Dn–sigma relation

The D_n–σ relation is an empirical scaling relation for elliptical and lenticular galaxies. It relates a galaxy's characteristic diameter, denoted by D_n, to the velocity dispersion of its stars, denoted by the Greek letter σ (sigma). Because stellar velocity dispersion is independent of a galaxy's distance, while its observed angular diameter varies with distance, the relation can be used as a standard ruler for estimating extragalactic distances.

Messier 49, a giant elliptical galaxy in the Virgo Cluster. The D_n–σ relation connects a characteristic photometric diameter of an early-type galaxy with the velocity dispersion of its stars.

The relation is commonly expressed as a power law,

$D_{n}\propto \sigma ^{\gamma },$

or, in logarithmic form,

$\log D_{n}=\gamma \log \sigma +C,$

where γ is the fitted slope and C is a calibration-dependent zero point. The measured coefficients depend on the photometric band, the adopted surface-brightness threshold, the galaxy sample, and the direction of the statistical regression. Typical determinations give values of γ between approximately 1.1 and 1.3.

The D_n–σ relation is closely related to the fundamental plane of elliptical galaxies. Both relations combine structural and dynamical properties of early-type galaxies and have been used to estimate galaxy-cluster distances, measure peculiar velocities, map large-scale matter flows, and constrain the Hubble constant.

Definition

The quantity D_n is the physical diameter of a galaxy within an aperture whose mean surface brightness reaches a specified value. In early applications using the Johnson B band, the conventional threshold was a mean surface brightness of approximately 20.75 magnitudes per square arcsecond.

If L(<r) is the luminosity enclosed within an angular radius r, the mean intensity within that radius is

$\langle I\rangle (r)={\frac {L(<r)}{\pi r^{2}}}.$

The angular radius r_n is selected so that

$\langle \mu \rangle (r_{n})=\mu _{n},$

where ⟨μ⟩ is the mean surface brightness expressed in magnitudes per square arcsecond and μ_n is the adopted surface-brightness threshold. The corresponding angular diameter is

$d_{n}=2r_{n}.$

Once the distance Δ to the galaxy is known or estimated, its physical diameter is

$D_{n}=\Delta d_{n},$

under the small-angle approximation, with d_n expressed in radians.

The second variable, σ, is the line-of-sight stellar velocity dispersion measured from the broadening of absorption lines in the galaxy's integrated spectrum. It is usually reported in kilometers per second. The observed velocity dispersion is normally corrected to a standardized aperture because measurements taken through different angular apertures sample different fractions of a galaxy.

For a calibrated sample, the physical diameter predicted from the measured velocity dispersion can be compared with the observed angular diameter. The angular-diameter distance is then estimated from

$\Delta ={\frac {D_{n}(\sigma )}{d_{n}}}.$

The method is therefore a standard-ruler technique rather than a conventional standard candle method.

Mathematical form

A general form of the relation is

$\log _{10}\left({\frac {D_{n}}{D_{0}}}\right)=\gamma \log _{10}\left({\frac {\sigma }{\sigma _{0}}}\right)+\beta ,$

where:

• D₀ is a reference diameter;
• σ₀ is a reference velocity dispersion;
• γ is the slope; and
• β is the zero point.

The reference quantities make the logarithms dimensionless. Their numerical values are chosen for convenience and do not change the physical meaning of the relation.

A normalized power-law form may be written as

${\frac {D_{n}}{D_{0}}}=10^{\beta }\left({\frac {\sigma }{\sigma _{0}}}\right)^{\gamma }.$

The numerical coefficients are not universal constants. Changing the photometric band, surface-brightness threshold, aperture correction, sample selection, regression method, or adopted distance scale changes the fitted values.

Using R-band photometry for 452 elliptical and lenticular galaxies in 28 clusters, Bernardi and collaborators obtained the direct relation

$\log D_{n}=1.203\log \sigma +1.406,$

after correcting for sample incompleteness and establishing the zero point with distant clusters assumed to be at rest in the cosmic microwave background reference frame. The units and normalization in this expression are specific to the authors' observational system. Their measured scatter corresponded to a typical individual distance uncertainty of approximately 20 percent.

Angular-diameter distance

Under the small-angle approximation,

$D_{n}=\Delta \theta _{n},$

where:

• D_n is the characteristic physical diameter;
• Δ is the angular-diameter distance; and
• θ_n is the measured angular diameter in radians.

