Astrophysical black hole

An astrophysical black hole is an astronomical object interpreted as a black hole through evidence about its mass, compactness and interaction with surrounding matter or gravitational waves. A black hole of stellar mass can form after stellar core collapse; a supermassive black hole can grow through black-hole accretion and mergers. The event horizon prevents light from escaping from within, but matter outside can radiate and reveal the central gravity. For Sagittarius A*, astrometry and radial velocities of nearby stars constrain a very large mass inside their small orbits. Such dynamical evidence complements observations of accretion and of binary black holes, which probe different aspects of the black-hole interpretation.^[ref]

GRS 1915+105 is an X-ray binary containing an accreting stellar-mass black hole. Gas supplied by its companion forms an accretion disk; changes in that gas produce strong, variable X-ray emission. Its radio-emitting jets make it an example of a microquasar.

The thermal disk spectrum has been used to estimate the black-hole spin by fitting a relativistic disk model. The inference depends on quantities including mass, distance, inclination and the disk's physical state. It therefore illustrates how a bright spectrum constrains a compact object through a model, rather than providing a direct measurement of the event horizon.^[ref]

An inference of a compact object's gravitating mass from specified tracers and a dynamical or radiation model. Resolved stellar or gas orbits, spectroscopic binary dynamics, and reverberation mapping supply complementary approaches with different distance, geometry and equilibrium assumptions. A short variability timescale constrains size but is not a unique mass measurement.

A binary black hole is a binary system whose two components are black holes. Gravitational waves carry away orbital energy and angular momentum, gradually reducing the separation; sufficiently close systems progress through inspiral, merger and the settling of the final black hole. The increasing orbital angular velocity produces a signal whose gravitational-wave frequency rises, called a chirp. In GW150914, two LIGO interferometers recorded a signal sweeping from about 35 to 250 hertz in September 2015. Comparing that measured gravitational wave strain with general relativity waveforms inferred the component masses and the merger's properties. This permits the study of systems that need no luminous surrounding gas to be detected.^[ref]

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.

An inference of black-hole angular momentum, usually quoted through $a_{\ast }$. Accretion-based techniques use the dependence of the stable inner orbit and relativistic radiation transfer on spin in the Kerr black hole model. Disk truncation, inclination, illumination and atmosphere assumptions can imitate a change of spin and must enter the uncertainty model.

An inference from radiation reflected by an irradiated accretion disc, including iron-line and continuum features. Doppler effects, gravitational redshift and ray propagation determine its profile. Associating the innermost reflecting radius with the innermost stable circular orbit can constrain spin, conditional on illumination, ionization, abundance, inclination and disk structure.

A fit of a thermal accretion-disk spectrum that infers the inner emitting radius and identifies it with the innermost stable circular orbit. A relativistic thin-disk model requires independently constrained mass, distance and inclination, together with spectral hardening and absorption. Use of a suitable disk-dominated state is an essential physical check, not merely a statistical cut.

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 is the inflow of matter onto a black hole, converting some of its gravitational binding energy into heat, electromagnetic radiation and outflows. Matter with substantial angular momentum generally forms an accretion disk; internal stresses transfer angular momentum outward so that matter can move inward. The released energy can be radiated before the material crosses the event horizon. In a thin-disk model the innermost stable circular orbit marks the transition toward rapid plunging motion, although the detailed inner flow depends on the accretion state.^[ref]

The relation $L=\eta \dot{M}{c}^2$ defines the radiative efficiency of black-hole accretion, where $L$ is emitted luminosity, $\dot{M}$ the mass accretion rate and $c$ the speed of light. A radiatively inefficient accretion flow carries much of its available energy inward or into outflows instead of emitting it, so luminosity alone need not measure the mass supply.

An oscillatory feature with finite width in a source's variability power spectrum, rather than a perfectly periodic clock. Orbital, precession and disc-mode interpretations connect different frequencies to the same compact object. Identifying an observed peak with an innermost stable circular orbit orbital frequency is therefore a model assumption, not a direct black-hole mass or spin measurement.

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.

For equatorial test-particle circular orbits in the extremal-spin limit, the prograde Kerr black hole innermost stable circular orbit has Boyer-Lindquist radius tending to $GM/c^2$ and specific energy tending to $1/\sqrt{3}$. The retrograde limit has radius $9GM/c^2$ and specific energy $5/(3\sqrt{3})$. The prograde radius approaching the horizon coordinate does not make the massive limiting orbit identical to the null horizon generator.^[ref]

The Schwarzschild coordinate rotation frequency at the marginally stable circular orbit, measured per unit time of a stationary clock at infinity. It scales inversely with black-hole mass. A finite-radius static clock adds a gravitational time-dilation factor; a quasi-periodic brightness signal need not directly equal this orbital frequency.

