Galaxy scaling relations

• Comment: Far too much is unsourced, strong opinions are expressed which is WP:OR and it reads like a textbook. Ldm1954 (talk) 17:26, 31 July 2026 (UTC)

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The Hubble sequence classifies galaxies by visible morphology. Different galaxy populations follow partly distinct scaling relations because their structures and internal motions differ.

Galaxy scaling relations are empirical or theoretically motivated correlations among the observable and physical properties of galaxies. They connect quantities such as luminosity, stellar mass, size, surface brightness, color, rotational velocity, velocity dispersion, gas mass, star formation rate, metallicity, and central supermassive black hole mass. Some relations contain two variables, while others define a plane or higher-dimensional surface in a space of galaxy properties.

The best-known examples include the Tully–Fisher relation for rotationally supported disk galaxies, the Faber–Jackson relation and fundamental plane for pressure-supported early-type galaxies, the stellar mass–size relation, the mass–metallicity relation, the star-forming main sequence, the Kennicutt–Schmidt law, and the M–σ relation between black-hole mass and bulge velocity dispersion. These correlations are used to estimate astronomical distances, compare galaxy populations, reconstruct peculiar motions, constrain the distribution of dark matter, and test models of galaxy formation and evolution.

Scaling relations do not imply that all galaxies obey a single universal law. Their slopes, zero points, and intrinsic scatter can depend on galaxy morphology, wavelength, environment, redshift, stellar population, measurement method, and sample selection. They are therefore generally treated as statistical descriptions of galaxy populations rather than exact relations for individual objects.

Definition and mathematical description

A scaling relation expresses how one measured or inferred galaxy property changes with one or more other properties. A two-variable power law may be written as

$Y=A{X}^{\alpha },$

where:

• X and Y are galaxy properties;
• A is the normalization or zero point; and
• α is the logarithmic slope.

Taking the logarithm gives a linear expression,

${\log }_{10}Y=\alpha {\log }_{10}X+\beta ,$

where

$\beta ={\log }_{10}A.$

Observed values generally display scatter around the fitted relation. For the ith galaxy, a simple model is

$y_i=\alpha x_i+\beta +{ϵ}_i,$

where x_i and y_i are logarithmic measurements and ε_i represents observational uncertainty and intrinsic variation.

If the residuals are approximated by a Gaussian distribution, their intrinsic scatter may be represented by

${ϵ}_i\sim {𝒩}(0,{\sigma }_{\mathrm{i}\mathrm{n}\mathrm{t}}^2),$

where σ_int is the intrinsic standard deviation after measurement errors have been considered.

Some scaling relations involve three variables. A plane may be represented by

$z=ax+by+c.$

The fundamental plane of early-type galaxies is an example. More complicated relations may be curved, broken, redshift dependent, or described by a multidimensional probability distribution rather than a single line.

Residuals

The residual of a measured point from a two-variable relation is

$\mathrm{\Delta }{y}_i=y_i-(\alpha x_i+\beta ).$

Residuals are studied to identify additional physical parameters. For example, residuals from a mass–metallicity relation may correlate with star formation rate, gas fraction, or galaxy size. Such behavior can indicate that a two-dimensional relation is a projection of a more complex multidimensional distribution.

Redshift evolution

A relation that changes over cosmic time may be written as

$\log Y=\alpha (z)\log X+\beta (z),$

where z is redshift. A frequently used parameterization of the evolving zero point is

$\beta (z)={\beta }_0+\gamma {\log }_{10}(1+z),$

although the appropriate expression depends on the relation and dataset.

Physical basis

Galaxy scaling relations arise because galactic properties are connected by gravity, conservation of angular momentum, star formation, stellar evolution, gas accretion, radiative cooling, chemical enrichment, mergers, and energetic feedback from stars and active galactic nuclei. Their interpretation usually requires both the gravitational influence of dark-matter halos and the behavior of ordinary baryonic matter.

A galaxy's observable properties are not independent. Its stellar mass influences its luminosity, gravitational potential, metallicity, and ability to retain gas. Its dark-matter halo affects its rotational velocity, size, and accretion history. The conversion of gas into stars determines its surface brightness, color, and chemical composition.

Virial scaling

For a gravitationally bound system near dynamical equilibrium, the virial theorem gives approximately

$2K+U=0,$

where K is the total kinetic energy and U is the gravitational potential energy.

For a pressure-supported galaxy with characteristic radius R and velocity dispersion σ, its dynamical mass can be approximated by

$M_{\mathrm{d}\mathrm{y}\mathrm{n}}\approx k\frac{{\sigma }^{2}R}{G},$

where:

• M_dyn is the dynamical mass;
• G is the gravitational constant; and
• k is a dimensionless factor that depends on structure, orbital distribution, and the definition of radius.

For a rotationally supported galaxy,

$M(<R)\approx \frac{V_{\mathrm{c}}^{2}R}{G},$

where V_c is the circular velocity.

These expressions provide a foundation for several relations involving size, velocity, luminosity, and mass. Departures from simple virial predictions reflect variations in stellar populations, dark-matter fraction, gas content, structure, and feedback.

