Astrophysical black hole

An astrophysical black hole is a black hole studied through its formation, accretion, mergers, dynamical environment, and electromagnetic or gravitational-wave signatures.

The black-hole sphere of influence is a characteristic radius where the central black hole appreciably changes surrounding stellar dynamics. Equating its Newtonian gravitational potential scale $G{M}_{\bullet }/r$ to the ambient squared velocity dispersion gives $r_{\inf }\simeq G{M}_{\bullet }/{\sigma }^2$. Dynamical mass measurements depend on angular resolution $r_{\inf }/D$, tracer brightness, and degeneracies in the orbital distribution, not solely on black hole mass.

Black-hole spin is the angular momentum of a black hole, commonly expressed by the dimensionless Kerr parameter $a_{\ast }=cJ/(G{M}^2)$. It changes the innermost stable circular orbit, accretion efficiency, and relativistic frame dragging.

Coherent prograde gas accretion increases the angular momentum of a black hole. For a zero-torque thin disc, the rest mass $d{m}_0$ entering from the innermost stable circular orbit changes the hole by $dM=E_{\mathrm{ISCO}}d{m}_0$ and $dJ={\ell }_{\mathrm{ISCO}}d{m}_0$. Its dimensionless spin obeys $d{a}_{\ast }/d\log M=c{\ell }_{\mathrm{ISCO}}/(GM{E}_{\mathrm{ISCO}})-2a_{\ast }$. Starting from zero spin and neglecting captured radiation, the formal extremal endpoint has $M_f=\sqrt{6}{M}_i$ and requires rest-mass supply approximately $1.8464M_i$. Photon capture leads instead to the Thorne spin limit.

The Thorne spin limit is the equilibrium spin produced by competition between coherent thin-disc matter accretion and capture of disc photons by a Kerr black hole. Preferential capture of counterrotating photon angular momentum prevents the standard radiatively efficient model from reaching $a_{\ast }=1$. The commonly quoted limit is approximately $0.998$, with some dependence on emission and accretion assumptions. Thorne's original calculation derives this result.

Black-hole accretion converts gravitational binding energy of inflowing matter into heat, radiation, outflows, and growth of the black hole.

Collisionless matter approaching a black hole generally cannot dissipate enough orbital energy to remain bound, so direct capture requires sufficiently small angular momentum. For nonrelativistic incident speed $v$, a Schwarzschild black hole has direct test-particle capture cross-section approximately $16\pi G^2{M}^2/(c^2{v}^2)$. A Newtonian pericentre estimate using the Schwarzschild radius instead gives $4\pi G^2{M}^2/(c^2{v}^2)$. Both show that capture at a given mass density is suppressed by order $(v/c{)}^2$ relative to supersonic Bondi--Hoyle--Lyttleton accretion. Tidal disruption events provide a distinct way to accrete some stellar debris.

For a singular isothermal sphere of dispersion $\sigma$ and gas fraction $f_g$, placing all gas inside radius $R$ onto a hole in time $R/\sigma$ gives ${\dot{M}}_{\mathrm{dyn}}\sim 2f_g{\sigma }^3/G$. This is an optimistic supply bound that omits angular momentum transport, star formation, and active-galactic-nucleus feedback. A numerical comparison with the Eddington accretion rate requires a gas fraction, reference radiative efficiency, opacity, and crossing-time convention.

The inflow time is the characteristic time for accreting matter to move radially by its current radius, $t_{\mathrm{in}}\sim R/|u_R|$.

The Paczyński-Wiita potential is a Newtonian pseudo-potential that reproduces the Schwarzschild innermost stable circular orbit at $r=3r_S$.

For $\beta =16c_s^2/c^2$, the exterior sonic point of the barotropic spherical accretion equation lies outside $r_G=2GM/c^2$. Its mass density is ${\rho }_c={\rho }_{\infty }\exp [4/(1+\sqrt{1+\beta })-1/2]$. The selected smooth transonic accretion rate is $4\pi r_c^2{\rho }_c{c}_s$. Subsonic formal solutions are not assigned this rate without transonic selection.

