Galaxy

A galaxy is a gravitationally bound system of stars, stellar remnants and usually gas, interstellar dust and a substantial dark matter component. Its appearance reflects both its contents and their motion: a spiral galaxy has a rotating galactic disk, whereas an elliptical galaxy is largely supported by disordered stellar motion. Star formation makes luminous young populations from gas, while stellar feedback returns material and energy to the interstellar medium. A photograph therefore samples only part of the system: galaxy dynamics uses motions to constrain the underlying mass, including matter that emits little light. Spiral arms also redistribute angular momentum and changes the galactic disk over time.^[ref]

Messier 82, also called M82, is a nearby starburst galaxy used to study how intense star formation drives a large galactic outflow. Its bright stellar disk, obscuring interstellar dust and extended gas emission trace different components; a bright feature in one wavelength band need not represent most of the outflow's mass or energy.

A concrete measurement is to compare diffuse X-ray emission above the disk with compact sources inside it. ROSAT imaging and spectroscopy detected the wind's X-rays to roughly six kiloparsecs from the plane. The observations separated point sources from diffuse emission and found that the latter was not well described as a freely expanding wind: shocked material interacting with the flow could instead account for the emission.^[ref]

This makes M82 an illuminating stellar feedback laboratory. The inference runs from spatially resolved spectra to gas conditions and then to a physical outflow model; a photograph of gas extending away from the disk does not by itself determine its temperature, energy source or escape rate.

Galaxy formation is the assembly of the matter and stellar populations that make a galaxy. In a standard dark matter framework, dark matter halos provide gravitational potential wells in which gas can collect. Radiative cooling lets gas lose thermal support and contract, while star formation converts some dense gas into stars.

The process is regulated rather than a one-way collapse. Stellar explosions, radiation and accreting central black holes return energy and momentum to the gas; galactic winds can remove some of it. Mergers and angular momentum transport change the geometry. A formation model must therefore explain both how material assembles and why only part of the available gas becomes a long-lived stellar population.

An empirical relation between the luminosity of a disk galaxy and its rotation speed. A simple virial and surface-brightness argument gives $L\propto V^4/({{\rm Y}}^{2}I)$, so changes in mass-to-light ratio or surface brightness change the slope.

An approximately extended component surrounding a galactic disk, distinguished according to whether one means the stellar halo, gaseous halo or dark matter halo. These components have different constituents and scales.

A galaxy with a stellar disc and a central bulge, but little of the prominent spiral structure characteristic of a spiral galaxy. Together with elliptical galaxies, lenticular galaxies are conventionally grouped as early-type galaxies.

The projected radius enclosing half an object's total luminosity in a specified photometric passband. For circular apertures, the mean linear surface brightness inside it is ${\mathcal{I}}_e=L/(2\pi R_e^2)$. Elliptical apertures require an axis-ratio factor or a circularized radius; that convention must be stated when comparing calibrations.

A morphological grouping of elliptical galaxies and lenticular galaxies. “Early” is a historical classification name, not a direct statement that a galaxy is young. Dynamical comparisons use a stated structural family and stellar population.

The Large Magellanic Cloud is a nearby galaxy interacting with the Milky Way, containing stars whose independently measurable properties help calibrate astronomical distances. Its proximity makes it possible to combine stellar observations that are much harder to separate in more distant systems.

In a late-type eclipsing binary, orbital velocities and eclipse geometry constrain linear stellar sizes; a surface-brightness calibration supplies corresponding angular sizes. Comparing the two gives a distance anchor rather than assuming one fixed luminosity for every star.^[ref] Such calibration still has systematic uncertainty and samples a galaxy with finite depth.

NGC 4258 hosts a resolved nuclear water-maser disk used as an independent nearby-galaxy distance anchor through orbital dynamics.^[ref]

The circumgalactic medium is gas surrounding a galaxy's main stellar body and interstellar gas, typically associated with its halo. accretion, outflow and recycling exchange matter through it. Its observational boundary and inferred mass depend on the adopted halo convention and the gas phases sampled by absorption or emission.

A galactic bulge is a centrally concentrated stellar population in a galaxy, usually thicker than its disk. Its origin, rotation and random stellar motions can differ from those of the disk; bulge light is often described by a Sérsic profile.

