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An emission line is a localized excess of specific intensity above the continuum. Heated chromospheric material can give a line source above the photospheric continuum source. Optically thin circumstellar gas can also add line photons through recombination or collisional excitation. Such line formation need not obey a thermal photospheric source prescription.

A nucleus bound in a solid can emit or absorb a photon without exciting lattice vibrations, transferring momentum to the solid as a whole. The resulting recoil energy is negligible compared with the isolated-nucleus recoil, making very narrow resonant emission and absorption lines observable. The Pound-Rebka experiment uses this effect to resolve the small frequency shift from gravitational redshift.

The striking feature is the combination of a large photon energy and a tiny natural linewidth: an atomic nucleus can act as an extraordinarily selective energy filter. The Mössbauer effect supplies a narrow shared emission line and absorption line even when nuclear recoil would separate the corresponding free-nucleus lines. A slow mechanical motion can then tune their overlap through the Doppler effect. The Pound-Rebka experiment is an application to general relativity; the discovery experiment below concerns the recoil-free process itself.

Let $v$ be the source velocity towards the absorber, positive for approach along the photon direction. A first-order Lorentz transformation of the photon energy gives $E^{\prime }=E+vp$; hence, for $|v|\ll c$,Moving the source shifts its emission line through the stationary absorption line. A minimum in transmitted photon counts marks their strongest overlap. The absorber is the narrow energy discriminator; the downstream particle detector only needs to count the transmitted gamma rays.

For a thin absorber with unsplit Lorentzian line shapes, the overlap is a convolution of the source and absorber profiles. The combined full width at half maximum is $W={\Gamma }_s+{\Gamma }_a$. A simple expected transmission model isHere $I_0$ is the off-resonance intensity, $v_0$ encodes a relative shift of the two transition energies, and $A$ includes the resonant absorber strength and $f_s{f}_a$. The convolution result can be checked by multiplying the Fourier transforms of two Cauchy distributions: their exponential decay constants add. The width of the absorption dip in velocity is $cW/E_{\gamma }$, rather than the width of one individual spectral line converted to velocity.

For the teaching system with ${\Gamma }_s={\Gamma }_a={\Gamma }_E$ and $v_0=0$, the dip's full width at half maximum is $2.9979\mathrm{mm}{\mathrm{s}}^{-1}$. Stipulate $A=0.25$ at zero temperature and $10,000$ expected off-resonance counts per equal-duration bin: the central bin then has $7,500$ expected counts, and the half-depth bins have $8,750$. These are model expectations, without sampled counting noise. Holding all other parameters fixed while heating both lattices to $k_{B}T=2\hslash \omega$ multiplies $A$ by $[f(T)/f(0){]}^2$. This isolates how a falling recoil-free fraction weakens the dip.

The thin-absorber model is deliberately limited. A thick absorber can saturate and broaden the dip; unresolved hyperfine structure can create multiple components. Differences in chemical surroundings can shift the centers, and uncontrolled motion can smear them. These effects must be modeled before interpreting an observed width as a natural linewidth. Their sensitivity also makes the Mössbauer effect useful: a small displacement of a narrow dip can reveal a small change in the local nuclear environment.

The main evidence combines a large luminosity with a compact source and a massive central gravitating object. Rapid continuum variability bounds the emitting size by a light-crossing scale, while broad emission lines and reverberation delays constrain the gas orbital speeds and characteristic radii. Their virial masses, together with stellar and gas dynamics in galactic nuclei, imply compact central masses far larger than ordinary stars. Thermal optical/ultraviolet disk emission, a high-energy coronal component, jets and relativistically broadened inner-disk features are naturally associated with an accretion flow. No one signature alone is a unique proof, but the combined compactness, mass and energy requirements strongly favor accretion onto supermassive black holes. Gravitational accretion releases a much larger fraction of rest energy than ordinary stellar burning, without requiring the luminosity to come from the black-hole interior.

