Extending the Shakura - Sunyaev Accretion Disk Model with Black Hole Spin: A Comparative Study with the Novikov - Thorne Framework Applied to Cygnus X-1

The Shakura–Sunyaev (1973, hereafter SS73) alpha-disk model describes accretion onto a compact object using a Newtonian treatment of gravity, while the Novikov–Thorne (1973), Page & Thorne (1974, hereafter NT73) model extends this description into full general relativity around a rotating (Kerr) black hole. In this work we take the black-hole spin parameter a as the explicit new variable linking the two frameworks: SS73 is recovered in the non-relativistic, non-spinning limit, while NT73 introduces spin-dependent relativistic corrections through the location of the innermost stable circular orbit (ISCO) and the orbital energy, angular momentum, and angular velocity of circular Kerr geodesics. We implement both models numerically and apply them to the well-studied stellar-mass black hole X-ray binary Cygnus X-1 (M = 21.2 solar masses, spin a > 0.9985), computing temperature profiles, multicolor disk-blackbody spectra, radiative efficiencies, and bolometric luminosities for a = 0, 0.5, and 0.9985. We find that increasing spin shrinks the ISCO from 6 r_g to 1.21 r_g, raises the radiative efficiency from 5.7% to 32.9%, and increases the bolometric luminosity by roughly a factor of 6.53 at fixed accretion rate, while hardening the emergent spectrum. The Newtonian SS73 profile is shown to systematically overestimate the inner-disk temperature relative to the relativistic a = 0 case, underscoring the importance of general-relativistic corrections even for a non-rotating hole.

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