ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O B-Formula I-Formula I-Formula B-TextGarbage O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O
tokens: and the horizontal axis shows the log of the effective temperature T eff . Each square represents one model. From left to right, the panels are evaluated for m 0 = 10 3 , 10 4 , and 10 5 , respectively. The dashed diagonal lines represent curves of constant Eddington ratio ( from left to right) and horizontal dashed curves show lines of constant mass ( from top to bottom) evaluated for the hottest annulus in the disk model. The white dashed curve in the left panel shows the variation as a function of radius for a relativistic α -disk model with a = 0, , and . The thick black solid curve corresponds to an analytical estimate (Equation ( 6 )) of where the annulus becomes photon starved. Note that we cap the color bar at f = 4 to better show the variation at lower values of f , but several of the hardest spectral models exceed this value. The black dashed lines show the values of Q and T eff computed at the radius where T eff reaches its maximum value in a relativistic accretion disk model for various values of the accretion rate and mass for a nonspinning black hole (Novikov Thorne 1973 ; Shakura Sunyaev 1973 ). For black hole spin a = 0, R max = 9.5 r g , where is the gravitational radius and G is Newton’s constant. We can then compute T eff , Q , and m 0 in the radiation pressure dominated limit via Here is the accretion rate scaled to , is the electron scattering opacity, is the Stefan–Boltzmann constant, η is the spin dependent radiative efficiency, α is the stress prescription parameter and . T rel , Q rel , and are relativistic correction factors (Novikov Thorne 1973 ). The diagonal lines in each panel correspond (from left to right) to , 0.1, and 1. The horizontal lines correspond (from top to bottom) to . Since the spectral hardening factor of a global disk model correlates closely with the value of f in the hottest annulus, this provides a good estimate of how f will vary for a global disk. Increasing the black hole spin shifts the diagonal lines to the right (i.e., toward higher T eff for a given M and ). In the left panel, we also show the variation of T eff and Q with radius in an accretion disk for and as a white, dashed curve. This shows that for a single disk model, f is generally largest at the radius with the highest T eff . Figure 4 shows the same quantities, but for lower values of m 0 . Here the ranges of T eff and Q with valid spectral models are narrower due to difficulties with convergence at these lower surface densities. The diagonal and horizontal dashed lines have the same meaning as in Figure 3 , but only cover masses of and 1000. Comparison with Figure 3 shows an accelerating trend for f to increase with decreasing m 0 . Equation ( 5 ) shows that the value of m 0 in the α -disk model is a function radius, mass, spin, and accretion rate. It is also sensitive to the details of the assumed stress prescription (Done Davis 2008 ) and thus likely to be the least robustly estimated parameter. Figure 4. Table of best-fit