Supermassive black hole (SMBH)

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.

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.