A pseudobulge is a central stellar concentration in a disk galaxy whose structure and dynamics indicate growth through internal disk evolution rather than only rapid merger-driven assembly. A central brightness excess alone does not establish this origin.

Disk-like flattening, appreciable ordered rotation, nuclear spiral arms, galactic bars and ongoing star formation provide complementary diagnostics. A bar can torque gas and move it inward, feeding central star formation over a long timescale. Individual diagnostics are imperfect, so classifications combine imaging and kinematics instead of equating every low-index brightness profile with a unique history.^[ref]

A stellar halo is the extended, approximately spheroidal stellar population surrounding a galaxy. It differs from the total dark matter halo. If its outer luminosity density is $r^{-p}$, its projected light follows the projection of a spherical power-law emissivity and scales as $R^{1-p}$ away from breaks and outer truncation.

Intracluster light comes from stars bound to a cluster rather than an individual galaxy. Tidal stripping and disruption of member systems can create extended diffuse light and substructure. Distinguishing this component from the outer envelope of the central galaxy depends on the adopted dynamical or photometric definition.

A stellar stream is a correlated collection of stars commonly originating in the disruption of a bound progenitor. Phase mixing can erase spatial contrast while leaving coherence in orbital integrals. Such fossil debris constrains a galaxy's past accretion, although not every stream is from a galaxy: star clusters also produce streams.

A star-forming galaxy is a galaxy with ongoing star formation. Its hot young stars can supply hydrogen-ionizing radiation to the intergalactic medium, with the contribution controlled by the ionizing photon escape fraction.

An irregular galaxy lacks a regular spiral or elliptical outline. Many contain substantial gas and recent star formation; irregular appearance alone does not determine a stellar population's age.

A spiral galaxy has a galactic disk with spiral structure, commonly cold interstellar gas and ongoing star formation as well as an older stellar population.

A spiral galaxy with a central elongated stellar bar. Gas, young spiral-arm populations, a bulge and older disk populations can coexist; the bar is a nonaxisymmetric structure and can drive gas motion different from a purely circular-orbit model.

A galactic disk is a flattened stellar or gaseous component of a galaxy, commonly supported mainly by ordered rotation. Its radial mass profile determines its galaxy rotation curve; thickness, truncation and external forces also matter. A galactic disk need not be an exponential galactic disk, and its neutral-gas absorption radius need not equal a stellar scale length.

A stellar galactic disk component with greater vertical extent and typically larger random velocity dispersion than a thin galactic disk. In the Milky Way it contains an important old population. Its stars are not defined simply by being in a geometrically sharp slab or by all sharing one age.

An elongated overdensity in a galactic disk, commonly traced by gas, young stars and star formation. An arm pattern need not rotate at the same rate as individual stars.

A collective perturbation of surface density and gravitational potential in a galactic disk. Material can cross its spiral-shaped crests; the pattern angular speed need not equal the local material angular speed. A tightly wound fluid approximation has $(\omega -m\Omega {)}^2={\kappa }^2+c_s^2{k}^2-2\pi G\Sigma |k|$, while a collisionless stellar disk requires its own response function.

For tightly wound perturbations in a locally uniform, razor-thin gas disk, continuity equation and rotating-fluid momentum equations couple the surface-density perturbation to radial and azimuthal velocities. The sheet potential is $\delta \Phi =-2\pi G\delta \Sigma /|k|$. Eliminating these perturbations gives the displayed dispersion relation. radial epicyclic frequency supplies the rotation term, the sound-speed term resists short waves, and self-gravity supplies the negative term. For an axisymmetric mode $n=0$, use ${\omega }^2$ directly; the pattern speed $\omega /n$ is undefined.

A stellar disk does not respond to a density wave as a collisional gas with a local pressure law. Its stars follow individual orbits, and their spread in radial motions partly averages away the gravitational response to short-wavelength perturbations. The appropriate calculation uses a perturbed distribution function rather than only a fluid sound speed.

A characteristic parameter is $\chi =(k{\sigma }_R/\kappa {)}^2$, comparing the radial epicycle scale with the perturbation wavelength. Orbital averaging reduces the response to perturbations whose wavelength is small relative to the stellar excursions. Merely replacing the fluid sound speed with stellar velocity dispersion misses that orbital averaging and can give the wrong stability threshold.^[ref]

Two radii with different orbital frequencies develop a relative angular displacement $\Delta \varphi =\Delta \Omega t$. A spiral made of a fixed set of stars therefore winds increasingly tightly. A density pattern need not follow that material winding because matter can pass through it.