The distance is therefore

$\Delta ={\frac {D_{n}}{\theta _{n}}}.$

If D_n is measured in kiloparsecs and the angular diameter is measured in arcseconds, the distance in megaparsecs can be expressed as

$\Delta \;(\mathrm {Mpc} )=0.206265{\frac {D_{n}\;(\mathrm {kpc} )}{d_{n}\;(\mathrm {arcsec} )}}.$

Distance modulus

If the relation yields a distance Δ in parsecs, the corresponding distance modulus is

$\mu =5\log _{10}\left({\frac {\Delta }{10\ \mathrm {pc} }}\right).$

For galaxy clusters, distances are generally derived from several member galaxies. Combining multiple measurements reduces the random uncertainty in the mean cluster distance, although systematic errors and correlated environmental effects remain.

Peculiar velocity

Once a redshift-independent distance has been obtained, a galaxy's radial peculiar velocity can be approximated at low redshift by

$v_{\mathrm {pec} }\approx cz-H_{0}\Delta ,$

where:

• v_pec is the radial peculiar velocity;
• c is the speed of light;
• z is the observed redshift;
• H₀ is the Hubble constant; and
• Δ is the independently estimated distance.

More detailed calculations include cosmological redshift corrections, transformations between reference frames, and corrections for survey selection effects.

Observational measurements

An elliptical galaxy in the Coma Cluster. Early-type galaxies in rich clusters have frequently been used to calibrate the D_n–σ relation.

Determining a D_n–σ distance requires both surface photometry and spectroscopy.

Surface photometry

The galaxy is imaged through a specified photometric filter. Foreground stars, neighboring galaxies, detector artifacts, and the background sky are removed or modeled. Astronomers then fit elliptical isophotes to the galaxy's light distribution and construct a surface brightness profile.

The cumulative luminosity within successive apertures is used to determine the radius at which the mean enclosed surface brightness equals the adopted threshold. Corrections may be applied for:

• Galactic extinction;
• cosmological surface-brightness dimming;
• K-corrections;
• atmospheric seeing;
• detector response;
• galaxy ellipticity and orientation;
• foreground contamination; and
• uncertainties in the sky background.

The angular diameter measured from this procedure is distance dependent: a physically identical galaxy appears smaller at a greater distance.

The mean surface brightness within an angular diameter d_n can also be expressed in magnitude units as

$\langle \mu \rangle _{D_{n}}=m(<D_{n}/2)+2.5\log _{10}\left[\pi \left({\frac {d_{n}}{2}}\right)^{2}\right],$

where m(<D_n/2) is the apparent magnitude enclosed within the aperture.

Stellar velocity dispersion

The central stellar velocity dispersion is measured by comparing the galaxy's absorption-line spectrum with spectra of template stars or stellar-population models. Doppler motions within the galaxy broaden features such as the calcium H and K lines, the G band, the magnesium b triplet, and iron absorption lines.

The measured dispersion may be represented schematically by

$\sigma _{\mathrm {obs} }^{2}=\sigma _{\star }^{2}+\sigma _{\mathrm {inst} }^{2},$

where σ_obs is the observed line broadening, σ_★ is the intrinsic stellar velocity dispersion, and σ_inst is the broadening introduced by the instrument. After measuring the instrumental contribution,

$\sigma _{\star }={\sqrt {\sigma _{\mathrm {obs} }^{2}-\sigma _{\mathrm {inst} }^{2}}}.$

In practice, the velocity dispersion is obtained by fitting broadened stellar templates or synthetic spectra to the galaxy spectrum. Common techniques include:

• direct pixel-space fitting;
• Fourier cross-correlation;
• Fourier quotient methods; and
• combinations of stellar-population templates.

Corrections are required when the spectrograph aperture covers different physical regions in galaxies at different distances. Velocity dispersion usually decreases with distance from the center, so measurements must be transformed to a standard aperture before they are compared.

Sample selection

Most applications use elliptical and lenticular galaxies because their stellar systems are largely supported by random stellar motion rather than ordered disk rotation. Galaxies may be excluded when they display:

• prominent dust lanes;
• strong ongoing star formation;
• disturbed or interacting morphologies;
• recent merger signatures;
• strong emission lines;
• substantial disk contamination; or
• uncertain cluster membership.