Surface brightness and luminosity

The luminosity of a galaxy can be expressed schematically as

$L\sim I{R}^2,$

where I is a characteristic surface brightness and R is a characteristic radius. More precisely, the luminosity enclosed within the effective radius R_e is

$L(<R_e)=\pi R_e^2⟨I{⟩}_e$

for a circularized aperture, where ⟨I⟩_e is the mean intensity within R_e.

Combining luminosity, size, and dynamical relations produces several observed galaxy planes and sequences.

Angular momentum

The specific angular momentum of a galaxy is

$j=\frac{J}{M},$

where J is total angular momentum and M is mass. Disk size and rotation depend partly on the angular momentum acquired by a dark-matter halo and retained by its baryons. Relations among stellar mass, size, rotational velocity, and specific angular momentum therefore provide constraints on gas accretion and disk formation.

Feedback and regulation

Energy and momentum from supernovae, stellar winds, radiation, and active galactic nuclei can heat or expel gas. Feedback changes the efficiency with which halo mass is converted into stars and can affect galaxy size, metallicity, gas fraction, and star formation.

Scaling relations are therefore frequently used as benchmarks for numerical simulations and semi-analytic models. A successful model must reproduce not only individual relations but also their scatter, covariance, morphology dependence, and evolution with redshift.

Classification

Galaxy scaling relations can be grouped by the principal properties they connect.

Major categories of galaxy scaling relations

Category	Representative quantities	Examples
Structural	Size, luminosity, surface brightness, concentration	Mass–size relation; Kormendy relation
Dynamical	Rotation velocity, velocity dispersion, dynamical mass	Tully–Fisher relation; Faber–Jackson relation; fundamental plane
Baryonic	Stellar mass, gas mass, baryonic mass	Baryonic Tully–Fisher relation; gas-fraction relation
Star-forming	Star formation rate, gas surface density, stellar mass	Star-forming main sequence; Kennicutt–Schmidt law
Chemical	Stellar mass, gas metallicity, stellar metallicity	Mass–metallicity relation
Black-hole and host-galaxy	Black-hole mass, bulge mass, luminosity, velocity dispersion	M–σ relation; black-hole–bulge-mass relation
Angular-momentum	Specific angular momentum, mass, morphology	Fall relation

The categories overlap. For example, the fundamental plane is simultaneously structural and dynamical, while the baryonic Tully–Fisher relation connects dynamics with the total observable baryonic mass.

Disk-galaxy relations

The barred spiral galaxy NGC 1300. Disk-galaxy relations commonly connect luminosity or baryonic mass with rotational velocity.

Tully–Fisher relation

Main article: Tully–Fisher relation

The Tully–Fisher relation connects the luminosity of a spiral or disk galaxy with a measure of its rotational velocity. It was introduced by R. Brent Tully and J. Richard Fisher in 1977 as a means of estimating extragalactic distances.

A general luminosity form is

$L\propto V_{\mathrm{r}\mathrm{o}\mathrm{t}}^{\alpha },$

where V_rot is the characteristic rotational velocity and α depends on the wavelength, sample, and fitting procedure.

In absolute-magnitude form,

$M_{\lambda }=a_{\lambda }{\log }_{10}W+b_{\lambda },$

where:

• M_λ is the absolute magnitude in passband λ;
• W is a corrected spectral-line or rotation-curve width; and
• a_λ and b_λ are calibrated coefficients.

The rotational velocity is often measured from a neutral-hydrogen 21-centimeter line profile, optical emission lines, or a spatially resolved rotation curve. Corrections are applied for disk inclination, internal extinction, turbulence, instrumental resolution, and redshift broadening.

The relation is used as a secondary rung of the cosmic distance ladder. A galaxy's rotational velocity predicts its intrinsic luminosity; comparison with observed brightness then yields its distance.

Baryonic Tully–Fisher relation

The baryonic Tully–Fisher relation replaces luminosity with the total baryonic mass,

$M_{\mathrm{b}\mathrm{a}\mathrm{r}}=M_{\star }+M_{\mathrm{g}\mathrm{a}\mathrm{s}},$

where M_★ is stellar mass and M_gas includes atomic and molecular gas, with a correction for helium and heavier elements.

It is commonly written as

$M_{\mathrm{b}\mathrm{a}\mathrm{r}}=A{V}_{\mathrm{f}}^{\alpha },$

where V_f is the approximately flat outer rotational velocity. Many studies find a slope near four, although the fitted value depends on the adopted stellar mass-to-light ratio, gas corrections, velocity definition, and sample.

The relation is especially useful for gas-rich dwarf galaxies, whose luminosities may not accurately represent their total baryonic content.

Disk size–mass relation

Disk galaxies exhibit a correlation between stellar mass and characteristic size. It can be approximated as

$R\propto M_{\star }^{\alpha },$

where R may be the exponential scale length or effective radius.