For ideal zero-torque equatorial thin discs, efficiency is $1-E_{\mathrm{ISCO}}$. The formal extremal retrograde, Schwarzschild and extremal prograde limiting radii are respectively $9,6,1$ in units $GM/c^2$, giving efficiencies $1-\sqrt{25/27}$, $1-\sqrt{8/9}$ and $1-1/\sqrt{3}$. Photon capture and non-ideal accretion can change the realized efficiency.

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.

The thermal sphere-of-influence scale $r_B=GM/c_s^2$ compares gravitational potential per unit mass with the ambient gas sound-speed squared. A convention based on escape speed instead introduces a factor of two. The scale is not a capture cross-section or an accretion rate without a flow and dissipation model.

With planet-to-star mass ratio $q$ and disk aspect ratio $h$, $r_B/a=q/h^2$ and $r_H/a=(q/3{)}^{1/3}$. Equating the two gives $q=h^3/\sqrt{3}$. The thermally limited radius grows faster with planet mass than the tidally limited radius; an envelope outer scale is often the smaller of the two.

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.

For steady Isothermal Bondi accretion from gas at rest at infinity, define $\lambda =\dot{M}{c}_s^3/(4\pi G^2{M}^2{\rho }_{\infty })$, $x=r/r_s$ and $y=u/c_s>0$, where $r_s=GM/(2c_s^2)$. The Bernoulli function and mass conservation give $y^2/2-\ln y=2\ln x+2/x-\ln (4\lambda )$. The left side has minimum $1/2$ at $y=1$, and the radius-dependent part on the right has minimum two at $x=1$. Existence over all positive radii therefore requires the displayed bound. Equality permits the smooth accelerating-inward transonic branch and gives the usual isothermal Bondi accretion rate. Smaller positive rates have an everywhere-subsonic reservoir-connected branch, and zero rate gives a hydrostatic atmosphere. An absorbing inner boundary inside the sonic radius selects the critical inflow; steady isothermal equations and the outer density alone do not specify that inner condition.

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.

A steady full-sphere radial inflow has mass flux $\dot{M}=4\pi r^2\rho |v_r|$ by conservation of mass. With Newtonian free-fall speed from rest at infinity, $|v_r|=\sqrt{2GM/r}$, this becomes the displayed expression. The assumed velocity then requires $\rho \propto r^{-3/2}$ along the steady flow. A local density and speed do not determine the total mass accretion rate of a general nonspherical inflow, and this estimate does not impose the sonic boundary condition of Bondi accretion.

Consider steady spherical accretion of a gas obeying a polytropic equation of state in isentropic flow, with fixed background potential $\Phi =-A{r}^{-1/2}$, where $A=16\pi G{\Psi }_0$, with reservoir mass density ${\rho }_0$ and sound speed $c_0$ at infinity. At a regular sonic point, radial momentum balance requires $c_c^2=A/(4\sqrt{r_c})$. The Bernoulli function isSetting $u=c_s=c_c$ at the sonic point givesA finite positive critical sound speed therefore requires $1<\gamma <9/7$. Mass conservation gives $\dot{M}=4\pi r_c^2{\rho }_c{c}_c$, while the constant-entropy gas law gives ${\rho }_c/{\rho }_0=(c_c^2/c_0^2{)}^{1/(\gamma -1)}$. Since $r_c\propto A^2{c}_0^{-4}$ at fixed $\gamma$, these equations yield $\dot{M}\propto {\rho }_0{G}^4{\Psi }_0^4{c}_0^{-7}$. The steep sound-speed dependence comes from both the selected capture scale and the critical flow speed. The background potential is prescribed; gas self-gravity and finite outer boundaries require a different model.

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 ${\dot{M}}_{\mathrm{Edd}}=10L_{\mathrm{Edd}}/c^2$.

For radiatively efficient release of Newtonian surface binding energy, $L\sim GM\dot{M}/R_{\ast }$ and $L_E=4\pi GMc/\kappa$ imply the displayed rate. Its efficiency is $\eta \sim GM/(R_{\ast }{c}^2)$, so at fixed opacity the maximum rate scales with stellar radius. This differs from defining an Eddington supply rate with an arbitrarily fixed reference efficiency. Geometry, advection and mass loss can modify the simple limit.