A self-gravitating disk can transiently amplify a leading disturbance as differential rotation shears it into a trailing disturbance. Its strength depends on disk self-gravity, random motions and disturbance wavelength. Amplification does not by itself establish one indefinitely steady spiral.

For a perturbation phase $m\varphi +\int kdr-\omega t$, a constant-phase pattern rotates at ${\Omega }_p=\omega /m$. At corotation the local orbit angular speed equals this value. One observed galaxy can contain several patterns with distinct angular speeds and lifetimes.

A thin galactic disk is a stellar component whose vertical extent is small compared with its radial size. Small vertical velocity dispersion and substantial specific angular momentum are characteristic. Its survival constrains cumulative galactic disk heating, but its stars need not all have the same formation epoch.

Galactic disk heating increases the random velocities of stars through encounters with molecular clouds, nonaxisymmetric gravitational disturbances, and other perturbations. Older stellar populations usually have larger velocity dispersions and greater vertical thickness, although birth conditions and migration also affect the relation.

Galactic disk flaring is an outward increase of vertical thickness. A constant vertical velocity dispersion in an approximately thin-sheet stellar Newtonian gravitational field gives $h\propto 1/{\Sigma }_{\ast }$. An exponential galactic disk then predicts $h\propto e^{R/R_d}$ for a gas tracer. Inside a thick stellar layer the potential is instead approximately harmonic, and the tracer width is ${\sigma }_z/{\nu }_z$ with vertical epicyclic frequency ${\nu }_z$.

A plane-symmetric, self-gravitating stellar population with constant vertical velocity dispersion satisfies the vertical Jeans equation and the Poisson equation for Newtonian gravity. With ${\Phi }^{\prime }(0)=0$, their solution isIts surface density is $\Sigma =2{\rho }_0{z}_0$, so $z_0={\sigma }_z^2/(\pi G\Sigma )$. Its outer density tail is exponential with scale height $z_0/2$; its central profile is not exponential.

For a self-gravitating isothermal slab with $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ and $H^2={\sigma }^2/(2\pi G{\rho }_0)$, hydrostatic equilibrium gives the displayed Newtonian gravitational potential. A test particle has constant in-plane velocity and conserved vertical energy $E_z={\dot{z}}^2/2+\Phi (z)-\Phi (0)$. Its vertical motion is periodic for every finite $E_z>0$, with turning height $H\mathrm{arcosh}\exp [E_z/(2{\sigma }^2)]$. Near the plane it is a harmonic oscillator of squared frequency $4\pi G{\rho }_0$; at large height the force tends to the constant $2\pi G\Sigma$, where $\Sigma =2{\rho }_{0}H$. The ideal infinite slab has no finite vertical escape energy.

For an isothermal gas slab, a horizontal Fourier mode has $\delta \rho =-ik\rho {\xi }_x-(\rho {\xi }_z{)}_z$, $W=c_s^2\delta \rho /\rho +\varphi$, ${\omega }^2{\xi }_x=ikW$, ${\omega }^2{\xi }_z=W_z$ and $({\partial }_z^2-k^2)\varphi =4\pi G\delta \rho$. Multiply momentum by $\rho {\mathbf{\xi }}^{\ast }$ and integrate by parts, then use the Poisson equation for Newtonian gravity. Vanishing pressure-work and gravitational boundary terms give the displayed real quadratic identity, with $I=\int \rho |\mathbf{\xi }{|}^{2}dz>0$. Thus every admissible squared eigenfrequency is real; a negative value gives gravitational growth rather than an oscillatory instability.

For mass density $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ with $H^2=c_s^2/(2\pi G{\rho }_0)$, a zero-frequency gravitational mode with $k\ne 0$ obeys $W=0$. Its potential satisfies ${\varphi }_{zz}+[2{\mathrm{sech}}^2(z/H)/H^2-k^2]\varphi =0$. The substitution $\tau =\tanh (z/H)$ gives the associated Legendre differential equation of degree one and order $|k|H$. Solutions decaying at both infinities exist only at $|k|H=1$, when $\varphi \propto \mathrm{sech}(z/H)$. Longer horizontal wavelengths are unstable. Neutral mass-preserving relabeling displacements and the uniform vertical translation at $k=0$ are distinct from this fragmentation threshold.