For a fully ionized pure-hydrogen plasma, each electron receiving Thomson radiation force is coupled electrostatically to a proton carrying almost all the mass. With spherical isotropic luminosity $L$, the outward force per electron is ${\sigma }_{T}L/(4\pi r^{2}c)$, while gravity per electron–proton pair is approximately $GM{m}_p/r^2$. Equality defines the Eddington luminosity:Here $G$ is the gravitational constant, $m_p$ the proton mass, $c$ the speed of light, $M$ the central mass and ${\sigma }_T$ the Thomson cross-section. The estimate assumes electron-scattering opacity, tight electron–ion coupling and no other dominant force. Geometry, beaming, other opacities and photon trapping can change the relevant luminosity limit.

If ${\dot{M}}_0$ is the supplied rest-mass rate and a fraction $ϵ$ of its rest energy is radiated, $L=ϵ{\dot{M}}_0{c}^2$. The associated Eddington accretion rate is consequentlyThis is a supply rate. The black-hole growth rate is ${\dot{M}}_{\mathrm{BH}}=(1-ϵ){\dot{M}}_0$ if the remaining supplied energy increases the hole's mass. Confusing these rates loses a factor relevant to the growth-time calculation.

The order-of-magnitude efficiency $ϵ\sim 0.1$ is motivated by orbital binding energy released near the innermost stable orbit. A Newtonian orbit at $6GM/c^2$ has binding energy per unit mass $c^2/12\simeq 0.083c^2$; the exact nonspinning thin-disk value is $1-\sqrt{8/9}\simeq 0.057$. Black-hole spin changes the innermost stable orbit and can raise or lower the efficiency. Thus $0.1$ is a useful radiatively efficient accretion estimate, not a universal constant; an advective or jet-dominated flow can have a different radiative efficiency.

Using the stipulated luminosity and ${\sigma }_T=6.65\times 10^{-29}{\mathrm{m}}^2$,The source's redshift and age are treated as its supplied data; its record wording is historical, not a present-day claim about the most distant known quasar.

At continuous Eddington luminosity and fixed efficiency, growth is exponential:This is the Salpeter time, including retained-mass efficiency. Even granting all $0.77\mathrm{Gyr}$ since the big bang permits only about 15.4 e-folds, requiring a seed of at least roughly $400M_{\odot }$. A $100M_{\odot }$ seed would take about $0.84\mathrm{Gyr}$ under these assumptions, before allowing any formation delay; a $10^5{M}_{\odot }$ seed takes about $0.49\mathrm{Gyr}$. These numbers motivate early massive seeds, sustained high duty-cycle accretion, lower radiative efficiencies, super-Eddington phases or contributions from mergers. They do not prove a unique seed channel, since the luminosity-to-mass conversion and accretion history are model-dependent.

Use the notation $I_{\nu }(\mu ,\tau )$ throughout, with optical depth increasing inward and $0<\mu \le 1$ for an outward ray. This removes the change in argument order used in the printed surface-intensity notation. Multiplying the radiative transfer equation by its integrating factor givesIntegrating from the surface to a deeper boundary $b$ yieldsFor a semi-infinite atmosphere with no exponentially growing homogeneous term, the boundary term vanishes as $b\to \infty$. The formal solution of the radiative transfer equation is thereforeThe usual deep-atmosphere boundary condition is necessary; a first-order differential equation alone does not fix the emerging specific intensity.

For a linear source function, the zeroth and first exponential moments give $I_{\nu }(\mu ,0)=a_0+a_1\mu$. Equivalently $I_{\nu }(\mu ,0)=S_{\nu }(\tau =\mu )$, the Eddington-Barbier relation. If the source rises inward, rays near the limb sample cooler, shallower layers, producing limb darkening.

For the finite polynomial, put $u=t/\mu$ and integrate each term. Repeated integration by parts gives ${\int }_0^{\infty }{u}^j{e}^{-u}du=j!$, so the polynomial source function moments for emergent intensity giveThe limiting grazing-ray specific intensity is $a_0$. If the displayed polynomial is only a local Taylor approximation to a more general source, this formula applies to that approximation and a source-function remainder also contributes; a finite local expansion is not automatically an exact description at every optical depth.

For a constant source in an overlying layer of optical thickness $\tau$, the radial ray has the exact transfer relationExpanding the exponentials gives the constant source transfer through a thin layer:This finite-layer result does not require the specific intensity below the layer to equal its source function. The sign of $S_{\nu }-I_{\nu }(1,\tau )$ already distinguishes emission from absorption.