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 $t_{\mathrm{Sal}}=\eta t_E/[(1-\eta )f_{\mathrm{Edd}}]$.

For fixed efficiency, growth from $M_i$ to $M_f$ within $\Delta t$ at no more than the Eddington luminosity requires $[\eta /(1-\eta )]t_E\log (M_f/M_i)\le \Delta t$. Solving gives an upper bound, because greater radiative efficiency reduces retained mass growth at a fixed luminosity. A duty cycle below unity tightens the requirement; mergers and super-Eddington episodes invalidate its stated premises.

At constant radiative efficiency of black-hole accretion $\eta$ and the Eddington luminosity, a black hole grows exponentially with the Salpeter time $t_S=\eta t_E/(1-\eta )$, where $t_E={\sigma }_{T}c/(4\pi G{m}_p)$. Its doubling time is $t_S\log 2$. The factor $1-\eta$ accounts for the supplied rest energy that escapes as radiation instead of increasing the hole's mass.

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.

Sagittarius A* is the compact radio source at the centre of the Milky Way. Nearby infrared stars allow its central mass concentration to be studied through their orbits.^[ref]

The useful idea is to weigh an unseen object through its effect on a visible tracer. A curved stellar orbit carries more information than a single fast-moving star: its changing position and velocity constrain a common centre, a length scale and a clock. The Galactic-centre stellar-orbit experiments combine astrometry with Doppler spectroscopy; the Galactic-centre point-mass orbit model below shows how the two measurements complement one another.

Consider a negligible-mass star moving in the stationary Newtonian gravitational potential of a point mass. Let its semi-major axis be $a$, orbital eccentricity $0\le e<1$, and orbital period $P$. Write $\mu =GM$ for the gravitational parameter. The eccentric anomaly $E$ is determined by Kepler's equation,where $\tau$ is the time of periapsis. Coordinates in the orbital plane, with the point mass at the origin, areThe focus is the central mass; the ellipse's geometric centre is displaced by $-ae$ along the first coordinate. The varying denominator makes the star move faster near periapsis, even though the mean anomaly increases uniformly.

Let $z$ increase away from the observer, with $x,y$ spanning the sky plane. A rotation $R=R_z(\Omega )R_x(i)R_z(\omega )$ sends the orbital-plane coordinates to $\mathbf{r}=R{\mathbf{r}}^{\prime }$. Here $i$ is orbital inclination, $\Omega$ the longitude of ascending node, and $\omega$ the argument of periapsis. For distance $D\gg a$, the small-angle model gives angular positions ${\theta }_x=x/D$, ${\theta }_y=y/D$ in radians. Doppler spectroscopy instead supplies the line-of-sight velocity $v_{\mathrm{los}}=\dot{z}+v_{\mathrm{sys}}$, with a fitted systemic offset $v_{\mathrm{sys}}$.

Define $\alpha =a/D$, the angular scale of the deprojected fitted orbit. It is not generally the semi-major axis of the apparent ellipse on the sky. Projection also need not send a true focus to a focus of the apparent ellipse. Fitting the rotated orbit preserves that distinction; measuring a bounding ellipse alone loses it.

Angular astrometry determines the combination displayed by substituting $a=D\alpha$ into Kepler's third law. Keeping the fitted shape, orientation and orbital period fixed, the rescalingleaves the entire angular time series unchanged. A longer physical orbit around a larger mass can trace precisely the same angular path. More angular samples improve the fitted combination; by themselves they cannot break this scaling symmetry of the model.

The radial-velocity semi-amplitude isFor $\sin i\ne 0$, combining it with the angular fit gives the orbital parallax distanceThe spectral velocity scale supplies the missing physical ruler. For a face-on orbit, $\sin i=0$, this radial signal vanishes and that method cannot supply a distance. In a partial or noisy data set, fitted orientation and systemic motion also correlate with the inferred scale.

Choose $D=7.5\mathrm{kpc}$, $\alpha =0.110$ arcseconds, $P=18$ years, $e=0.70$ and $i=55$ degrees. The small-angle conversion gives $a=825\mathrm{AU}$. Taking one model year as $365.25$ days gives $\mu \simeq 2.30\times 10^{26}{\mathrm{m}}^3{\mathrm{s}}^{-2}$ and $K\simeq 1566km/s$.

