= Solution
The <alpha disk> relation and Keplerian angular frequency give
$$
c_s\sim\left(\frac{\nu\Omega}{\alpha}\right)^{1/2}
\propto
R^{-3/8}f_{\rm Edd}^{3/20}
\left(\frac\alpha{0.1}\right)^{-1/10}M^{11/40}.
$$
Combining this with $\Omega\propto M^{1/2}R^{-3/2}$ and the surface-density scaling from part (d), the <Toomre stability criterion> becomes
$$
Q(R)\propto
R^{-9/8}f_{\rm Edd}^{-11/20}
\left(\frac\alpha{0.1}\right)^{7/10}M^{-7/40}.
$$
It decreases strictly with radius, so it crosses unity at a unique <self-gravitating radius of an accretion disk>. Expressing radius in units of $R_S\propto M$ gives
$$
Q\propto
\left(\frac R{R_S}\right)^{-9/8}
f_{\rm Edd}^{-11/20}
\left(\frac\alpha{0.1}\right)^{7/10}M^{-13/10}.
$$
Solving $Q(R_{\rm sg})=1$ therefore yields
$$
\boxed{
\frac{R_{\rm sg}}{R_S}
=C_2f_{\rm Edd}^{-22/45}
\left(\frac\alpha{0.1}\right)^{28/45}
\left(\frac{M}{10^6M_\odot}\right)^{-52/45}
},
$$
and
$$
(k_5,k_6,k_7)=
\left(-\frac{22}{45},\frac{28}{45},-\frac{52}{45}\right).
$$
Let $x_{\rm ISCO}=R_{\rm ISCO}/R_S$. Requiring a non-self-gravitating annulus outside the <innermost stable circular orbit> gives $R_{\rm sg}>R_{\rm ISCO}$, and equality defines
$$
M_{\rm crit}
=10^6M_\odot
\left(\frac{C_2}{x_{\rm ISCO}}\right)^{45/52}
f_{\rm Edd}^{-11/26}
\left(\frac\alpha{0.1}\right)^{7/13}.
$$
For a nonspinning hole, $x_{\rm ISCO}=3$, and with $C_2\simeq10^5$, $\alpha\simeq0.1$, and $f_{\rm Edd}\simeq1$, this is approximately $8\times10^9M_\odot$, conventionally quoted as order $10^{10}M_\odot$. Above this mass the disk would become self-gravitating essentially as soon as stable circular orbits begin, so the assumed smooth <Shakura--Sunyaev thin disk> cannot provide a broad luminous accretion region.
The corresponding <Eddington luminosity> is of order $10^{48}\,\mathrm{erg\,s^{-1}}$, comparable to the upper envelope of quasar luminosities. This supports self-gravity as one contributor to the observed luminous-mass ceiling. It is not an absolute upper bound on black-hole mass: mergers, radiatively inefficient growth, nonstandard gas supply, spin-dependent inner radii, and fragmented or episodic accretion can all build a more massive hole without maintaining this particular steady thin disk.
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