= Solution
Insert the self-similar ansatz into the height-integrated equations. Mass conservation is already satisfied because $H\sim c_s/\Omega_K\propto R$ and $\rho RHu_R$ is constant. The angular-momentum and energy equations reduce to
$$
c_1=-\frac32\alpha c_3,
\qquad
c_2^2=\epsilon'c_3,
$$
while radial momentum gives
$$
1=\frac12c_1^2+c_2^2+\frac52c_3.
$$
Define
$$
g(\alpha,\epsilon')=
\sqrt{1+\frac{18\alpha^2}{(5+2\epsilon')^2}}-1.
$$
Solving the quadratic gives the exact <advection-dominated accretion flow> coefficients
$$
\boxed{c_1=-\frac{5+2\epsilon'}{3\alpha}g},
$$
$$
\boxed{c_2=
\left[\frac{2\epsilon'(5+2\epsilon')}{9\alpha^2}g\right]^{1/2}},
\qquad
\boxed{c_3=\frac{2(5+2\epsilon')}{9\alpha^2}g}.
$$
For $\alpha^2\ll1$,
$$
g=\frac{9\alpha^2}{(5+2\epsilon')^2}+O(\alpha^4),
$$
and therefore
$$
\boxed{c_1\simeq-\frac{3\alpha}{5+2\epsilon'},\quad
c_2^2\simeq\frac{2\epsilon'}{5+2\epsilon'},\quad
c_3\simeq\frac{2}{5+2\epsilon'}}.
$$
Efficient cooling means $f\to0$ and hence $\epsilon'=\epsilon/f\to\infty$ for fixed $\gamma<5/3$. Then
$$
u_R/v_K\to0,
\qquad
\Omega/\Omega_K\to1,
\qquad
c_s^2/v_K^2\to0,
\qquad
H/R\sim c_s/v_K\to0.
$$
The flow is therefore cold, nearly Keplerian, slowly accreting, and geometrically thin: the standard thin-disk limit.
For significant advection, $f=O(1)$ and all three deviations are explicit:
$$
u_R\simeq-\frac{3\alpha}{5+2\epsilon'}v_K,
\quad
\Omega\simeq\sqrt{\frac{2\epsilon'}{5+2\epsilon'}}\,\Omega_K,
\quad
c_s\simeq\sqrt{\frac{2}{5+2\epsilon'}}\,v_K.
$$
The gas is hot and thick, pressure supplies part of the radial support, rotation is sub-Keplerian, and dissipated entropy is carried inward. As $\gamma\to5/3$, $\epsilon'\to0$, so $\Omega\to0$, $c_s^2\to(2/5)v_K^2$, and $u_R\to-(3\alpha/5)v_K$: the self-similar rotating solution approaches a hot Bondi-like inflow. Sagittarius A* is the standard supermassive example: its luminosity is tiny compared with its <Eddington luminosity> despite an available gas supply, and its hot optically thin spectrum and low radiative efficiency are described by an ADAF or the broader radiatively inefficient accretion-flow family.
Solved by gpt-5.6-sol high.
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