Advection-Dominated Accretion Flows (ADAFs)

7 Advection-Dominated Accretion Flows (ADAFs)

The ADAF, or advection-dominated accretion flow, solution also involves advective cooling. In fact, it carries it to an extreme – nearly all of the viscously dissipated energy is advected into the black hole rather than radiated. Unlike the slim disk solution, which is usually invoked at high luminosities, the ADAF applies when the luminosity (and generally the mass accretion rate) are low.

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).

Figure 12: Profiles of temperature, optical depth, ratio of scale height to radius, and advection factor (the ratio of advective cooling to turbulent heating) of a hot, one- $T$ ADAF (solid lines). The parameters are $M = 10M ⊙$ , $M˙ = 10−5LEdd ∕c2$ , $α = 0.3$ , and $β = Pgas∕(Pgas + Pmag) = 0.9$ . The outer boundary conditions are $Rout = 103RS$ , $T = 109 K$ , and $v∕cs = 0.5$ . Two- $T$ solutions with the same parameters and $δ = 0.5$ (dashed lines) and 0.01 (dot-dashed lines) are also shown for comparison, where $δ$ is the fraction of the turbulent viscous energy that directly heats the electrons. Image reproduced by permission from [321 ], copyright by AAS.

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 [106 ], 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 [224 ]. Again we present the solution with the following scaling: $2 r∗ = rc ∕GM$ , $m = M ∕M ⊙$ and $m˙ = M˙ c2∕LEdd$ .

10 −1 −1∕2 v = [− 3.00 × 10 cm s ]αc1r ∗ , Ω = [2.03 × 105 s− 1]c m −1r− 3∕2, 2 20 2−2 ∗−1 cS = [9.00 × 10 cm s ]c3r∗ , ρ = [1.07 × 10−5 g cm −3]α −1c−1c−1∕2m −1m˙r −3∕2, 1 3 ∗ P = [9.67 × 1015 g cm −1 s−2]α−1c−1 1c13∕2m −1m˙r −∗5∕2, 8 − 1∕2 1∕2 −1∕2 1∕4 − 1∕2 1∕2 −5∕4 B = [4.93 × 10 G]α (1 − βm ) c1 c3 m m˙ r∗ , q+ = [2.94 × 1021 erg cm −3 s−1]𝜖′c13∕2m − 2 ˙mr −∗4, −1 −1 − 1∕2 τes = [1.75]α c1 m˙r ∗ , (101 )

where $v$ is the radial infall velocity and $q+$ is the viscous dissipation of energy per unit volume. The constants $c 1$ , $c 2$ , and $c 3$ are given by

(5 + 2 𝜖′) ′ c1 = --3α2---g(α, 𝜖) [ ′ ′ ]1∕2 c = 2𝜖(5-+-2𝜖-)g(α,𝜖′) 2 9α2 ′ c3 = 2(5 +-2𝜖)g(α, 𝜖′), 9α2

where

( ) ′ -1-- 5-∕3 −-γg 𝜖 = fadv γg − 1 [ ]1∕2 ′ --18-α2--- g(α, 𝜖) ≡ 1 + (5 + 2𝜖′)2 − 1,

and the parameter $fadv$ represents the fraction of viscously dissipated energy which is advected. The remaining amount, $1 − fadv$ , is radiated locally.

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 [6 ] 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 [219 ].