Let $r_S=2GM/c^2$. In the Paczyński-Wiita potential, circular orbit balance and the radial epicyclic frequency formula giveThus the innermost stable circular orbit lies at $r=3r_S$. Its specific orbital energy is $-c^2/16$, giving a zero-torque radiative efficiency of black-hole accretion of $1/16$. The model reproduces the Schwarzschild inner orbit exactly but approximates its relativistic energy and frequency. Abramowicz's derivation explains the potential's relation to the Schwarzschild effective potential.

The limiting inward departure from the marginal Paczyński-Wiita circular orbit at $r=3a$ conserves the displayed specific angular momentum and specific orbital energy. In the Paczyński-Wiita potential, its inward speed satisfies $v_r^2=GM(3a-r{)}^3/[4a{r}^2(r-a)]$. The exactly marginal circular orbit does not depart without a perturbation; the formula describes the separatrix limit of inward releases.

For a steady accretion disk with arbitrary rotation law and a zero-torque inner boundary condition at $R=3R_S$, the outward viscous torque in an accretion disk is $\mathcal{G}=\dot{M}(j-j_{\mathrm{in}})$. The dissipation per unit planar area summed over both disk faces is $D=-\mathcal{G}{\Omega }^{\prime }/(2\pi R)=\nu \Sigma (R{\Omega }^{\prime }{)}^2$. Integrating $2\pi RD$ gives $L=\dot{M}GM/(8R_S)=\dot{M}{c}^2/16$ in the Paczyński-Wiita potential. The radiative flux from one face is $D/2$, preventing a factor-of-two error in the luminosity.

The radiative efficiency is $\eta =L/(\dot{M}{c}^2)$. For a zero-torque thin disk it is $1-E_{\mathrm{ISCO}}$, the binding energy per unit rest mass released before matter crosses the innermost stable circular orbit.

The maximal radiative efficiency is the available binding energy per unit accreted rest-mass energy when all energy released outside the capture boundary escapes as radiation. A relativistic zero-torque disk has ${\eta }_{\max }=1-E_{\mathrm{ISCO}}$.

The innermost stable circular orbit is the smallest stable circular timelike orbit around a black hole. Its radius and orbital energy depend on black-hole spin and on whether the orbit is prograde or retrograde.

Bondi accretion is steady spherical accretion from a uniform gas reservoir onto a compact gravitating object. In its adiabatic form, compression raises internal energy without providing the irreversible shear heating of a radiatively efficient disk.

For isentropic spherical accretion with this adiabatic index, define $\lambda =\dot{M}{c}_{\infty }^3/(4\pi G^2{M}^2{\rho }_{\infty })$ and $x=r{c}_{\infty }^2/(GM)$. The steady-flow equation is $\sqrt{\lambda }[{\mathcal{M}}^{3/2}/2+3{\mathcal{M}}^{-1/2}/2]=1+3x/2$. The bracket has minimum two at Mach one. Reaching arbitrarily small positive radii therefore requires $\lambda \le 1/4$. For smaller rates the outer-rest branch remains subsonic; at the maximal rate it approaches Mach one only as $r\to 0$, not at a finite sonic point. Supersonic outer branches fail the specified resting finite-density condition, and larger rates terminate at a positive minimum radius. An additional inner boundary prescription is needed to select a unique rate among the globally defined subsonic solutions.

Relativistic spherical accretion is steady radial gas inflow in Schwarzschild spacetime, with conserved rest-mass flux and relativistic Bernoulli function. A regular solution passes a finite sonic point outside the event horizon. For an isentropic gas with $\gamma =5/3$ and cold asymptotic sound speed $c_{\infty }\ll c$, the critical radius is approximately $3GM/(4c{c}_{\infty })$ and the leading mass accretion rate is $\pi G^2{M}^2{\rho }_{\infty }/c_{\infty }^3$. Relativistic corrections resolve the zero-radius critical point of the Newtonian $\gamma =5/3$ model without eliminating accretion.

Bondi--Hoyle--Lyttleton accretion is gravitational capture by a compact object moving through a gas. In the pressureless supersonic limit, ballistic streamlines focus into a downstream wake.

