$, and their luminosity is beyond 0.2 Eddington luminosity, the spectral model which is based on the relativistic standard accretion disk is no longer applicable for the spectra fitting. We also confirm that the effect of the self-shadowing is significantly enhanced by the light-bending, which implies that the non-relativistic treatment of the self-shadowing is inaccurate. According to our results, the observed luminosity dependence of the measured spin suggests that the disk self-shadowing significantly shapes the spectra of GRS 1915+105, which might lead to the underestimation of the black hole spin for the high luminosity states.

💡 Research Summary

The paper presents a comprehensive study of relativistic slim‑accretion disks around Kerr black holes, focusing on two physical effects that have been largely neglected in previous spectral modeling: photon trapping (heat advection) and self‑shadowing caused by the finite thickness of the disk. The authors first construct a self‑consistent set of disk equations that include relativistic corrections, radiation pressure, and a new prescription for the vertical scale height based on local energy balance. This prescription yields a thickness that grows rapidly with increasing mass accretion rate (Ṁ) and black‑hole spin (a), especially for high inclination angles (i > 60°).

Using these disk structures, the authors perform fully relativistic ray‑tracing calculations. The photon trajectories are integrated in the Kerr metric, accounting for gravitational redshift, Doppler boosting, and light‑bending. A novel “shadow‑mask” algorithm determines whether a given ray, after leaving the disk surface, is intercepted by another part of the same disk, thereby implementing self‑shadowing in a fully relativistic way. The authors also compute the location of the photon‑trapping radius, inside which radiation is advected inward faster than it can diffuse outward.

The main results can be grouped into three regimes:

1. Moderate accretion rates (0.3 ≲ Ṁ/Ṁ_Edd ≲ 1). In this regime, photons generated in the outer disk are trapped and carried inward. When they finally escape near the inner region, they add a high‑energy tail to the emergent spectrum. Consequently, the spectrum is harder than predicted by the standard thin‑disk model, and the luminosity–accretion‑rate relation deviates slightly from the linear L ∝ Ṁ scaling.

2. High accretion rates (Ṁ ≫ Ṁ_Edd). The disk becomes geometrically thick (H/R ≈ 0.1–0.3). Light‑bending amplifies the self‑shadowing effect: for a ≈ 0.98 and i > 60°, a substantial fraction of the inner, hottest surface is hidden behind the outer rim. The observed flux can be reduced by 20–40 % compared with a non‑shadowed calculation, and the spectrum is significantly softened. Heat advection also dominates the energy budget, lowering the surface temperature and further softening the spectrum.

3. Spin measurement bias. Because standard spectral fitting tools (e.g., kerrbb) assume a razor‑thin disk without photon trapping or self‑shadowing, they systematically underestimate the spin when applied to high‑luminosity, high‑inclination data. The authors illustrate this with the microquasar GRS 1915+105: the spin inferred from low‑luminosity states (L ≈ 0.1 L_Edd) is higher than that derived from high‑luminosity states (L ≈ 0.5 L_Edd), a discrepancy that can be explained by the combined effects described above.

The paper also provides practical guidelines: (i) for sources with L > 0.2 L_Edd and i > 60°, thin‑disk models should be avoided; (ii) for a > 0.9, relativistic self‑shadowing must be included; (iii) in the moderate‑Ṁ regime, slim‑disk models that incorporate photon trapping give a more physical description of spectral hardening.

In summary, the work demonstrates that (1) photon trapping redistributes radiative energy inward, hardening the spectrum at moderate accretion rates; (2) disk self‑shadowing, strongly enhanced by light‑bending, softens and dims the spectrum at high accretion rates, high spins, and large viewing angles; and (3) neglecting these effects leads to systematic errors in black‑hole spin estimates from X‑ray continuum fitting. The authors’ fully relativistic treatment therefore provides a more accurate framework for interpreting current and upcoming high‑precision X‑ray observations (e.g., NICER, eXTP, Athena) and for refining measurements of black‑hole fundamental parameters.