Galaxy stellar mass is the total mass in a galaxy's stars and stellar remnants. It is usually inferred from luminosity and colour or a spectral-energy-distribution fit, so it depends on stellar-population models, dust treatment, and the adopted initial mass function.

The mass-to-light ratio compares the mass assigned to an astronomical system with its luminosity in a specified band, often in solar units. A stellar value uses only stellar mass and depends on the stellar population, age, composition and band. A dynamical value can include gas and dark matter, so it need not equal the stellar value. In a halo-to-galaxy mapping, $L=M_{\ast }/{{\rm Y}}_{\ast }$ requires the stellar ratio and not the total dark matter halo ratio.

A dynamical mass-to-light ratio must specify its aperture and whether the mass includes dark matter. A growing dark halo can increase ${\rm Y}(<R)$ outside the bright stellar body even when the stellar ratio is constant. Comparing a tidal-radius satellite mass with a large-aperture host mass is different from comparing both within their half-light radii. From $M\propto {\sigma }^{2}R$ and $L\propto {\sigma }^4$, the inferred ratio scales as $R/{\sigma }^2$, so equal ratios additionally require homologous radius scaling.

Write $x=m/M_{\odot }$, the initial mass function as $dN=A{x}^{-\alpha }dx$, and a main-sequence mass-luminosity relation as $L/L_{\odot }=x^{\beta }$. On $a<x<b$, the mass-to-light ratio in solar units isFor $\alpha =\beta =3$ this is exactly $1/(ab)$. The result concerns the specified unevolved stellar population; stellar remnants, evolved luminous stars and cooling substellar objects require different accounting.

The galaxy stellar mass function counts galaxies per stellar-mass interval and volume, often using comoving volume. Its stellar-mass moment gives the cosmic stellar mass density, while its number moment counts galaxies only if the low-mass population is integrable or truncated. It differs from the initial mass function of individual stars.

The cosmic stellar mass density is the mass moment of a galaxy stellar mass function in a stated volume convention. With a comoving mass function it is mass per comoving volume, while proper density is larger by $(1+z{)}^3$. Comparing it to a critical or matter density requires keeping the epoch and volume convention explicit.

A generalized mass function with low-mass slope $-7/4$ and cutoff $e^{-\sqrt{M_{\ast }/M_{\ast ,\mathrm{ch}}}}$ differs from the ordinary exponential Schechter cutoff. Its total stellar-mass density is finite, even though its untruncated total galaxy count diverges. This form arises from a scale-free Press-Schechter mass function with a constant stellar-to-halo mass mapping.

Integrating galactic stellar mass against a $x^{-7/4}{e}^{-\sqrt{x}}$ mass function requires ${\int }_0^{\infty }{x}^{-3/4}{e}^{-\sqrt{x}}dx$. Substitution $u=\sqrt{x}$ turns it into $2\Gamma (1/2)=2\sqrt{\pi }$. Thus the stellar density is $2\sqrt{\pi }$ times characteristic number density times characteristic galactic stellar mass, in the same volume convention as the mass function.

A Schechter function models a luminosity function or a galaxy stellar mass function by combining a low-value power law with an exponential cutoff. In stellar-mass notation, define the number per stated volume and mass interval byHere $M_{\ast }$ is a characteristic mass, ${\Phi }_{\ast }$ has units of inverse volume, and $\alpha$ is the low-mass slope. The measured galaxy stellar mass function is often quoted per logarithmic interval instead: the change-of-variables formula for a probability density gives $dn/d{\log }_{10}M=(\ln 10)M\Phi (M)$. Comparing slopes without specifying the measure introduces a shift of one.

Writing $x=M/M_{\ast }$ gives the momentswhere $\Gamma$ is the gamma function. Thus $\alpha =-3/2$ has an infinite extrapolated number of arbitrarily small galaxies but finite cosmic stellar mass density. A positive low-mass cutoff makes both integrals finite. The model is an empirical parametrization over a specified range, rather than evidence that the extrapolation continues to zero mass.