Dividing by the nominal solar mass parameter, $(GM{)}_{\odot }^{\mathrm{N}}=1.3271244\times 10^{20}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, gives $1.73\times 10^6$ nominal solar masses. The nominal parameter is an exact conversion convention, rather than an assertion that the Sun's measured mass is exact.^[ref]

The chosen periapsis radius is $r_p=a(1-e)=247.5\mathrm{AU}$. Its corresponding Schwarzschild radius, $r_s=2\mu /c^2$, is about $0.0342\mathrm{AU}$. Thus this illustrative orbit remains roughly $7200$ Schwarzschild radii from the centre at closest approach. Inferring a very concentrated mass from such an orbit is different from directly resolving its event horizon. Every number here follows from the chosen orbit, not from a fit to the actual Galactic-centre measurements.

For a bound Kepler orbit, specific orbital energy $v^2/2-\mu /r$ is negative. Therefore $\mu >v^{2}r/2$. A projected separation $R_{\perp }\le r$ and a systemic-motion-corrected line-of-sight speed $|v_{\mathrm{los}}|\le v$ give the displayed lower bound. One position and one spectral velocity already constrain the mass in this model; a fitted orbit gives much more. The bound assumes a bound orbit in a point-mass potential; it is not an unrestricted enclosed-mass formula for an arbitrary extended potential.

Rainer Schödel, Reinhard Genzel and collaborators combined NTT imaging with VLT adaptive-optics infrared observations. Silicon-monoxide masers registered stellar images to the radio position of Sagittarius A*. Their 2002 observations included S2's closest approach; a fitted Kepler orbit constrained a concentrated central mass. Rejecting a particular extended cluster also required its density and survival assumptions.^[ref]

Andrea Ghez and collaborators combined Keck speckle imaging and laser-guide-star adaptive optics with infrared spectra. Image fitting supplied angular positions; spectral lines supplied velocities. The 2008 analysis treated optical distortion, coordinate alignment and blended sources: a shifting photocentre can imitate motion. Joint angular and spectral fitting constrained both distance and mass, with assumptions about central radial motion affecting the result. Apparatus calibration and confusion tests are therefore part of the dynamical inference.^[ref]

A star used in the Galactic-centre stellar-orbit experiments. The two designations refer to the same object in the SIMBAD catalogue.^[ref]

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 the compact luminous centre of a galaxy powered mainly by black-hole accretion onto a supermassive black hole. Its radiation can dominate the host at some wavelengths even though the emitting region is small compared with the galaxy. Observers identify activity using combinations of emission lines, a compact radio wave source, energetic X-rays and brightness changes. Different components respond on different timescales: reverberation mapping measures the delay between changes in the central radiation and the response of surrounding line-emitting gas, giving a light-travel estimate of its distance. Combining that distance with the Doppler broadening of the emission lines constrains the central mass if the emitting gas is gravitationally bound. Activity can be episodic; the existence of a central black hole alone does not imply a luminous nucleus.^[ref]

Time-dependent inference of an emitting region's response to a variable illuminating continuum. For an active nucleus, a broad-line delay estimates a light-travel scale and a line width estimates a velocity scale. A virial mass estimate then requires a geometrical and kinematic factor $f_{\mathrm{geom}}$; the lag alone neither fixes the geometry nor establishes purely gravitational motion.

A blazar is an active galactic nucleus whose relativistic jet is viewed close to its direction of motion. Relativistic beaming enhances the apparent radiation from approaching material and changes observed variability timescales. Its extreme brightness is therefore affected by orientation as well as by the power supplied by black-hole accretion.

A source's brightness and spectrum do not directly identify the particles responsible for every emission component. Radio through gamma-ray observations and their timing help distinguish candidate mechanisms, while TXS 0506+056 illustrates the additional information supplied by a possible association with high-energy neutrinos.

TXS 0506+056 is a blazar studied through a gamma-ray flare associated in direction and time with the high-energy neutrino alert IceCube-170922A. The event motivated coordinated observations across the electromagnetic spectrum, connecting a variable relativistic source with a possible source of energetic neutrinos.^[ref]

An association is a statistical inference involving localization, timing and backgrounds. It does not directly image the neutrino-production site or uniquely determine the jet's particle composition. The physical interpretation must account for the measured photons and neutrino evidence together.