Cluster membership is normally established using redshift, sky position, and the velocity distribution of the cluster. Contamination by foreground or background galaxies can bias the inferred cluster distance.

Physical interpretation

The relation reflects the connection between the sizes, stellar motions, luminosities, and mass distributions of early-type galaxies. A simplified interpretation follows from the virial theorem. For a gravitationally supported galaxy of characteristic mass M, radius R, and velocity dispersion σ,

$M\sim k{\frac {\sigma ^{2}R}{G}},$

where G is the gravitational constant and k is a dimensionless structural coefficient.

The luminosity may be written approximately as

$L\sim IR^{2},$

where I is a characteristic surface brightness. Combining these expressions gives

${\frac {M}{L}}\sim {\frac {k\sigma ^{2}}{GIR}}.$

If early-type galaxies were perfectly homologous systems with constant mass-to-light ratios, their structural parameters would follow a simple virial scaling. Observed galaxies depart from this idealization because stellar populations, dark-matter fractions, orbital distributions, rotational support, and internal structures vary systematically with galaxy mass.

The D_n–σ relation reduces some of this complexity by defining the diameter at a fixed mean surface brightness. It can be interpreted as a projection of the three-parameter fundamental plane onto a two-dimensional relation.

The detailed slope and zero point may be affected by:

• variations in stellar mass-to-light ratio;
• differences in stellar age and metallicity;
• variations in dark-matter fraction;
• structural non-homology;
• rotational support;
• orbital anisotropy; and
• observational selection effects.

The relation is therefore empirical rather than a universal physical law with fixed coefficients.

Relation to other galaxy scaling laws

Fundamental plane

Main article: Fundamental plane (elliptical galaxies)

The fundamental plane describes a relation among the effective radius R_e, mean surface brightness ⟨I⟩_e, and central velocity dispersion σ of early-type galaxies:

$R_{e}\propto \sigma ^{a}\langle I\rangle _{e}^{b}.$

In logarithmic form, it is commonly written as

$\log R_{e}=a\log \sigma +b\log \langle I\rangle _{e}+c.$

The measured exponents depend on the passband and galaxy sample but differ from the simplest virial prediction. Because D_n incorporates a surface-brightness condition into the definition of the diameter, the D_n–σ relation can be understood as a nearly edge-on projection of the fundamental plane.

The fundamental plane generally uses three measured quantities and can produce less scatter, while the D_n–σ relation is simpler to apply and was particularly useful in photographic and spectroscopic surveys.

Faber–Jackson relation

Main article: Faber–Jackson relation

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

$L\propto \sigma ^{\alpha },$

with α historically approximated as 4.

Both relations use velocity dispersion, but the Faber–Jackson relation predicts luminosity, whereas the D_n–σ relation predicts a characteristic linear diameter. Incorporating surface-brightness information generally reduces the scatter compared with a luminosity–dispersion relation alone.

Tully–Fisher relation

Main article: Tully–Fisher relation

The Tully–Fisher relation is primarily applied to spiral galaxies and relates luminosity or baryonic mass to rotational velocity. The D_n–σ relation plays an analogous distance-indicator role for pressure-supported early-type galaxies, whose stellar motions are dominated by velocity dispersion rather than ordered disk rotation.

M–sigma relation

Main article: M–sigma relation

The D_n–σ relation should not be confused with the M–sigma relation, which connects the mass of a galaxy's central supermassive black hole with the stellar velocity dispersion of its bulge.

Historical development

Earlier scaling relations

Before the formal development of the D_n–σ relation, astronomers had recognized correlations among the luminosities, sizes, surface brightnesses, and stellar motions of elliptical galaxies. The Faber–Jackson relation, introduced in 1976, demonstrated a strong correlation between luminosity and central velocity dispersion.

Improved photometry and spectroscopy during the 1970s and 1980s showed that including a galaxy's structural properties could substantially reduce the scatter in distance estimates.

Diameter-based relations were developed partly because estimating total galaxy luminosities was difficult. The outer regions of elliptical galaxies gradually merge into the sky background, making total magnitudes sensitive to the adopted photometric model and the accuracy of sky subtraction. A diameter defined at a standardized mean surface brightness could be measured more consistently.

Introduction in 1987

Two major studies published in 1987 established the modern framework. Dressler and six collaborators introduced the D_n–σ relation as a distance estimator based on observations of elliptical galaxies. In a separate study, S. Djorgovski and Marc Davis showed that the global properties of elliptical galaxies occupy a narrow plane in parameter space, now called the fundamental plane.