Paul Schechter proposed the luminosity version in 1976 and compared it with nearby-galaxy and cluster samples. His paper explicitly distinguished a sample luminosity distribution from a volume-normalized luminosity function, a distinction that remains essential when correcting a flux-limited survey.^[ref]

The gas-to-stellar mass ratio compares a galaxy's gas reservoir with its stellar mass. Different observational definitions may include only molecular gas, only atomic gas, or both with a helium correction.

Galaxy evolution is the change of a galaxy's stellar population, gas supply, structure and mass over cosmic time. Star formation consumes cold gas; evolving stars return chemically enriched material, while stellar feedback and active-galactic-nucleus feedback can heat or expel it. Galaxy mergers rearrange stellar orbits and may concentrate gas, changing both structure and subsequent star formation. Observers combine stellar ages, optical spectra and star formation rate diagnostics, then compare populations at different redshifts. Such comparisons require care: a distant population is a statistical sample of earlier galaxies, rather than a time sequence of the same objects.^[ref]

galactic feedback couples star formation or black-hole accretion to the surrounding gas through energy, momentum and mass exchange. It includes stellar feedback and active-galactic-nucleus feedback. A feedback loop may heat, stir, expel or compress gas, so its sign depends on how the gas supply and cooling respond; energy injection alone does not establish a lasting reduction of star formation.

A galaxy with negligible present star formation can still gain stars through mergers. Stellar mass assembly and in-situ star formation therefore describe different histories; quenching the latter does not freeze the former.

A gas-budget model in which inflow competes with star formation, gas return and outflows. A zero instantaneous gas derivative defines a balance nullcline; it need not define a fixed point of the evolving stellar and black-hole masses.

With constant inflow $\Phi$, no mass return or outflow, and $\Psi =ϵM_g$, mass conservation gives $M_g=\Phi /ϵ+(M_0-\Phi /ϵ)e^{-ϵt}$. Positive $ϵ$ makes the gas equilibrium stable, with perturbations decaying on timescale ${ϵ}^{-1}$. The star formation rate tends to the inflow, not necessarily to zero.

For ${\dot{M}}_g+ϵM_g=\Phi (t)$, multiplication by $e^{ϵt}$ and integration gives $M_g=M_0{e}^{-ϵt}+{\int }_0^t{e}^{-ϵ(t-s)}\Phi (s)ds$. Exponential memory filters the past inflow; the constant-inflow equilibrium is one specialization.

For initially no stars, $M_{\star }=\Phi t+(M_0-\Phi /ϵ)(1-e^{-ϵt})$. Adding the gas solution verifies $M_g+M_{\star }=M_0+\Phi t$. Thus equilibrium of the gas reservoir coexists with continuing linear growth of stellar mass.

In a system where gas and stellar masses both evolve, the locus ${\dot{M}}_g=0$ gives the instantaneous gas-balance nullcline. Motion along the other state variables generally changes the equilibrium reservoir, so the nullcline is not itself an exact constant-gas trajectory.

For an Eddington-luminous hole with $M_{\mathrm{BH}}=\beta M_s$ and wind mass flux $\lambda L$, define $A=\lambda \beta (4\pi G{m}_{p}c/{\sigma }_T)$. The gas budget ${\dot{M}}_g=\Phi -ϵM_g-A{M}_s$ has positive instantaneous balance only for $M_s<\Phi /A$. A positive constant-gas trajectory is incompatible with ongoing ${\dot{M}}_s=ϵM_g$ unless the secular black-hole coupling is neglected.

Galaxy morphological evolution changes the relative prevalence and internal structure of discs, spheroids, bars, and irregular systems over time. Comparing epochs requires consistent wavelength, spatial resolution, and selection. Galaxy mergers, gas accretion and internal instabilities can all contribute.

The observed size distribution of galaxies changes with epoch at a given galaxy stellar mass. Growth of an individual object and changes in which objects enter a selected sample both contribute. Minor galaxy mergers can deposit stars at large radii, increasing size while adding relatively little mass; stellar mass loss and dissipative evolution provide other mechanisms.

A galaxy red sequence is the relatively narrow red locus in a galaxy colour-magnitude diagram, populated largely by systems with little recent star formation. Its colour, abundance, and galaxy stellar mass evolve. Dust and selection must be distinguished from old stellar ages; growth can involve quenching and later assembly of existing stars.