In a fixed central Newtonian gravitational potential, pressureless free fall has speed $u=a{R}^{-1/2}$, and randomly moving stars have typical gas-relative speed $V=b{R}^{-1/2}$. A steady, undepleted spherical mass flux $A$ gives $\rho =A{R}^{-3/2}/(4\pi a)$. The Bondi--Hoyle--Lyttleton accretion rate therefore becomes $\dot{M}=\lambda M^2$ with $\lambda =G^{2}A/(a{b}^3)$, independent of radius. Integrating $d(1/M)/dt=-\lambda$ gives the displayed solution before $\lambda M_{0}t=1$. Its finite-time divergence signals failure of the fixed potential, undepleted gas supply and negligible feedback assumptions; it is not a physically unlimited mass supply.

If stars are neither created nor lost and all follow quadratic accretion in a central Keplerian potential, conservation of number gives $F(M,t)dM=F_0(M_0)d{M}_0$. Inverting the trajectory gives $M_0=M/(1+\lambda tM)$ and its Jacobian determinant is $(1+\lambda tM{)}^{-2}$. This proves the displayed mass-density-in-mass-space transformation. In a broad high-mass range with $\lambda tM\gg 1$, whose inverse images remain in a narrow interval where $F_0$ varies little, the power law is $F\propto M^{-2}$. A strictly monochromatic initial population remains monochromatic. Finite initial support imposes a finite upper cutoff before the earliest formal blow-up, so the power law is an intermediate asymptotic subject to the accretion model's limits.

In a cold pressureless stream of asymptotic speed $v$, conservation of specific angular momentum and the eccentricity vector makes a streamline at impact parameter $d$ cross the downstream axis at $b=d^2{v}^2/(2GM)$. Its axial velocity there remains $v$, while opposing transverse velocities collide in a shock.

If transverse kinetic energy is dissipated at the downstream axis, the remaining specific orbital energy is $v^2/2-GM/b$. Binding requires $b<2GM/v^2$, corresponding to incident impact parameter $d<2GM/v^2$. Integrating the captured mass flux gives the Bondi--Hoyle--Lyttleton accretion rate.

In the ballistic supersonic limit, streamline collisions bind gas with impact parameter $b<2GM/v_{\infty }^2$, giving

Isothermal Bondi accretion has $p=c_s^2\rho$ with constant sound speed. Its regular transonic solution crosses the Bondi sonic point and approaches free fall near the accretor.

The regular spherical isothermal inflow from mass density ${\rho }_{\infty }$ at rest at infinity has sonic mass density $e^{3/2}{\rho }_{\infty }$. Combining it with the Bondi sonic point and mass conservation gives $\dot{M}=\pi e^{3/2}{G}^2{M}^2{\rho }_{\infty }/c_s^3$.

For inward speed $u>0$, steady spherical mass conservation and radial momentum balance give

With $x=r/r_s$ and $\mathcal{M}=|u_r|/c_s$, the radial isothermal equation integrates to ${\mathcal{M}}^2/2-\ln \mathcal{M}=2/x+2\ln x+C$. A regular transonic branch has $C=-3/2$, and the reservoir-fed accretion branch crosses $(x,\mathcal{M})=(1,1)$ with slope $-1$.

The isothermal Bondi sonic point is $r_s=GM/(2c_s^2)$, where the regular transonic solution has $u=c_s$.

At the critical point, both the coefficient of $du/dr$ and the remaining right-hand side of the Isothermal Bondi equation vanish, allowing a finite derivative through the sonic transition.

The mass accretion rate is the mass entering an accretor per unit time. In a steady spherical flow, $\dot{M}=4\pi r^2\rho u$ is independent of radius.

For a background gravitating density ${\rho }_{\mathrm{dm}}={\Psi }_0{r}^{-5/2}$, the potential is $\Phi =-16\pi G{\Psi }_0{r}^{-1/2}$. A steady polytropic inflow has a regular sonic point only for $1<\gamma <9/7$, and its accretion rate scales as $\dot{M}\propto {\rho }_0{G}^4{\Psi }_0^4{c}_0^{-7}$.

The Newtonian free-fall speed from rest at infinity toward a point mass is $v_{\mathrm{ff}}=\sqrt{2GM/r}$.

For an adiabatic gas with $\gamma =5/3$, asymptotic density ${\rho }_0$, and sound speed $v_{s0}$ accreting onto a Newtonian point mass $M$, the largest globally subsonic steady rate is

A radiatively inefficient accretion flow carries a substantial fraction of locally generated energy with the gas instead of radiating it before accretion. It can occur at low density because emission is weak or at high optical depth because photons are trapped.