For an ideal exponential disk,

$I(R)=I_0{e}^{-R/R_d},$

where I₀ is central intensity and R_d is the scale length. Its total luminosity is

$L=2\pi I_0{R}_d^2.$

Disk sizes reflect the angular momentum of accreted gas, the spin of the host dark-matter halo, the galaxy's mass-assembly history, and feedback.

Specific-angular-momentum relation

The relation between stellar specific angular momentum and stellar mass is often represented as

$j_{\star }\propto M_{\star }^{\alpha }.$

Disk and spheroidal galaxies occupy partly distinct sequences, with disk galaxies generally having greater specific angular momentum at a given stellar mass. The relation has been used to connect present-day morphology with the acquisition, loss, and redistribution of angular momentum during galaxy formation.

Radial-acceleration relation

The radial acceleration relation connects the observed centripetal acceleration in rotating galaxies with that predicted from the observed distribution of baryonic matter. The observed acceleration is

$g_{\mathrm{o}\mathrm{b}\mathrm{s}}=\frac{V^2(R)}{R},$

while the baryonic contribution is calculated from the measured distributions of stars and gas. The relation is closely connected to regularities in galaxy rotation curves and the baryonic Tully–Fisher relation.

Its physical interpretation is debated in the context of baryonic feedback, dark-matter halo response, and alternative gravitational models.

Early-type galaxy relations

The giant elliptical galaxy Messier 87. Relations for pressure-supported galaxies commonly involve luminosity, radius, surface brightness, and stellar velocity dispersion.

Faber–Jackson relation

Main article: Faber–Jackson relation

The Faber–Jackson relation connects the luminosity of an elliptical galaxy with its central stellar velocity dispersion:

$L\propto {\sigma }^{\gamma }.$

The original 1976 study by Sandra Faber and Robert Jackson found an exponent near four for its sample, leading to the commonly quoted approximation

$L\propto {\sigma }^4.$

The slope varies with luminosity range, passband, sample, and regression method.

The relation is analogous in purpose to the Tully–Fisher relation, but it uses random stellar motions rather than ordered rotation. It has greater scatter than the fundamental plane because it omits explicit information about galaxy size and surface brightness.

Fundamental plane

Main article: Fundamental plane (elliptical galaxies)

The fundamental plane is a three-parameter relation among effective radius R_e, central velocity dispersion σ, and mean surface brightness ⟨I⟩_e:

$R_e\propto {\sigma }^a⟨I{⟩}_e^b.$

In logarithmic form,

$\log R_e=a\log \sigma +b\log ⟨I{⟩}_e+c.$

It was identified independently in 1987 by two research groups.

A simplified virial model with structural homology and constant mass-to-light ratio predicts

$R_e\propto {\sigma }^2⟨I{⟩}_e^{-1}.$

Observed coefficients differ from these values. This difference is called the tilt of the fundamental plane and is attributed to systematic changes in stellar populations, stellar initial mass function, dark-matter fraction, orbital structure, and structural non-homology.

The fundamental plane is used to estimate relative distances and peculiar velocities. Because surface brightness and velocity dispersion are approximately independent of distance while angular size is distance dependent, the plane predicts a physical size that can be compared with the measured angular radius.

D_n–sigma relation

Main article: Dn-sigma relation

The D_n–sigma relation connects a characteristic physical diameter of an elliptical or lenticular galaxy with its stellar velocity dispersion:

$D_n\propto {\sigma }^{\gamma }.$

The diameter D_n is defined at a prescribed mean surface brightness. The relation can be interpreted as a projection of the fundamental plane and was widely used to measure galaxy-cluster distances and peculiar velocities during the 1980s and 1990s.

Kormendy relation

Main article: Kormendy relation

The Kormendy relation connects the effective radius of an elliptical galaxy or bulge with its mean surface brightness within that radius:

$⟨\mu {⟩}_e=\alpha \log R_e+\beta .$

It is a projection of the fundamental plane that does not include velocity dispersion. The relation is named after John Kormendy, who described the correlation in 1977.

It is used to compare elliptical galaxies, classical bulges, dwarf spheroidals, and other spheroidal systems, although different populations do not necessarily occupy a single continuous relation.

Color–magnitude relation

Early-type galaxies in clusters exhibit a relatively narrow sequence in color–magnitude or color–mass diagrams, commonly called the red sequence. More luminous or massive early-type galaxies are generally redder.

A simplified form is

$C=aM+b,$

where C is a color index and M is absolute magnitude. The slope is associated mainly with systematic differences in metallicity, while the scatter constrains variations in stellar age, chemical enrichment, and recent star formation.

Stellar mass, size, and surface density

The Hubble Ultra-Deep Field contains galaxies observed across a wide range of cosmic time. Comparing mass–size and other relations at different redshifts reveals changes in galaxy structure.

Galaxies display correlations among stellar mass, effective radius, surface mass density, and morphology. The stellar mass–size relation is often represented by

$R_e=R_0{\left(\frac{M_{\star }}{M_0}\right)}^{\alpha }.$

Early-type and late-type galaxies have different slopes and normalizations. At a fixed stellar mass, quiescent spheroidal galaxies are generally more compact than star-forming disks in the nearby universe.