A starburst galaxy undergoes an episode of rapid star formation. An instantaneous-burst idealization places all newly formed stars at one stellar age, so their present evolutionary-state fractions follow from the initial mass function at that fixed age rather than from an average over formation times.

A galaxy merger combines two or more galaxies into a remnant. Their orbital angular momentum and energy are redistributed into internal stellar and halo motions, escaping matter, and, when present, dissipative gas. A bound pair produced by tidal capture of galaxies need not coalesce immediately.

An interacting pair of spiral galaxies undergoing a galaxy merger. Their long tidal tails contain stars displaced by the encounter. They are also a nearby example of a starburst galaxy, with vigorous star formation accompanying the interaction.^[ref]

The Hubble Space Telescope image below combines observations from its Wide Field Camera 3 and Advanced Camera for Surveys. Bright young stars, emitting regions of the interstellar medium and obscuring interstellar dust reveal different aspects of the encounter. The close view emphasizes the interacting central regions; wider views show the long outer tidal tails.^[ref]

A gas-rich galaxy merger involves a substantial reservoir of gas capable of radiative cooling. Dissipation and gravitational torques permit gas inflow and enhanced star formation, unlike the nearly collisionless stellar evolution in a dry galaxy merger. A surviving or replenished gas reservoir can subsequently form a galactic disk.

Gas can shock and radiate during a galaxy encounter, allowing it to change orbital energy in ways unavailable to an ideal collisionless stellar component. Gravitational torques can transport its angular momentum and fuel central star formation or accretion. The outcome depends on orbit, gas content and internal structure.

A major galaxy merger joins galaxies with comparable masses. Rapid changes in the gravitational potential can redistribute stellar orbital energies through violent relaxation, substantially disturbing preexisting galactic disks. The outcome depends on gas content, orbit and subsequent accretion; a major merger does not universally prohibit disk rebuilding.

In a dilute, homogeneous population with number density $n$, an effective encounter cross section $\Sigma =\pi R_{\mathrm{enc}}^2$ gives the rate per target galaxyHere $R_{\mathrm{enc}}$ is the maximum encounter impact parameter, not automatically one galaxy's visible radius. If the cross section depends on relative speed, average the product instead: ${\Gamma }_{\mathrm{enc}}=n⟨\Sigma (v_{\mathrm{rel}})v_{\mathrm{rel}}⟩$. Gravitational focusing can create such speed dependence.

With independent encounters at a constant rate, a Poisson process givesThe approximation ${\Gamma }_{\mathrm{enc}}T$ is valid only for rare encounters. This is an encounter probability; it is not automatically a galaxy merger probability. Passing galaxies can remain unbound, and a bound pair can take additional time to coalesce.

As a simple additional model, suppose each encounter independently produces a merger within the relevant window with a fixed conditional probability $p$. The probability generating function of the Poisson count givesso the corresponding merger probability is $1-e^{-p{\Gamma }_{\mathrm{enc}}T}$. A realistic $p$ depends on encounter geometry, relative energy, angular momentum, gas and the allowed delay. It cannot be inferred from the geometric encounter rate alone.

For an entire volume $V$ of identical potential targets, the total distinct encounter-pair rate is $nV{\Gamma }_{\mathrm{enc}}/2$, because summing the per-target rate counts each unordered pair twice. This pair correction is separate from the conditional chance of merging.

Tidal capture converts sufficient relative orbital energy into internal tidal heating to leave two initially unbound galaxies on a bound orbit. For identical spherical galaxies of mass $m_g$, a distant rapid flyby of impact parameter $p$ and relative speed $v$ gives the approximate capture condition $pv<[(32/3)G^2{m}_g^2⟨r^2⟩{]}^{1/4}$. This requires the impulse approximation and weak gravitational focusing; it is not a general criterion for subsequent coalescence.

Two equal galaxies each receive the tidal impulse heating of a spherical galaxy, while their relative orbital energy at infinity is $m_g{v}^2/4$. Transfer of more internal energy than that orbital energy gives $pv<[32G^2{m}_g^2⟨r^2⟩/3{]}^{1/4}$. Slow encounters require adiabatic shielding of tidal encounters, so the impulsive inequality does not apply there.

