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Supermassive black holes are compact galactic nuclei with masses typically of millions to billions of solar masses. Their gravitational influence is measured from the motion of surrounding matter; accretion luminosity is evidence of feeding, not a necessary property of every black hole. Keplerian stellar orbits near the Galactic Centre and molecular-maser rotation curves in other nuclei provide particularly direct compact-mass constraints. A rough black-hole sphere of influence is $r_{\mathrm{infl}}\sim G{M}_{\bullet }/{\sigma }_{\star }^2$; resolving it helps dynamical mass estimates.

Accretion converts binding energy into radiation,The latter is the Eddington luminosity for isotropic radiation opposed by electron-scattering opacity, not an absolute luminosity ceiling for every geometry. The retained black-hole mass grows at $(1-\eta ){\dot{M}}_{\mathrm{in}}$. Sustained feeding can produce an active galactic nucleus, whereas gas-poor nuclei may be faint. Loss of gas angular momentum is a central feeding problem.

Black-hole mass correlates with classical-bulge stellar velocity dispersion through the M-sigma relation. Accretion-powered radiation, winds and jets can heat or remove gas, linking nuclear growth to galaxy evolution. Such correlations do not establish one universal feedback history, and disks and pseudobulges do not follow the same tight relations as classical bulges. Seed formation, accretion and mergers all contribute; seed masses and feeding histories remain model dependent.^[ref]

The singular isothermal sphere has one-dimensional velocity dispersion $\sigma$, density ${\rho }_{\mathrm{tot}}(r)={\sigma }^2/(2\pi G{r}^2)$, and enclosed mass $M(<R)=2{\sigma }^{2}R/G$. The gas mass inside $R$ is thereforeTaking the dynamical crossing time to be $t_{\mathrm{dyn}}\simeq R/\sigma$, the dynamical upper bound on black-hole fuelling isThe cancellation of $R$ is special to the singular isothermal sphere. At $\sigma =200\mathrm{km}{\mathrm{s}}^{-1}$, this is approximately

Use the stated M-sigma relation to set $M_{\mathrm{BH}}\simeq 3\times 10^8{M}_{\odot }$. Defining the Eddington accretion rate with a reference radiative efficiency of black-hole accretion ${\eta }_0$,Combining this with $M_{\mathrm{BH}}\propto {\sigma }^4$ givesThus 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 $R/(\sqrt{2}\sigma )$ instead changes the coefficient by $\sqrt{2}$, so this is an order-of-magnitude estimate rather than a sharp universal bound. The problem does not specify $f_{\mathrm{gas}}$ or the rate convention: without these choices, the numerical claim cannot be proved for every gas fraction. Defining the rate as $L_{\mathrm{Edd}}/c^2$ 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:Assuming the initial gas had $c_s\sim \sigma$, its density has not yet changed, and its polytropic coefficient remains comparable, heating to $c_{s,\mathrm{feed}}\sim 10\sigma$ givesThe capture radius $G{M}_{\mathrm{BH}}/c_s^2$ 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.