The mean stellar surface density within the effective radius may be defined as

${\mathrm{\Sigma }}_e=\frac{M_{\star }}{2\pi R_e^2}.$

The factor of two reflects that the effective radius encloses approximately half the total light, assuming stellar mass follows light. Other definitions use different geometric or mass-profile corrections.

Mass–size relations are used to study:

• disk growth;
• dissipative collapse;
• galaxy mergers;
• environmental transformation;
• compact quiescent galaxies at high redshift;
• the buildup of outer stellar envelopes; and
• the connection between galaxies and dark-matter halos.

The relation evolves with redshift. Massive quiescent galaxies at high redshift are, on average, more compact than similarly massive quiescent galaxies in the nearby universe.

Star-formation relations

Star-forming main sequence

Most actively star-forming galaxies occupy a relatively narrow relation between star formation rate and stellar mass, known as the star-forming main sequence:

$\mathrm{S}\mathrm{F}\mathrm{R}\propto M_{\star }^{\alpha }.$

In logarithmic form,

$\log \mathrm{S}\mathrm{F}\mathrm{R}=\alpha \log M_{\star }+\beta .$

The normalization increases strongly with redshift, indicating that galaxies of a given stellar mass formed stars more rapidly in the earlier universe. The slope and the presence of high-mass curvature depend on sample selection, star-formation indicators, dust correction, and treatment of inactive galaxies.

The specific star formation rate is

$\mathrm{s}\mathrm{S}\mathrm{F}\mathrm{R}=\frac{\mathrm{S}\mathrm{F}\mathrm{R}}{M_{\star }}.$

It represents the current star formation rate per unit existing stellar mass. Its inverse gives an approximate stellar-mass growth timescale if the present rate were maintained.

Galaxies substantially above the main sequence are often classified as starbursts, while those below it may be transitioning toward or occupying the quiescent population.

Kennicutt–Schmidt law

Main article: Kennicutt–Schmidt law

The Kennicutt–Schmidt law relates the surface density of star formation to the surface density of gas:

${\mathrm{\Sigma }}_{\mathrm{S}\mathrm{F}\mathrm{R}}=A{\mathrm{\Sigma }}_{\mathrm{g}\mathrm{a}\mathrm{s}}^N.$

In a widely used disk-averaged calibration, the exponent N is approximately 1.4, although its value changes with gas tracer, spatial scale, galaxy population, and treatment of atomic and molecular gas.

The molecular-gas depletion time is

$t_{\mathrm{d}\mathrm{e}\mathrm{p}}=\frac{M_{\mathrm{m}\mathrm{o}\mathrm{l}}}{\mathrm{S}\mathrm{F}\mathrm{R}},$

or, in surface-density form,

$t_{\mathrm{d}\mathrm{e}\mathrm{p}}=\frac{{\mathrm{\Sigma }}_{\mathrm{m}\mathrm{o}\mathrm{l}}}{{\mathrm{\Sigma }}_{\mathrm{S}\mathrm{F}\mathrm{R}}}.$

A nearly linear molecular-gas relation corresponds to an approximately constant depletion time. Variations occur with galactic environment, starburst activity, metallicity, and spatial scale.

Resolved star-formation relations

Integral-field spectroscopy and resolved gas mapping have revealed local or kiloparsec-scale counterparts of global scaling relations. Within individual galaxies, local stellar mass surface density, star formation rate surface density, gas content, and metallicity are correlated.

Resolved relations help determine whether a global correlation arises from local physical regulation or from combining regions with different properties. They also reveal radial gradients and distinguish central bulges from star-forming disks.

Chemical-abundance relations

Mass–metallicity relation

Main article: Mass–metallicity relation

The mass–metallicity relation connects galaxy stellar mass with the chemical abundance of its gas or stars. For star-forming galaxies, gas-phase metallicity is commonly expressed using the oxygen abundance

$12+{\log }_{10}\left(\frac{\mathrm{O}}{\mathrm{H}}\right).$

More massive galaxies generally have greater gas-phase metallicities, with the relation flattening at high stellar mass.

The relation reflects several processes:

• enrichment by successive generations of stars;
• loss of metal-enriched gas through galactic winds;
• dilution by infalling low-metallicity gas;
• star formation efficiency;
• gas fraction; and
• the depth of the gravitational potential.

In a simplified closed-box chemical-evolution model,

$Z=y\ln \left(\frac{1}{f_{\mathrm{g}\mathrm{a}\mathrm{s}}}\right),$

where:

• Z is metallicity;
• y is the nucleosynthetic yield; and
• f_gas is the gas fraction.

Real galaxies are not closed systems, so inflows and outflows modify this relation.

Fundamental metallicity relation

At fixed stellar mass, galaxies with higher star formation rates often show lower gas-phase metallicities. This led to proposals for a three-variable relation among stellar mass, metallicity, and star formation rate.