A quasar is an extremely luminous active galactic nucleus whose compact central emission can outshine its host galaxy and be observed across cosmological distances. Its apparent resemblance to a star originally concealed its nature: a small angular image does not establish a small physical size or a nearby distance. In 1963, Maarten Schmidt identified familiar Balmer series emission lines in the unusual optical spectrum of 3C 273, shifted together to a redshift of 0.158. The optical spectra came from a spectrograph on the 200-inch telescope at Palomar. Interpreting the shift as cosmological implied a distant, extraordinarily luminous galactic nucleus. Modern black-hole accretion models provide the compact power source for that luminosity.^[ref]

3C 273 is a bright quasar whose star-like optical appearance initially concealed its extragalactic character. In 1963, Maarten Schmidt identified its emission lines as a familiar series shifted by a common redshift of about $z=0.158$.

The identification connected the spectrum to a large cosmological distance and hence an enormous luminosity. The lesson is methodological: an object's unresolved angular size does not establish that it is an ordinary nearby star. Interpreting spectral-line patterns can change the inferred physical scale.^[ref]

For a small bound system at cosmological redshift $z$, let $R_{\perp }$ be the projected source-to-cloud distance and $s$ the cloud displacement toward a distant observer. The extra source–cloud–observer path is $\sqrt{R_{\perp }^2+s^2}-s$, and cosmological time dilation multiplies its local delay by $1+z$. A projected separation alone does not determine the delay; $s=0$ is an explicitly transverse estimate.

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 is the transfer of energy and momentum from an active galactic nucleus to surrounding gas. Radiation can heat or accelerate the gas, while astrophysical jets and galactic outflows inject mechanical energy. Heating may offset radiative cooling and reduce the cold material available for star formation or further black-hole accretion; compression can instead favour local star formation. Evidence includes cavities in hot X-ray-emitting plasma that coincide with radio wave emission, together with surrounding shock waves. Their sizes and the ambient pressure constrain the mechanical work, while estimated ages convert an energy budget into a mean power. Such measurements test whether nuclear activity can regulate gas on scales much larger than the central black hole.^[ref]

For an approximately pressure-balanced cavity containing relativistic fluid with $\gamma =4/3$, internal energy plus displacement work is $4pV$. Dividing by an estimated inflation or buoyancy age gives a mechanical-power diagnostic. Projection, pressure gradients, the equation of state, age estimates and additional shock energy affect the inference; cavity power is not identical to locally dissipated heat.

If a hole gains mass $\Delta M_{\mathrm{BH}}$ at constant radiative efficiency $\eta =L/({\dot{M}}_{\mathrm{in}}{c}^2)$, the radiated energy is $\eta \Delta M_{\mathrm{BH}}{c}^2/(1-\eta )$. For negligible seed mass, final hole mass $\alpha M$, and total three-dimensional galaxy dispersion $\sigma$, its ratio to $|W|\simeq M{\sigma }^2$ is $\eta \alpha (c/\sigma {)}^2/(1-\eta )$. The net virial unbinding energy is half $|W|$ when surface terms are negligible.

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 $L/c$.

A swept gas shell of fraction $f$ at radius $R$ in a galaxy of one-dimensional dispersion $\sigma$ has mass $2f{\sigma }^{2}R/G$. Interior non-gas gravity supplies weight $4f(1-f){\sigma }^4/G$. Fully intercepted single-scattering Eddington momentum overcomes it when $M_{\mathrm{BH}}>f(1-f){\sigma }^4{\sigma }_T/(\pi m_p{G}^2)$. This neglects shell self-gravity, central-hole gravity and stored outward kinetic energy; weaker coupling raises the required luminosity.

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 quantitative momentum boost needs an energy-transfer convention as well as a momentum input. For one-speed outflow diagnostics, ${\dot{p}}_s={\dot{M}}_s{v}_s$ and ${\dot{E}}_s={\dot{M}}_s{v}_s^2/2$. A driving wind similarly has ${\dot{E}}_w={\dot{p}}_w{v}_w/2$. If a fraction $f_E={\dot{E}}_s/{\dot{E}}_w$ appears in the shell's inferred kinetic energy flux, thenThus the estimate below assumes a wind momentum input of order $L/c$ and an order-one shell energy fraction, with $f_E\sim 1/2$ giving its stated factor. Large momentum transfer is compatible with energy conservation because the swept shell moves much more slowly than the driving wind. Cooling, thermal energy retained inside the bubble and the observational definition of the shell rates change the coefficient.

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.