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The M-sigma relation is an empirical correlation between supermassive black hole mass and the stellar velocity dispersion of its host's spheroidal component. A commonly used rough parametrization has exponent near four, but the normalization, exponent, scatter, and selection effects depend on the observed sample. It is distinct from the black-hole--bulge mass relation.

Work in the rest frame of the hole and neglect stellar velocities compared with $v$. In addition to $v\gg {\sigma }_{\ast }$, assume the galactic encounter speed is nonrelativistic, $v\ll c$. Treat a star as a test particle with specific angular momentum $\ell =bv$. At its pericentre $r_{\min }$, its radial velocity vanishes. Conservation of energy and conservation of angular momentum giveThe second term is gravitational focusing. For the rough Newtonian estimate in the question, assume capture when the pericentre reaches the Schwarzschild radius, $r_S=2G{M}_{\mathrm{BH}}/c^2$. It follows thatThe approximation keeps the focusing term because $v\ll c$.

Let ${\rho }_{\ast }=m{n}_{\ast }$ be the stellar mass density. Multiplying the geometric collision cross-section by the incoming mass flux gives the collisionless black-hole capture rateFor a supersonic gas with negligible sound speed, Bondi--Hoyle--Lyttleton accretion has capture radius $r_a=2G{M}_{\mathrm{BH}}/v^2$ and rateConsequentlyFor comparable densities and $v=200\mathrm{km}{\mathrm{s}}^{-1}$, the suppression is approximately $4\times 10^{-7}$. Gas can collide, dissipate orbital energy, and be captured from a region of size $GM/v^2$; collisionless particles generally leave on an unbound orbit unless their angular momentum is small enough for direct capture. Chandrasekhar dynamical friction can slow the hole by transferring energy to a stellar wake, but does not mean that the stars producing that wake are swallowed.

There is an important accuracy limitation in the requested numerical coefficient. Newtonian orbital mechanics evaluated at a relativistic event horizon cannot give the exact capture boundary. In Schwarzschild spacetime, a slowly incoming massive particle has dimensionless energy $\mathcal{E}\simeq 1$ and timelike geodesic effective potentialThe double turning point satisfying $V=1$ and $dV/dr=0$ occurs at $r=4r_g$ and $\ell =4G{M}_{\mathrm{BH}}/c$. The exact low-speed boundary is therefore $b_{\mathrm{cap}}\simeq 4G{M}_{\mathrm{BH}}/(cv)$, twice the rough value; it makes the rate ratio $4({\rho }_{\ast }/{\rho }_g)(v/c{)}^2$. The key suppression scaling is unchanged.

Direct capture of ordinary stars or dark matter is therefore an inefficient route to rapid growth of a supermassive black hole under these conditions. Tidal disruption events can capture stellar debris without swallowing an intact point-like star, and special dense environments can modify the supply. These qualifications do not make collisionless capture equivalent to dissipative gas accretion.

Interpret $\dot{m}$ as the rest-mass supply rate through a coherent disc; the hole's gravitating mass grows more slowly because radiation carries away binding energy. Initially the hole is a Schwarzschild black hole, whose innermost stable circular orbit is at $R_{\ast }=6G{M}_0/c^2$. The relativistic orbital constants there areThey follow by extremizing the timelike geodesic effective potential for a circular orbit and imposing marginal stability. Assume negligible torque inside the inner edge, so the gas carries this specific angular momentum into the hole. Its initial spin-up torque isfor $M_0=10^8{M}_{\odot }$ and $\dot{m}=0.1M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$. In SI units this is $9.7\times 10^{41}\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-2}$. A purely Newtonian estimate $\ell \sim \sqrt{G{M}_0{R}_{\ast }}$ gives a similar order of magnitude, but the relativistic value is appropriate at the innermost stable circular orbit.

The maximal Kerr black hole angular momentum at the initial mass is $J_{\max ,0}=G{M}_0^2/c\simeq 8.8\times 10^{64}\mathrm{g}\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. Holding both that mass and the initial torque fixed would giveInitially ${\dot{M}}_{\mathrm{BH}}=E_{\mathrm{ISCO}}\dot{m}\simeq 0.094M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$, so the corresponding frozen-coefficient estimate of gravitating mass gain is about $2.7\times 10^7{M}_{\odot }$. These are useful initial spin-growth scales, not a self-consistent time and mass increase for reaching extremality: the target angular momentum itself grows as $M^2$, and the inner orbit changes as the spin increases.