Dressler subsequently extended the D_n–σ method to the bulges of disk galaxies and used it to derive an independent estimate of the Hubble constant.

Seven Samurai survey

The relation became especially prominent through the work of a group of astronomers informally called the Seven Samurai: David Burstein, Roger Davies, Alan Dressler, Sandra Faber, Donald Lynden-Bell, Roberto Terlevich, and Gary Wegner.

The group measured distances and redshifts for approximately 400 elliptical galaxies. Their analysis revealed a coherent large-scale motion toward a region they called the Great Attractor.

The result was influential in studies of the large-scale velocity field and demonstrated the importance of redshift-independent galaxy distances. It also stimulated discussion of statistical biases, calibration errors, and the interpretation of coherent galaxy motions.

Development of the fundamental plane

The recognition of the fundamental plane provided a broader framework for understanding the D_n–σ relation. Instead of considering galaxy diameter and velocity dispersion alone, the fundamental plane explicitly includes surface brightness as an independent variable.

Studies of the Coma Cluster found that the fundamental plane could have less scatter than the corresponding D_n–σ relation. Jørgensen, Franx, and Kjærgaard reported scatter of approximately 11 percent for the fundamental plane, compared with approximately 17 percent for the D_n–σ relation in their Coma sample.

Later surveys

Subsequent surveys standardized photometric and spectroscopic measurements across larger samples. The ENEAR survey used early-type galaxies across much of the sky to construct a homogeneous peculiar-velocity catalogue.

Bernardi and collaborators derived a template relation from 452 elliptical and lenticular galaxies in 28 clusters and reported an individual distance uncertainty of approximately 20 percent.

The development of larger digital imaging and spectroscopic surveys later made the full fundamental plane and other multivariable scaling relations easier to measure. As a result, the standalone D_n–σ method became less dominant in modern distance-scale work, although it remains historically and conceptually important.

Calibration

The zero point must be calibrated before the relation can provide absolute distances. A calibration may be established through one or more of the following approaches:

• assigning distances to nearby clusters using Cepheid variables;
• using surface brightness fluctuation distances;
• comparing with Type Ia supernova distances;
• fixing the mean motion of sufficiently distant clusters relative to the cosmic microwave background; or
• adopting an independently measured value of the Hubble constant.

The Hubble Space Telescope Key Project calibrated the fundamental-plane and D_n–σ relations in the Leo I Group, Virgo Cluster, and Fornax Cluster using Cepheid distances to spiral galaxies associated with those systems. The resulting relations were applied to more distant clusters to estimate the Hubble constant.

Calibration is complicated because Cepheid variables occur in young stellar populations and are therefore uncommon in elliptical galaxies. Distances to elliptical galaxies are often transferred from spiral galaxies believed to belong to the same cluster or group. Differences in spatial depth, cluster substructure, and group membership can introduce systematic uncertainty.

A typical calibration procedure includes:

1. selecting a homogeneous sample of early-type galaxies;
2. transforming photometry to a common passband and magnitude system;
3. correcting angular diameters for Galactic extinction, seeing, redshift, and cosmological dimming;
4. correcting velocity dispersions to a common instrumental and aperture system;
5. fitting a direct, inverse, or orthogonal regression;
6. modeling sample incompleteness and selection bias;
7. determining relative cluster distances; and
8. anchoring the zero point to an external distance scale.

Applications

The central region of the Coma Cluster, which contains numerous elliptical galaxies. Cluster samples reduce random distance errors by combining measurements from several galaxies.

Galaxy-cluster distances

Early-type galaxies are abundant in rich clusters and often contain little dust or ongoing star formation. These properties make them suitable for homogeneous photometric and spectroscopic studies.

A distance can be estimated for each suitable cluster member and then combined to derive the mean cluster distance. The statistical uncertainty decreases approximately as

$\sigma _{\bar {d}}\approx {\frac {\sigma _{d}}{\sqrt {N}}},$

where σ_d is the individual distance scatter and N is the number of independent galaxies. Correlated systematic errors do not decrease in the same manner.

The method is particularly useful for clusters because averaging several galaxies reduces the effect of intrinsic scatter. It also provides distances to early-type galaxies, which cannot normally be measured using methods that depend on young stars or active star-forming regions.

