= Solution
{c}
The <singular isothermal sphere> has one-dimensional <velocity dispersion> $\sigma$, density $\rho_{\rm tot}(r)=\sigma^2/(2\pi Gr^2)$, and enclosed mass $M(<R)=2\sigma^2R/G$. The gas mass inside $R$ is therefore
$$
M_g(<R)=\frac{2f_{\rm gas}\sigma^2R}{G}.
$$
Taking the dynamical crossing time to be $t_{\rm dyn}\simeq R/\sigma$, the <dynamical upper bound on black-hole fuelling> is
$$
\boxed{\dot M_{\rm dyn}\sim\frac{M_g(<R)}{t_{\rm dyn}}
=\frac{2f_{\rm gas}\sigma^3}{G}.}
$$
The cancellation of $R$ is special to the <singular isothermal sphere>. At $\sigma=200\,\mathrm{km\,s^{-1}}$, this is approximately
$$
\dot M_{\rm dyn}\simeq3.8\times10^3f_{\rm gas}\,M_\odot\,\mathrm{yr^{-1}}
\simeq6.1\times10^2\left(\frac{f_{\rm gas}}{0.16}\right)M_\odot\,\mathrm{yr^{-1}}.
$$
Use the stated <M-sigma relation> to set $M_{\rm BH}\simeq3\times10^8M_\odot$. Defining the <Eddington accretion rate> with a reference <radiative efficiency of black-hole accretion> $\eta_0$,
$$
\dot M_{\rm Edd}=\frac{L_{\rm Edd,es}}{\eta_0c^2}
=\frac{4\pi GM_{\rm BH}}{\eta_0\kappa_{\rm es}c}
\simeq6.6\left(\frac{\eta_0}{0.1}\right)^{-1}
\left(\frac{\kappa_{\rm es}}{0.40\,\mathrm{cm^2\,g^{-1}}}\right)^{-1}
M_\odot\,\mathrm{yr^{-1}}.
$$
Combining this with $M_{\rm BH}\propto\sigma^4$ gives
$$
\boxed{\frac{\dot M_{\rm dyn}}{\dot M_{\rm Edd}}
\simeq92\left(\frac{f_{\rm gas}}{0.16}\right)
\left(\frac{\eta_0}{0.1}\right)
\left(\frac{\kappa_{\rm es}}{0.40\,\mathrm{cm^2\,g^{-1}}}\right)
\left(\frac{\sigma}{200\,\mathrm{km\,s^{-1}}}\right)^{-1}.}
$$
Thus a gas fraction of order $0.1$--$0.16$ and the usual $\eta_0=0.1$ give roughly $60$--$90$ times the <Eddington accretion rate>, below $100$ at the requested dispersion. Using a crossing time $R/(\sqrt2\sigma)$ instead changes the coefficient by $\sqrt2$, so this is an order-of-magnitude estimate rather than a sharp universal bound. The problem does not specify $f_{\rm gas}$ or the rate convention: without these choices, the numerical claim cannot be proved for every gas fraction. Defining the rate as $L_{\rm Edd}/c^2$ would make this ratio ten times larger.
This is extremely optimistic because it gives every gas element a direct path to the hole on a crossing time. In a real <galaxy>, gas must shed <angular momentum>, avoid conversion into stars through <star formation>, cool sufficiently to flow inward, and survive <active-galactic-nucleus feedback>. A rotating <accretion disk> generally transports material on a <viscous timescale>, much longer than the orbital crossing time. Most available gas need not reach the central hole. If the hole is supplied at approximately the <Eddington accretion rate>, only about one percent of this optimistic dynamical supply reaches it; the actual percentage depends on its luminosity and duty cycle, so the bound alone is not a measurement of the fuelling efficiency.
For the temperature jump, use <Bondi accretion> as the local hot-gas estimate:
$$
\dot M_B\propto\rho_\infty c_s^{-3}.
$$
Assuming the initial gas had $c_s\sim\sigma$, its density has not yet changed, and its polytropic coefficient remains comparable, heating to $c_{s,\rm feed}\sim10\sigma$ gives
$$
\boxed{\frac{\dot M_{B,\rm feed}}{\dot M_{B,\rm before}}\sim10^{-3}.}
$$
The capture radius $GM_{\rm BH}/c_s^2$ also shrinks by about $100$. Subsequent expansion can lower the density and suppress the <mass accretion rate> further. The result illustrates <thermal suppression of black-hole accretion> and negative <active-galactic-nucleus feedback>: vigorous activity can shut off its own gas supply. The factor $10^{-3}$ is conditional on the fixed-density comparison; the previous dynamical upper bound is not itself an actual pre-heating Bondi rate.
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