A consistent endpoint estimate is obtained from black-hole spin-up by coherent accretion. Let $a_{\ast }=cJ/(G{M}^2)$ and $r=R_{\mathrm{ISCO}}{c}^2/(GM)$, where $M$ is the current mass. On the prograde Kerr black hole inner-orbit branch, $1\le r\le 6$, the orbital constants may be writtenIgnoring photon capture, an accreted rest mass $d{m}_0$ changes the hole by $dM=Ed{m}_0$ and $dJ=\ell d{m}_0$. Differentiating the spin definition givesSubstitution of the orbital constants gives $dr/d\log M=-2r$, henceAt the formal extremal limit $r=1$, the result isFor constant rest-mass supply, use $r=6(M_0/M{)}^2$ to integrate the elapsed time:Thus the idealized coherent-disc endpoint isIf the quoted $0.1M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$ were instead the actual gravitating mass-growth rate, the corresponding endpoint time would be $\Delta M_{\mathrm{BH}}/{\dot{M}}_{\mathrm{BH}}\simeq 1.45\times 10^9\mathrm{yr}$. The two rate conventions cannot be interchanged.

The exact extremal endpoint is an idealization. Capture of disc photons exerts a counteracting torque and gives the Thorne spin limit, $a_{\ast }\simeq 0.998$, for the standard radiatively efficient thin-disc model. Therefore physical disc accretion approaches a large spin below unity rather than producing an exactly extremal hole. Thorne's original spin-evolution calculation describes that correction. The independently integrated expressions above explain why the frozen-mass estimate is too short for the ideal endpoint.

The qualitative conclusion is robust: a hole that gains most of its mass through a persistent, coherently corotating Shakura--Sunyaev thin disk is expected to have substantial black-hole spin. Very low spins require a different angular-momentum history, such as short episodes with changing orientations, counterrotating accretion, or mergers. Coherent feeding, not merely the existence of a disc during each episode, is the decisive assumption.

The Soltan argument offers a population-level observational test. From the bolometric luminosity function of active galactic nuclei, infer the total energy radiated per comoving volume over cosmic history,Compare it with the increase $\Delta {\rho }_{\mathrm{BH}}$ in supermassive black hole mass density attributed to radiative accretion. For an effective radiative efficiency of black-hole accretion ${\eta }_{\mathrm{eff}}$,In a zero-torque relativistic thin disc, $\eta =1-E_{\mathrm{ISCO}}$ rises from about $0.057$ at zero spin to $0.423$ in the photon-free prograde extremal limit. The measured effective efficiency therefore constrains population spin models under the thin-disc assumption. Soltan's original population argument explains how integrated quasar light constrains accumulated black-hole mass.

This is an accreted-rest-mass-weighted efficiency constraint, not a direct measurement of the arithmetic mean spin of every black hole. Obscured emission, bolometric corrections, seed masses, nonradiative growth, and disc orientation introduce uncertainty, and the efficiency-spin relation is nonlinear. These qualifications are essential when interpreting an inferred average spin for the whole population.

This mode is most plausible for a nearly stationary supermassive black hole immersed in the hot, pressure-supported, X-ray-emitting atmosphere of a massive elliptical galaxy. Such gas is comparatively smooth, its random and rotational speeds can be smaller than its sound speed, and radiative cooling can be slow on the inflow scale.

The ideal Bondi accretion assumptions can fail in several independent ways. The gas may carry enough specific angular momentum to circularize into an accretion disk; a galaxy's stars and dark matter can dominate the gravitational potential outside the black hole's sphere of influence; the hole can move relative to the gas; cooling, conduction, or external heating can destroy adiabaticity or isothermality; a multiphase or clumpy medium is not a uniform reservoir; magnetic fields, turbulence, and viscosity add stresses absent from the spherical model; and AGN feedback, winds, or jets can expel or recirculate the inflow. Time dependence and self-gravity provide further failures. Thus a Bondi estimate is best interpreted as an idealized supply rate, not automatically as the rate crossing the event horizon.

The Soltan argument compares the time-integrated luminosity density of the cosmological AGN population with the present comoving mass density in dormant supermassive black holes. If $U_{\mathrm{AGN}}$ is the emitted energy density corrected for obscuration and bolometric output, accretion with population-averaged efficiency $\bar{\eta }$ predictsIn practice $U_{\mathrm{AGN}}$ comes from integrating AGN luminosity functions over luminosity and cosmological redshift, with corrections for obscured sources and missed wavebands, while ${\rho }_{\mathrm{BH}}$ is inferred from local galaxy--black-hole scaling relations.

The inferred efficiency is of order the canonical thin-disk value, about ten per cent, so most cosmic black-hole mass was accumulated in radiatively efficient, optically thick accretion episodes. Radiatively inefficient flows can dominate low-luminosity activity or brief extreme phases, and mergers redistribute existing mass, but neither naturally accounts for the observed integrated AGN radiation while supplying most of the final mass.