Peculiar-velocity surveys

Comparing D_n–σ distances with galaxy redshifts allows astronomers to measure deviations from uniform Hubble expansion. These peculiar velocities trace the gravitational effects of matter concentrations and cosmic voids.

The method contributed to early evidence for coherent flows on scales of tens to hundreds of megaparsecs, including the streaming motion associated with the Great Attractor.

Peculiar-velocity measurements can be combined to reconstruct the local gravitational and density fields. Because elliptical galaxies are concentrated in galaxy clusters, the relation is especially useful for tracing the motions of dense environments.

Hubble constant

When the relation is calibrated with independently known distances, it can be applied to distant clusters whose peculiar velocities are small relative to their recession velocities. The Hubble constant may then be approximated by

$H_{0}\approx {\frac {v}{\Delta }},$

where v is the recession velocity after appropriate corrections and Δ is the independently estimated distance.

Dressler's 1987 application to bulges of disk galaxies produced an independent estimate of the cosmic expansion rate. Later work combined D_n–σ and fundamental-plane distances with Cepheid calibrations from the Hubble Space Telescope Key Project.

Galaxy structure and evolution

The slope, scatter, and residuals of the relation provide information about variations in stellar populations, mass-to-light ratios, internal dynamics, and structural homology.

Differences among photometric bands or environments can indicate the effects of:

• stellar age;
• metallicity;
• galaxy mergers;
• cluster environment;
• dark-matter content; and
• rotational support.

The relation is therefore useful not only as a distance indicator but also as a diagnostic of the physical properties of early-type galaxies.

Scatter and uncertainty

The relation has both intrinsic and observational scatter. Important sources include:

• uncertainties in sky subtraction and surface photometry;
• errors in stellar velocity dispersion;
• differences in stellar age and chemical composition;
• internal rotation and orbital anisotropy;
• dust or recent star formation;
• variations in galaxy structure and mass-to-light ratio;
• contamination by galaxies that are not dynamically typical early-type systems;
• cluster depth and membership errors; and
• sample-selection effects.

Studies of Coma Cluster galaxies found that the fundamental plane could have smaller scatter than the corresponding D_n–σ projection. Residuals were also found to correlate with surface brightness and other galaxy properties.

A surface-brightness correction has been proposed because galaxies with different light-profile shapes or stellar populations may occupy systematically different positions in the relation.

Individual-galaxy distance uncertainties are commonly between approximately 15 and 25 percent, depending on the data quality, calibration, galaxy sample, and method of fitting. Cluster distances are generally more precise because several galaxy measurements can be averaged.

Statistical biases

Malmquist bias

Distance-indicator samples are vulnerable to Malmquist bias. At a given observed distance, brighter or larger galaxies are more likely to enter a magnitude- or diameter-limited catalogue than fainter or smaller galaxies. Scatter in the relation can therefore produce systematically biased inferred distances.

Corrections require a model of the sample selection, the intrinsic distribution of galaxy properties, and the spatial density field. Failure to account for the bias can create or exaggerate apparent large-scale peculiar motions.

Incompleteness bias

If the observed catalogue excludes galaxies below a luminosity, diameter, or signal-to-noise threshold, the measured slope and zero point can be biased. Bernardi and collaborators explicitly corrected their cluster template for incompleteness.

The effect becomes more important at large distances because only the brightest, largest, or highest-surface-brightness galaxies remain detectable.

Regression direction

A direct regression minimizes residuals in log D_n, whereas an inverse regression minimizes residuals in log σ. An orthogonal regression minimizes the distances perpendicular to the fitted line.

The different procedures generally yield different slopes when both variables contain measurement errors and the sample is truncated. The preferred regression depends on the scientific application and the selection properties of the survey.

A fit intended to describe the intrinsic physical relation may not be the same as the fit that gives the least biased galaxy distances.

Environmental effects

Cluster and field galaxies may differ in age, metallicity, merger history, and stellar-population properties. If these differences alter the zero point or slope, applying a single universal relation can introduce systematic errors.

Studies commonly test residuals against:

• cluster richness;
• local galaxy density;
• galaxy color;
• magnesium absorption strength;
• surface brightness;
• rotational support; and
• morphological subtype.

