A simplified approach for reproducing fully relativistic spectra in X-ray binary systems: Application to Cygnus X-1

Abstract

Context. General relativistic effects are strong near the black hole of an X-ray binary and significantly impact the total energy released in the innermost accretion disk's region. The simple pseudo-Newtonian solution in the standard disk model cannot replenish for effects such as light-bending, gravitational redshift, and Doppler boost. These can heavily affect the observed spectra depending on the rotation of the black hole and the disk's inclination. Aims. A proper relativistic treatment would be troublesome, unappealing, or require more advanced computational tools (e.g., the kerrbb code). Our goal is to fully incorporate the black hole's spin and all the general relativistic effects on the observed spectra coming from X-ray binary systems while maintaining the simplicity of the standard Shakura-Sunyaev disk model. Methods. We propose a way to replicate general relativistic spectra as predicted by the Novikov-Thorne model and the kerrbb numerical code by assuming a standard accretion disk with a shifted inner boundary that depends on the black hole spin and the source's viewing angle. An essential aspect in employing this approach for a broader range of disk inclinations is the derivation of spin-dependent temperature profiles for the accretion disk, obtained from some of the most efficient pseudo-Newtonian potentials around Kerr black holes. We then applied this method to Cygnus X-1, fitting the observational data obtained during its soft and hard spectral states. Results. The fully relativistic spectra are reproduced to an excellent approximation, with an error margin of 0.03–4%, by a standard disk model with a modified innermost radius within the range R_in = (0.2 ‑ 2)R_ISCO, depending on the source's viewing angle and black hole spin. This approach produces observed spectra as predicted by general relativity without the need for the ray-tracing method and complex numerical calculations. Thus, it emerges as a more straightforward alternative way of estimating black hole spins through the continuum-fitting method by successfully blending the general relativity properties with the Newtonian simplicity in a more complete way than the pseudo-Newtonian solutions. Relativistic effects near the black hole make an otherwise standard accretion disk with inclination θ < 60° seem truncated to larger radii to a distant observer. On the other hand, an edge-on view of the disk gives the perspective of being pulled closer to the central object than the respective innermost radius of the stable circular orbit. In addition, we show that the observational data of Cygnus X-1 can be satisfactorily fitted by employing a reasonably simple lepto-hadronic jet model and a hybrid thermal–nonthermal corona along with the relativistic-equivalent standard thin accretion disk.