An advection-dominated accretion flow is a hot, geometrically thick, radiatively inefficient flow in which most dissipated energy remains in the gas and is carried inward rather than radiated locally.

A two-temperature accretion flow has ions and electrons at substantially different temperatures because their Coulomb equilibration time exceeds the inflow time. Ions can retain most viscous heat while electrons, which radiate more effectively, remain cooler.

The electron--proton thermal equilibration time is the timescale on which Coulomb collisions exchange energy between electron and proton populations. Its inverse dependence on number density helps make dilute accretion flows two-temperature and radiatively inefficient.

A slim accretion disk is an optically thick, moderately geometrically thick flow at a high accretion rate. Its inflow time can be shorter than its photon-diffusion time, so radial advection carries trapped radiation inward.

A slim accretion disk is optically thick and retains an important inward advective heat flux. It is distinct from the optically thin version of a radiatively inefficient accretion flow.

Photon trapping occurs when the radiative diffusion time across an optically thick flow exceeds its inward advection time. The trapped radiation is carried inward, reducing the emergent radiative efficiency.

The photon-trapping radius is where the outward radiative diffusion time equals the inward inflow time. Radiation generated inside it is predominantly advected toward the accretor.

A luminous hot accretion flow has radiative cooling greater than local viscous heating, so compressive inward advection supplies the difference. In the convention $q_{\mathrm{adv}}=q^{+}-q^{-}$, it has $q_{\mathrm{adv}}<0$.

The accretion-flow advection balance compares local volumetric heating $q^{+}$, radiative cooling $q^{-}$, and energy transport with the gas. Positive $q_{\mathrm{adv}}$ means advection removes locally generated heat; zero gives local radiative balance; negative means advection supplies heat that is radiated locally.

The Eddington ratio is the luminosity divided by the Eddington luminosity. At fixed radiative efficiency it also measures the accretion rate relative to the Eddington rate.

An Eddington accretion rate converts the Eddington luminosity into a mass supply rate using an adopted reference efficiency ${\eta }_0$: ${\dot{M}}_{\mathrm{Edd}}=L_{\mathrm{Edd}}/({\eta }_0{c}^2)$. The common choice ${\eta }_0=0.1$ gives ${\dot{M}}_{\mathrm{Edd}}=10L_{\mathrm{Edd}}/c^2$.

The Salpeter time is the e-folding time for Eddington-limited black-hole growth. If $t_E={\sigma }_{T}c/(4\pi G{m}_p)$, then $t_{\mathrm{Sal}}=\eta t_E/[(1-\eta )f_{\mathrm{Edd}}]$.

An intermediate-mass black hole has a mass between stellar remnants and supermassive black holes, approximately $10^2$--$10^5{M}_{\odot }$.

A supermassive black hole has a mass of roughly $10^6{M}_{\odot }$ or more and commonly occupies the nucleus of a massive galaxy.

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.

The black-hole--bulge mass relation is the empirical correlation between a galaxy's central black-hole mass and the stellar mass of its spheroidal component. Locally, the black hole is typically of order $10^{-3}$--$10^{-2}$ of the bulge mass, with substantial scatter and possible redshift evolution.

An active galactic nucleus is a compact luminous galactic nucleus powered primarily by accretion onto a supermassive black hole.

A quasar is an extremely luminous active galactic nucleus whose compact central emission can be observed across cosmological distances.

A quasar light curve records its measured brightness as a function of time. Intrinsic accretion variability, measurement noise, and gravitational lensing can all contribute structure.

Active-galactic-nucleus feedback transfers radiative or mechanical energy and momentum from black-hole accretion to surrounding galactic gas.

Heating a gas reservoir reduces its Bondi accretion rate because ${\dot{M}}_B\propto {\rho }_{\infty }{c}_s^{-3}$. At fixed density, increasing the sound speed by ten reduces the rate by about a thousand and the gravitational capture radius by a hundred. Expansion and density loss can strengthen this negative feedback. Neither the density nor the cold pre-heating accretion rate is determined by the temperature change alone.

An active-galactic-nucleus wind bubble forms when a fast nuclear wind shocks against galactic gas. From inside outward it contains free wind, a reverse shock, shocked wind, a contact discontinuity, a swept-up shocked shell, a forward shock, and undisturbed interstellar gas.