One compressed variable used in such studies is

${\mu }_{\alpha }=\log M_{\star }-\alpha \log \mathrm{S}\mathrm{F}\mathrm{R},$

where α is chosen to minimize metallicity scatter. Whether a single redshift-invariant fundamental metallicity relation exists depends on calibration, sample, redshift range, and metallicity diagnostic.

Stellar metallicity and mass

Galaxy stellar metallicity also increases with stellar mass. Gas-phase and stellar metallicities measure different aspects of chemical history: gas-phase metallicity reflects the present interstellar medium, while stellar metallicity records the composition of stars formed over the galaxy's lifetime.

Gas-content relations

The atomic- and molecular-gas fractions of galaxies correlate with stellar mass, color, surface density, and specific star formation rate. A gas fraction may be defined as

$f_{\mathrm{g}\mathrm{a}\mathrm{s}}=\frac{M_{\mathrm{g}\mathrm{a}\mathrm{s}}}{M_{\mathrm{g}\mathrm{a}\mathrm{s}}+M_{\star }}.$

Lower-mass and actively star-forming disk galaxies generally have larger cold-gas fractions than massive quiescent galaxies. Molecular-gas fraction is strongly connected to star formation, while atomic hydrogen often extends well beyond the stellar disk.

Frequently studied relations include:

• atomic-gas fraction versus stellar mass;
• molecular-gas fraction versus specific star formation rate;
• gas depletion time versus offset from the star-forming main sequence;
• dust mass versus stellar or gas mass; and
• gas metallicity versus gas fraction.

These relations link the supply of star-forming material with the growth and chemical evolution of galaxies.

Black-hole and host-galaxy relations

M–sigma relation

Main article: M–sigma relation

The M–sigma relation connects the mass of a central supermassive black hole, M_BH, with the stellar velocity dispersion of the host galaxy's bulge:

$M_{\mathrm{B}\mathrm{H}}=M_0{\left(\frac{\sigma }{{\sigma }_0}\right)}^{\alpha }.$

In logarithmic form,

${\log }_{10}\left(\frac{M_{\mathrm{B}\mathrm{H}}}{M_{\odot }}\right)=\alpha {\log }_{10}\left(\frac{\sigma }{{\sigma }_0}\right)+\beta .$

The relation became firmly established around 2000 and is generally tighter than the original black-hole-mass–bulge-luminosity relation for some samples.

Its existence suggests a connection between black-hole growth and the formation of the galactic bulge. Proposed mechanisms include active-galactic-nucleus feedback, merger-driven growth, common dependence on gravitational potential, and statistical convergence through repeated mergers.

Black-hole mass and bulge mass

Supermassive black-hole mass also correlates with bulge stellar mass and bulge luminosity. A simplified expression is

$M_{\mathrm{B}\mathrm{H}}\propto M_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{g}\mathrm{e}}^{\alpha }.$

The relation differs between classical bulges, elliptical galaxies, pseudobulges, and potentially other morphological classes. Measurement samples can be biased toward black holes with large gravitational spheres of influence, affecting fitted slopes and normalizations.

Black-hole fundamental plane

Relations involving black-hole mass, radio luminosity, and X-ray luminosity are sometimes described as a fundamental plane of black-hole activity. These extend beyond galaxy structure alone and are generally discussed in the context of accretion physics and active galactic nuclei.

Distance measurement

Several galaxy scaling relations can serve as secondary distance indicators. They rely on a distance-independent observable to predict an intrinsic luminosity or physical size.

Luminosity-based relations

For a calibrated luminosity relation, the distance modulus is

$\mu =m-M,$

where m is apparent magnitude and M is the absolute magnitude predicted by the scaling relation.

The luminosity distance is then

$d_L=10^{(\mu +5)/5}\mathrm{p}\mathrm{c}.$

The Tully–Fisher relation and, less precisely, the Faber–Jackson relation can be applied in this manner.

Size-based relations

If a relation predicts a physical diameter D from a distance-independent measurement, comparison with the observed angular diameter θ gives

$d_A=\frac{D}{\theta },$

where d_A is the angular-diameter distance and θ is expressed in radians.

The D_n–sigma relation and fundamental plane can be used as standard-ruler methods.

Peculiar velocities

At sufficiently low redshift, a radial peculiar velocity may be approximated by

$v_{\mathrm{p}\mathrm{e}\mathrm{c}}\approx cz-H_{0}d,$

where d is a redshift-independent distance. Catalogues constructed from the Tully–Fisher relation and fundamental-plane methods are used to map local matter flows and gravitational structures.

Individual scaling-relation distances commonly have substantial scatter. Precision improves when several galaxies in a group or cluster are combined.

Galaxy populations and bimodality

Galaxy properties are distributed across partially distinct populations. In color–magnitude and color–mass diagrams, many galaxies lie either on a blue star-forming sequence or a red quiescent sequence, with a less populated intermediate region sometimes called the green valley.

This bimodality affects scaling relations. Star-forming and quiescent galaxies commonly follow different:

• mass–size relations;
• color–mass relations;
• star-formation relations;
• gas-fraction relations;
• surface-density distributions; and
• structural sequences.

