In thin and slim disk models, one often uses a vertically integrated form for many physical quantities. For example, instead of density one uses the surface density defined as,
- Mass conservation (continuity): 2.
- Radial momentum conservation:
- Angular momentum conservation: alpha viscosity (Section 3.2.1), and is the specific angular momentum at the horizon, which can not be known a priori. As we explain in the next section, it provides an eigenvalue linked to the unique eigensolution of the set of thin disk differential equations, once they are properly constrained by boundary and regularity conditions.
- Vertical equilibrium:
- Energy conservation:
, , , and is the ratio of specific heats of the gas.
Through a series of algebraic manipulations one can reduce the thin disk equations to a set of two ordinary differential equations for two dependent variables, e.g., the Mach number and the angular momentum . Their structure reveals an important point here,
The extra regularity conditions at the sonic point are satisfied only for one particular value of the angular momentum at the horizon , which is the eigenvalue of the problem that should be found. For a given the location of the sonic point depends on the mass accretion rate. For low mass accretion rates one expects the transonic transition to occur close to the ISCO. Figure 5 shows that this is indeed the case for accretion rates smaller than about , independent of , where we use the authors’ definition of . At a qualitative change occurs, resembling a “phase transition” from the Shakura–Sunyaev behavior to a very different slim-disk behavior. For higher accretion rates the location of the sonic point significantly departs from the ISCO. For low values of , the sonic point moves closer to the horizon, down to for . For the sonic point moves outward with increasing accretion rate, reaching values as high as for and . This effect was first noticed for small accretion rates by Muchotrzeb  and later investigated for a wide range of accretion rates by Abramowicz , who explained it in terms of the disk-Bondi dichotomy.
The topology of the sonic point is important, because physically acceptable solutions must be of the saddle or nodal type; the spiral type is forbidden. The topology may be classified by the eigenvalues of the Jacobi matrix,5 with the dotted and the solid lines, respectively. For the lowest values of only the saddle-type solutions exist. For moderate values of () the topological type of the sonic point changes at least once with increasing accretion rate. For the highest solutions, only nodal-type critical points exist.
Shakura and Sunyaev  noticed that a few physically reasonable extra assumptions reduce the system of thin disk equations (88) – (93) to a set of algebraic equations. Indeed, the continuity and vertical equilibrium equations, (88) and (92), are already algebraic. The radial momentum equation (90) becomes a trivial identity with the extra assumptions that the radial pressure and velocity gradients vanish, and the rotation is Keplerian, . The algebraic angular momentum equation (91) only requires that we specify . The Shakura–Sunyaev model makes the assumption that . This is equivalent to assuming that the torque vanishes at the ISCO. This is a point of great interest that has been challenged repeatedly [164, 104, 25]. Direct testing of this hypothesis by numerical simulations is discussed in Section 11.4.
The right-hand side of the energy equation (93) represents advective cooling. This is assumed to vanish in the Shakura–Sunyaev model, though we will see that it plays a critical role in slim disks (Section 6) and ADAFs (Section 7). Because the Shakura–Sunyaev model assumes the rotation is Keplerian, , meaning is a known function of , the first term on the left-hand side of Eq. (93), which represents viscous heating, is algebraic. The second term, which represents the radiative cooling (in the diffusive approximation) is also algebraic in the Shakura–Sunyaev model.
In addition to being algebraic, these thin-disk equations are also linear in three distinct radial ranges: outer, middle, and inner. Therefore, as Shakura and Sunyaev realized, the model may be given in terms of explicit algebraic (polynomial) formulae. This was an achievement of remarkable consequences – still today the understanding of accretion disk theory is in its major part based on the Shakura–Sunyaev analytic model. The Shakura–Sunyaev paper  is one of the most cited in astrophysics today (see Figure 6), illustrating how fundamentally important accretion disk theory is in the field.
The general relativistic version of the Shakura–Sunyaev disk model was worked out by Novikov and Thorne , with important extensions and corrections provided in subsequent papers [237, 265, 241]. Here we reproduce the solution, although with a more general scaling: and .
Outer region: , (free-free opacity)
Middle region: , (electron-scattering opacity)
Inner region: ,98), (99), (100), are defined in terms of and as :
The Shakura–Sunyaev and Novikov–Thorne solutions are only local solutions; this is because they do not take into account the full eigenvalue problem described in Section 5.2. Instead, they make an assumption that the viscous torque goes to zero at the ISCO, which makes the model singular there. For very low accretion rates, this singularity of the model does not influence the electromagnetic spectrum , nor several other important astrophysical predictions of the model. However, in those astrophysical applications in which the inner boundary condition is important (e.g., global modes of disk oscillations), the Novikov–Thorne model is inadequate. Figure 7 illustrates a few ways in which the model fails to capture the true physics near the ISCO.