Because of their low efficiency, ADAFs are much less luminous than the Shakura–Sunyaev thin disks. The solutions tend to be hot (close to the virial temperature), optically thin, and quasi-spherical (see Figure 12). Their spectra are non-thermal, appearing as a power-law, often with a strong Compton component. This makes them a good candidate for the Hard state observed in X-ray binaries (discussed in Section 12.3).
ADAFs were formally introduced in the Newtonian limit through a series of papers by Narayan and Yi [223, 224, 225], followed closely by Abramowicz [6, 7] and others , although the existence of this solution had been hinted at much earlier [134, 257]. In the same spirit as we gave the equations for the Novikov–Thorne solution in Section 5.3 for thin disks, we report the self-similar ADAF solution found by Narayan and Yi . Again we present the solution with the following scaling: , and .
The rapid advection in ADAFs generally has two effects: 1) dissipated orbital energy can not be radiated locally before it is carried inward and 2) the rotation profile is generally no longer Keplerian, although Abramowicz  found solutions where the dominant cooling mechanism was advection, even when the angular momentum profile was Keplerian. Fully relativistic solutions of ADAFs have also been found numerically [13, 41]. Further discussion of ADAFs is given in the review article by Narayan and McClintock .