Morphological classification and star-formation state are related but not identical. A red disk and a blue spheroid may not follow the same relations as typical members of their morphological class. Modern studies therefore often separate galaxies using several criteria, such as color, specific star formation rate, Sérsic index, kinematics, and visual morphology.

Environmental dependence

Galaxy environment influences the relative numbers of different galaxy types and can also modify individual scaling relations. Relevant environments include:

• isolated field regions;
• galaxy pairs;
• groups;
• rich clusters;
• filaments; and
• cosmic voids.

Processes such as ram-pressure stripping, tidal interactions, harassment, strangulation, and mergers can change gas content, morphology, star formation, size, and stellar structure.

For many nearby galaxy relations, much of the environmental difference results from changes in the mixture of galaxy populations rather than large shifts in the relation followed by each fixed type. Smaller residual environmental effects have nevertheless been measured.

Cluster early-type galaxies historically provided important samples for studies of the color–magnitude relation, fundamental plane, and D_n–sigma relation because many galaxies could be assumed to lie at approximately the same distance.

Evolution with cosmic time

Galaxy scaling relations evolve because galaxies accrete gas, form stars, merge, quench, and undergo structural transformation.

Observed evolutionary trends include:

• increasing star-forming-main-sequence normalization with redshift;
• decreasing typical size of massive quiescent galaxies at earlier epochs;
• evolution in gas fraction and depletion time;
• changes in the mass–metallicity relation;
• evolution in luminosity-based relations caused by aging stellar populations; and
• possible changes in black-hole–host relations.

Evolution must be distinguished from selection effects. At high redshift, surveys preferentially detect bright, massive, compact, or strongly star-forming systems. Measurements are also affected by cosmological surface-brightness dimming,

$I_{\mathrm{o}\mathrm{b}\mathrm{s}}=\frac{I_{\mathrm{e}\mathrm{m}}}{(1+z{)}^4},$

where I_obs is observed bolometric surface brightness and I_em is emitted surface brightness.

Changes in rest-frame wavelength, spatial resolution, signal-to-noise ratio, and galaxy classification can produce apparent evolution if they are not treated consistently.

Measurement methods

Galaxy scaling relations combine measurements from imaging, spectroscopy, radio observations, and inferred physical models.

Photometry

Imaging provides:

• apparent magnitude;
• color;
• surface-brightness profile;
• effective radius;
• disk scale length;
• Sérsic index;
• concentration;
• axis ratio; and
• morphological structure.

A commonly used Sérsic profile is

$I(R)=I_e\exp \{-b_n[{\left(\frac{R}{R_e}\right)}^{1/n}-1]\},$

where:

• I_e is the intensity at the effective radius;
• R_e is the effective radius;
• n is the Sérsic index; and
• b_n is chosen so that R_e encloses half the total model luminosity.

Different profile-fitting procedures can produce systematically different galaxy luminosities and sizes, particularly for massive galaxies with extended outer envelopes.

Spectroscopy

Spectroscopy provides:

• redshift;
• stellar velocity dispersion;
• emission-line velocity;
• gas-phase metallicity;
• stellar age and metallicity;
• star formation rate;
• dust attenuation; and
• active-galactic-nucleus diagnostics.

Integrated spectra combine light across a large portion of a galaxy, while integral field spectroscopy maps spatially resolved stellar and gas properties.

Radio and millimeter observations

The 21-centimeter line of neutral hydrogen measures atomic-gas mass and rotational line width. Molecular gas is commonly inferred from carbon-monoxide emission using a conversion factor:

$M_{{\mathrm{H}}_{\mathrm{2}}}={\alpha }_{\mathrm{C}\mathrm{O}}L{\text{'}}_{\mathrm{C}\mathrm{O}},$

where L′_CO is carbon-monoxide line luminosity and α_CO is the conversion factor. The factor varies with metallicity, gas conditions, and galaxy type.

Stellar-mass estimation

Stellar masses are usually inferred by comparing multiband photometry or spectra with models of stellar populations. A simplified expression is

$M_{\star }={\mathrm{{\rm Y}}}_{\star }L,$

where Υ_★ is the stellar mass-to-light ratio.

Systematic uncertainty arises from stellar-evolution models, dust, star-formation history, metallicity, and the adopted initial mass function.

Statistical fitting

The fitted slope and normalization of a relation depend on the statistical method. Ordinary least-squares regression of Y on X is not equivalent to regression of X on Y when both variables contain errors and intrinsic scatter.

Common approaches include:

• direct regression;
• inverse regression;
• orthogonal regression;
• maximum-likelihood fitting;
• hierarchical Bayesian modeling; and
• forward modeling of the survey selection function.

For independent Gaussian uncertainties in both coordinates, a simplified likelihood may include

$s_i^2={\sigma }_{y,i}^2+{\alpha }^2{\sigma }_{x,i}^2+{\sigma }_{\mathrm{i}\mathrm{n}\mathrm{t}}^2,$

and

$\ln {ℒ}=-\frac{1}{2}\sum _i[\frac{(y_i-\alpha x_i-\beta {)}^2}{s_i^2}+\ln (2\pi s_i^2)].$

This expression is simplified because it does not include covariance, correlated systematic errors, truncation, or a model of the population distribution.