An outflow is momentum driven when shocked driving material cools rapidly, so it cannot retain enough thermal energy to inflate a pressure-supported bubble. The swept shell then receives approximately the direct wind momentum flux, often of order $L/c$.

An outflow is energy driven when cooling of the shocked wind is slow. Its hot bubble stores mechanical energy and expands by pressure work on the swept-up shell.

A momentum boost is a shell momentum flux exceeding the direct input momentum flux. An energy-conserving wind of speed $v_w$ driving a shell of speed $v_s\ll v_w$ can give ${\dot{p}}_s/(L/c)$ of order $v_w/(2v_s)$.

A dusty radiation-pressure-driven shell is swept up when ultraviolet radiation from an active nucleus is absorbed by dust and reradiated in the infrared. Single scattering supplies up to roughly $L/c$ of momentum flux, while trapped infrared photons can enhance it by the infrared optical depth.

A fixed-mass shell with ultraviolet opacity ${\kappa }_{\mathrm{UV}}$ has dust transparency radius $R_{\mathrm{UV}}=\sqrt{{\kappa }_{\mathrm{UV}}{M}_{\mathrm{sh}}/(4\pi )}$. For an initially expanding shell driven by constant ultraviolet luminosity, a sharp opaque-to-transparent approximation givesHere ${\Gamma }_{\mathrm{ss}}=L{R}_0^2/(GM{M}_{\mathrm{sh}}c)$ is the initial single-scattering force ratio and ${\Gamma }_{\mathrm{IR}}={\kappa }_{\mathrm{IR}}L/(4\pi GMc)$. The trajectory must actually reach $R_{\mathrm{UV}}$. Large initial ultraviolet optical depth enhances the radiative work at fixed initial single-scattering ratio; the same conclusion does not follow unconditionally at fixed electron-scattering luminosity.

The critical luminosity of a dusty shell is the luminosity at which radiation pressure balances gravity. For shell mass $M_s$, enclosed gravitating mass $M$, and ultraviolet and infrared optical depths,

For a spherical swept shell of mass $M_s(R)$, radial speed $\dot{R}$, driving force $F_d$, and enclosed gravitating mass $M(<R)$,The derivative includes momentum needed to accelerate newly swept ambient mass.

The dust transparency radius is where a swept shell's ultraviolet optical depth equals one. For a gas fraction $f_g$ in a singular isothermal sphere, $R_{\tau }={\kappa }_{\mathrm{UV}}{f}_g{\sigma }^2/(2\pi G)$.

The Soltan argument compares the cosmological energy density radiated by active galactic nuclei with the present mass density in supermassive black holes. Their ratio constrains the mass-weighted mean radiative efficiency of black-hole growth.

A tidal disruption event occurs when a star passes close enough to a black hole that differential gravity tears it apart outside the capture boundary, producing returning debris and often a luminous flare.

For a star of mass $M_{\ast }$ and radius $R_{\ast }$ near a black hole of mass $M_{\mathrm{BH}}$, the Newtonian tidal disruption radius is $r_t\simeq R_{\ast }(2M_{\mathrm{BH}}/M_{\ast }{)}^{1/3}$ under a peak-to-center tidal-acceleration convention.

The Hills mass is the black-hole mass above which a given star crosses the capture boundary before being tidally disrupted. For a Schwarzschild capture radius $2G{M}_{\mathrm{BH}}/c^2$, it is $c^3{R}_{\ast }^{3/2}/(2G^{3/2}{M}_{\ast }^{1/2})$ with the corresponding tidal-radius convention.

A black-hole seed is an early black hole from which a later massive black hole can grow by accretion and mergers.

A Population III remnant seed forms when a massive metal-free star collapses, typically leaving a light seed of order $10$--$10^2{M}_{\odot }$ depending on stellar mass loss and pair-instability effects.

A runaway stellar-collision seed forms when mass segregation and repeated mergers in a dense young cluster create a very massive star that collapses to a black hole of roughly $10^2$--$10^4{M}_{\odot }$.

A direct-collapse black-hole seed forms when rapidly inflowing primordial gas avoids fragmentation, builds a supermassive star or quasistar, and collapses to a seed of roughly $10^4$--$10^6{M}_{\odot }$.