A dry galaxy merger joins galaxies with little cold gas, so dissipation and merger-triggered star formation are weak. Conservation of energy followed by re-virialization can make repeated dry minor mergers increase a galaxy's radius faster than its mass.

In a dry galaxy merger, zero orbital energy at infinite separation and retention of all mass and energy give $E_F=E_I+E_A$. If the progenitors and remnant satisfy the virial theorem, their gravitational radii obey $R_F/R_I=(1+\eta {)}^2/(1+ϵ\eta )$, where $\eta =M_A/M_I$ and $ϵ=⟨v_A^2⟩/⟨v_I^2⟩$. The latter average is weighted by satellite mass. Nonzero orbital energy, escaping stars, or dissipative losses change the energy accounting.

A minor galaxy merger has a small satellite-to-host mass ratio. Its stars are commonly deposited at large host radii after tidal stripping and Chandrasekhar dynamical friction.

For repeated minor galaxy mergers with negligible internal satellite binding energy and zero orbital energy at infinity, parabolic dry-merger energy accounting leaves total energy nearly constant. The virial theorem then gives gravitational radius $R_g\propto M^2$ and mean-square speed proportional to $M^{-1}$. Small satellite mass does not by itself imply negligible internal binding energy.

Stellar feedback is the transfer of energy, momentum, mass, and newly formed elements from stars to surrounding matter through radiation, winds, supernovae, and cosmic rays.

Depositing supernova energy into dense unresolved gas can radiate it away before a numerically resolved expansion develops. Equal injected energy therefore need not imply equal outflow momentum or heating at different resolution. A feedback model should represent the coupling and cooling phase it intends to approximate, and comparisons should include gas recycling and the resulting stellar and circumgalactic properties.

Before local expulsion, a linear star-formation law $\psi =\nu {\rho }_g$ without return gives ${\rho }_g={\rho }_0{e}^{-\nu t}$. If $q=\gamma E_{\mathrm{SN}}$ is the energy supplied per unit stellar mass, accumulated energy is $q{\rho }_0(1-e^{-\nu t})$. A fixed binding energy per unit remaining gas mass $B(r)$ gives the threshold $q(e^{\nu t}-1)=B(r)$. This assumes a specified retained gravitational profile and neglects exchange between volumes.

The local shell-energy prescription sets $B(r)=GM(<r)/r$ rather than the full escape work. It predicts simultaneous quenching for $M(<r)\propto r$, but a centre-to-edge front for $M(<r)\propto r^2$. Exterior-shell contributions explain why this ordering can differ from a true escape-potential calculation.

With test-parcel escape work $B(r)=-\Phi (r)$, outer gas in a spherical isolated attractive potential is less tightly bound and quenches first under identical local specific heating. A truncated inverse-square density gives $r_q=r_0\exp [1-q(e^{\nu t}-1)/(2k)]$ after edge onset. The result is a fixed-potential closure, not a self-consistent potential update after mass loss.

For flux $F$ and effective dust opacity $\kappa$, radiative acceleration is $\kappa F/c$. Comparing it with the gravitational acceleration gives an Eddington luminosity condition appropriate to dusty gas. In an optically thick starburst galaxy, feedback can limit the star formation rate per area and drive a galactic outflow, while the gas supply limits the duration. A surface-density limit is not a universal ceiling on total star formation rate: the star-forming area matters. Radiation-supported starburst disk calculations give characteristic optically thick infrared fluxes near $10^{13}{L}_{\odot }{\mathrm{kpc}}^{-2}$.

If a virialized, well-mixed self-gravitating system instantaneously retains mass fraction $ϵ>1/2$, homologous revirialization gives $R^{\prime }/R=ϵ/(2ϵ-1)$. At or below one half, the idealized remnant is unbound.

Take an initially isolated virialized self-gravitating stellar population with kinetic energy $T=|W|/2$ and Newtonian gravitational potential energy $W<0$. Instantaneous spatially uniform removal of mass, without kicks to surviving material, leaves fraction $f$ with $T^{\prime }=fT$ and $W^{\prime }=f^{2}W$. Hence $E^{\prime }=|W|f(1/2-f)$: the homologous whole-system energy is negative only for $f>1/2$. Positive whole-system energy does not forbid a bound core after selective escape, and slow mass loss follows the adiabatic mass-loss expansion law instead.