Covariance

Galaxy measurements are often correlated. Stellar mass and star formation rate may be derived from the same photometry, while luminosity and size may come from the same profile fit. Ignoring covariance can underestimate uncertainties or create artificial relations.

A covariance matrix may be written as

$\mathbf{𝐂}=\left(\begin{array}{cc}{\sigma }_x^2& \rho {\sigma }_x{\sigma }_y\\ \rho {\sigma }_x{\sigma }_y& {\sigma }_y^2\end{array}\right),$

where ρ is the correlation coefficient of the measurement errors.

Principal-component analysis

Relations involving several observables are sometimes examined with principal component analysis. The technique identifies directions of greatest variance in a multidimensional dataset and can reveal whether galaxies occupy an approximately one-dimensional sequence, a plane, or a thicker manifold.

Principal-component directions do not automatically have a causal physical interpretation and may be strongly influenced by sample selection and measurement scaling.

Scatter

Observed scatter contains several contributions:

${\sigma }_{\mathrm{o}\mathrm{b}\mathrm{s}}^2\approx {\sigma }_{\mathrm{i}\mathrm{n}\mathrm{t}}^2+{\sigma }_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}^2+{\sigma }_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}^2,$

where:

• σ_int represents genuine galaxy-to-galaxy variation;
• σ_meas represents observational uncertainty; and
• σ_model represents mismatch between the fitted functional form and the true population.

Intrinsic scatter can encode physical information. It may reflect differences in:

• formation time;
• merger history;
• gas accretion;
• feedback efficiency;
• halo spin;
• dark-matter concentration;
• stellar initial mass function;
• environment; and
• viewing orientation.

A small observed scatter does not necessarily imply a simple physical origin, because correlated processes or selection can compress the population into a narrow relation.

Selection effects and biases

Malmquist bias

In a flux-limited survey, intrinsically luminous objects can be detected at greater distances than faint objects. Scatter in a luminosity-based relation can therefore bias inferred distances and slopes. This is a form of Malmquist bias.

Eddington bias

If objects are more numerous on one side of a distribution than the other, measurement scatter moves more objects from the populous region into the sparse region than in the opposite direction. This Eddington bias can alter the high-mass or high-luminosity end of a relation.

Surface-brightness selection

Diffuse galaxies can be missed even when their total luminosities exceed a nominal survey threshold. This affects size–luminosity, surface-brightness, and baryonic relations.

Morphological selection

Different classification methods select different populations. Visual morphology, color, Sérsic index, spectral classification, and kinematic support are not interchangeable.

Aperture effects

Spectroscopic quantities measured through a fixed angular fiber sample different physical fractions of galaxies at different distances. Velocity dispersion, metallicity, and star formation can vary radially, requiring aperture corrections or resolved measurements.

Distance covariance

Two quantities can appear correlated because both depend on an adopted distance. Luminosity scales as distance squared, while physical size scales linearly with distance:

$L\propto d_L^2,$

$R\propto d_A.$

Distance uncertainty can therefore introduce correlated errors into mass–size and luminosity–size relations.

Calibration differences

Strong-line metallicity calibrations, stellar population models, dust corrections, gas-conversion factors, and photometric decompositions can shift zero points by amounts greater than the statistical uncertainty within a single analysis. Comparisons require consistent definitions and calibrations.

Use in galaxy-formation models

Scaling relations are central tests of galaxy-formation simulations. Models attempt to reproduce simultaneously:

• the stellar-to-halo mass relation;
• the galaxy stellar-mass function;
• the Tully–Fisher relation;
• the mass–size relation;
• the star-forming main sequence;
• gas fractions;
• metallicities;
• black-hole scaling relations; and
• the proportion of quiescent galaxies.

Matching one relation does not guarantee a physically accurate model. Parameters can compensate for one another, and a model may reproduce a mean relation while failing to reproduce scatter, residual covariance, morphology dependence, radial structure, or redshift evolution.

Hydrodynamic simulations directly follow dark matter, gas dynamics, radiative cooling, star formation, enrichment, and feedback within a cosmological volume. Semi-analytic models apply simplified prescriptions to dark-matter merger trees. Both approaches calibrate or test their physical assumptions using observed scaling relations.

Limitations of interpretation

Galaxy scaling relations are powerful summaries but have several limitations:

• Correlation does not establish causation.
• A projected two-variable relation may hide additional controlling variables.
• Different physical mechanisms can produce similar slopes.
• Apparent breaks may result from sample selection or calibration.
• A single power law may not describe the full mass range.
• Dwarf, low-surface-brightness, starburst, and interacting galaxies may deviate from relations calibrated with ordinary massive systems.
• Redshift evolution can be confused with changing sample composition.
• Strong covariance among inferred quantities can create artificially tight relations.
• Numerical agreement with a relation does not uniquely identify the correct galaxy-formation model.