An absorption line forms when a bound-bound transition increases opacity at selected frequencies. In a photosphere whose temperature decreases outward, optical depth of order one at the line frequency occurs higher than at a nearby continuum frequency. In an absorption-dominated region in local thermodynamic equilibrium, the source is the Planck function; the cooler line-forming layer then emits less specific intensity than the deeper continuum-forming layer. The result is a dark spectral feature. Scattering and departures from LTE modify the source, so this temperature-gradient picture is a useful mechanism rather than a universal identity for every line.

Two distinct situations can produce emission lines. First, a heated chromosphere has an outward temperature rise, so an optically thick line can sample gas hotter than the continuum photosphere beneath it. Secondly, a hot optically thin circumstellar envelope or stellar wind emits line photons through collisional excitation or recombination, with no comparably bright background along all rays; its added line flux can exceed the underlying continuum. Winds of hot luminous stars can also give a mixture of emission and absorption across a Doppler-broadened line profile. These examples explain the change in specific intensity through either an enhanced source function or additional emitting material.

A quasar is a luminous active galactic nucleus powered by gas accretion onto a supermassive black hole. The hot disc produces much of the optical/ultraviolet continuum; surrounding gas gives broad and narrow emission lines, while some systems have relativistic jets. A small emitting region is inferred from rapid variability, $R\lesssim c\Delta t_{\mathrm{obs}}/(1+z)$ when beaming is absent. Sustaining a high luminosity from such a compact region favours accretion over ordinary stellar power. For approximately spherical, ionized hydrogen with electron-scattering opacity,Comparing luminosity with this Eddington scale constrains mass only after assumptions about anisotropy, opacity and accretion state. The emitted power obeys $L=\eta \dot{M}{c}^2$; high binding-energy efficiency makes growth of a massive compact accretor energetically plausible.

Broad-line reverberation mapping estimates a gas radius from a continuum-to-line time lag, $R\sim c\tau$, and combines it with line velocity width to infer $M\sim f_{\mathrm{vir}}R{v}^2/G$. Geometry, inclination and nongravitational gas motions enter the factor $f_{\mathrm{vir}}$. Independent evidence for compact massive nuclei comes from resolved stellar or gas orbits in nearby galaxies. Andrea M. Ghez and collaborators' Galactic-centre stellar-orbit study fitted a common central potential and distance, demonstrating a compact dark mass of order four million solar masses; the Galactic centre is a low-luminosity example rather than a quasar.^[ref]

quasars are an accretion phase of massive nuclear black holes; luminosity, variability and dynamical evidence constrain different aspects of that interpretation. A finite-resolution dark-mass measurement by itself is not a direct image of an event horizon.

An active galactic nucleus is powered by accretion onto a central supermassive black hole. Binding-energy release gives $L\sim \eta \dot{M}{c}^2$, so a compact source can outshine its host without converting its entire mass into radiation. A disc produces a thermal continuum, a hot corona can Comptonize disc photons into X-rays, and surrounding gas produces broad and narrow emission lines. Dust can obscure and reradiate the central emission. Orientation explains part of the diversity between unobscured and obscured nuclei, but accretion state and intrinsic variability also matter.

relativistic jets channel energy and momentum into collimated magnetized outflows. Disc rotation and, in some models, black-hole rotation supply the energy; magnetic stresses accelerate the plasma and connect the engine to the outflow. synchrotron radiation traces relativistic electrons and magnetic fields, while inverse Compton scattering can produce higher-energy emission. Bright knots, terminal hotspots and large radio lobes show that the deposited energy is not confined to the unresolved nucleus.

For bulk speed $\beta c$ at viewing angle $i$, the Doppler factor is $\mathcal{D}=[\Gamma (1-\beta \cos i){]}^{-1}$, and an approaching feature hasLight-travel delay can therefore give apparent superluminal motion without any material or signal travelling locally faster than light. Beaming also makes approaching jets appear brighter and more rapidly variable; a one-sided image is not proof of a physically one-sided outflow. Spectral, polarization and resolved-motion measurements constrain the geometry and emission mechanisms. Jets can inflate cavities and heat surrounding gas, linking black-hole accretion to galaxy and cluster feedback.