Relations should therefore be interpreted with their measurement definitions, sample selection, residuals, and intrinsic scatter.

Historical development

Early galaxy classification emphasized morphology and luminosity. During the 20th century, increasingly accurate photometry and spectroscopy revealed that galaxy structure and dynamics were strongly correlated.

Important milestones include:

• identification of surface-brightness and luminosity regularities in galaxies;
• the Faber–Jackson relation in 1976;
• the Tully–Fisher relation in 1977;
• the Kormendy relation in 1977;
• establishment of the fundamental plane and D_n–sigma relation in 1987;
• development of black-hole–host-galaxy relations during the 1990s and early 2000s;
• precise measurement of the mass–metallicity relation in large redshift surveys;
• recognition of the star-forming main sequence;
• expansion from integrated to spatially resolved relations through integral-field and radio surveys; and
• measurement of relation evolution across much of cosmic history.

Large surveys such as the Sloan Digital Sky Survey, Galaxy And Mass Assembly survey, 6dF Galaxy Survey, Mapping Nearby Galaxies at Apache Point Observatory, and deep Hubble Space Telescope programs greatly increased the number and range of galaxies available for statistical study.

Comparison of selected relations

Selected galaxy scaling relations

Relation	Principal variables	Typical population	Main applications
Tully–Fisher relation	Luminosity and rotation velocity	Disk galaxies	Distances, disk dynamics
Baryonic Tully–Fisher relation	Baryonic mass and rotation velocity	Gas-rich and stellar disks	Baryon–halo connection
Faber–Jackson relation	Luminosity and velocity dispersion	Elliptical galaxies	Dynamics, approximate distances
Fundamental plane	Radius, surface brightness, velocity dispersion	Early-type galaxies	Distances, structure, evolution
Dn–sigma relation	Characteristic diameter and velocity dispersion	Early-type galaxies	Distances, peculiar velocities
Kormendy relation	Effective radius and surface brightness	Elliptical galaxies and bulges	Structural comparison
Stellar mass–size relation	Stellar mass and effective radius	Most galaxy types	Structural evolution
Star-forming main sequence	Stellar mass and star formation rate	Star-forming galaxies	Growth and quenching
Kennicutt–Schmidt law	Gas and star-formation surface densities	Star-forming regions and galaxies	Star-formation efficiency
Mass–metallicity relation	Stellar mass and chemical abundance	Star-forming galaxies	Chemical evolution and feedback
M–sigma relation	Black-hole mass and bulge velocity dispersion	Galaxies with measured central black holes	Black-hole and bulge evolution
Specific-angular-momentum relation	Specific angular momentum and mass	Disks and spheroids	Morphology and angular-momentum evolution

See also

• Cosmic distance ladder
• Dark matter halo
• Elliptical galaxy
• Galaxy color–magnitude diagram
• Galaxy formation and evolution
• Galaxy morphology
• Galaxy rotation curve
• Galaxy stellar mass function
• Hubble sequence
• Large-scale structure of the universe
• Spiral galaxy
• Standard candle
• Standard ruler

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Further reading

• Binney, James; Tremaine, Scott (2008). Galactic Dynamics (2nd ed.). Princeton: Princeton University Press. ISBN 978-0-691-13026-2. Search this book on
• Blanton, Michael R.; Moustakas, John (September 2009). "Physical properties and environments of nearby galaxies". Annual Review of Astronomy and Astrophysics. 47 (1): 159–210. arXiv:0908.3017. Bibcode:2009ARA&A..47..159B. doi:10.1146/annurev-astro-082708-101734.
• Bovy, Jo (2026). "Galaxy scaling relations". Dynamics and Astrophysics of Galaxies. Princeton University Press. Search this book on
• D'Onofrio, Mauro; Marziani, Paola; Chiosi, Cesare (2021). "Past, present and future of the scaling relations of galaxies and active galactic nuclei". Frontiers in Astronomy and Space Sciences. 8. arXiv:2109.06301. Bibcode:2021FrASS...8..157D. doi:10.3389/fspas.2021.694554.
• Mo, Houjun; van den Bosch, Frank; White, Simon (2010). Galaxy Formation and Evolution. Cambridge: Cambridge University Press. Bibcode:2010gfe..book.....M. doi:10.1017/CBO9780511807244. ISBN 978-0-521-85793-2. Search this book on

• Somerville, Rachel S.; Davé, Romeel (August 2015). "Physical models of galaxy formation in a cosmological framework". Annual Review of Astronomy and Astrophysics. 53: 51–113. arXiv:1412.2712. Bibcode:2015ARA&A..53...51S. doi:10.1146/annurev-astro-082812-140951.
• Saintonge, Amélie; Catinella, Barbara (August 2022). "The cold interstellar medium of galaxies in the local universe". Annual Review of Astronomy and Astrophysics. 60: 319–361. arXiv:2202.00690. Bibcode:2022ARA&A..60..319S. doi:10.1146/annurev-astro-021022-043545.

External links

• NASA/IPAC Extragalactic Database Level 5 knowledge base
• NASA Astrophysics Data System

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