Advantages and limitations

The principal advantages of the D_n–σ relation are:

• it applies to early-type galaxies, which are common in rich clusters;
• it uses a distance-independent spectroscopic measurement;
• it can reach beyond the distances at which individual stars can be resolved;
• it is generally less affected by internal dust than methods applied to many spiral galaxies;
• it permits distances to be measured across large regions of the sky;
• cluster distances can be improved by averaging several member galaxies; and
• it complements spiral-galaxy methods such as the Tully–Fisher relation.

Its limitations include:

• relatively large uncertainties for individual galaxies;
• dependence on photometric calibration and surface-brightness thresholds;
• sensitivity to selection effects and regression method;
• the need for sufficiently high-quality spectroscopy;
• possible dependence on stellar population and environment;
• sensitivity to aperture corrections;
• possible contamination from rotationally supported lenticular galaxies;
• the need for external calibration; and
• overlap with the more informative fundamental-plane relation.

The relation is therefore most effective when applied to homogeneous samples containing several early-type galaxies per group or cluster.

Comparison with related distance indicators

Selected extragalactic distance indicators

Method	Principal galaxy type	Main observables	General use
Tully–Fisher relation	Spiral galaxies	Rotational velocity and luminosity or baryonic mass	Secondary distance indicator
Dn–σ relation	Elliptical and lenticular galaxies	Characteristic diameter and stellar velocity dispersion	Secondary distance indicator and peculiar-velocity probe
Fundamental plane	Early-type galaxies	Effective radius, surface brightness, and velocity dispersion	Secondary distance indicator and galaxy-structure relation
Faber–Jackson relation	Elliptical galaxies	Luminosity and stellar velocity dispersion	Scaling relation and approximate distance indicator
Surface brightness fluctuation	Galaxies with measurable stellar brightness fluctuations	Pixel-to-pixel brightness variance	Secondary distance indicator
Type Ia supernova	Galaxies hosting a suitable supernova	Standardized peak luminosity	Cosmological distance indicator

Unlike the Tully–Fisher relation, which uses the ordered rotation of a disk galaxy, the D_n–σ relation uses the random stellar motions of a dynamically hot galaxy. Unlike the fundamental plane, it combines photometric structure into a single characteristic diameter rather than retaining radius and surface brightness as separate variables.

Nomenclature

The name is variously written as:

• D_n–σ relation;
• D_n–sigma relation;
• D_n–σ correlation;
• diameter–dispersion relation; and
• diameter–velocity-dispersion relation.

The subscript n refers to the adopted normalized mean surface-brightness level. It does not represent a mathematical power or the number of galaxies in the sample.

The relation should not be confused with:

• the M–sigma relation, which connects the mass of a supermassive black hole with the velocity dispersion of its host galaxy's bulge;
• the Σ–D relation, which connects radio surface brightness with the diameter of some supernova remnants; or
• the Faber–Jackson relation, which connects total luminosity with velocity dispersion.

See also

• Cosmic distance ladder
• Early-type galaxy
• Elliptical galaxy
• Faber–Jackson relation
• Fundamental plane (elliptical galaxies)
• Galaxy scaling relations
• Great Attractor
• Hubble–Lemaître law
• Peculiar velocity
• Standard ruler
• Surface brightness fluctuation
• Tully–Fisher relation
• Velocity dispersion

Further reading

• Bender, R.; Burstein, D.; Faber, S. M. (December 1992). "Dynamically hot galaxies. I. Structural properties". The Astrophysical Journal. 399: 462–477. Bibcode:1992ApJ...399..462B. doi:10.1086/171940.
• Jacoby, George H.; Branch, David; Ciardullo, Robin; Davies, Roger; Harris, William E.; Pierce, Michael J.; Pritchet, Christopher J.; Tonry, John L.; Welch, Douglas L. (July 1992). "A critical review of selected techniques for measuring extragalactic distances". Publications of the Astronomical Society of the Pacific. 104: 599–662. Bibcode:1992PASP..104..599J. doi:10.1086/133035.
• Strauss, Michael A.; Willick, Jeffrey A. (October 1995). "The density and peculiar velocity fields of nearby galaxies". Physics Reports. 261 (5–6): 271–431. arXiv:astro-ph/9502079. Bibcode:1995PhR...261..271S. doi:10.1016/0370-1573(95)00013-7.
• Dressler, Alan (1994). Voyage to the Great Attractor: Exploring Intergalactic Space. New York: Alfred A. Knopf. ISBN 978-0-394-58899-5.

External links

• Measurement of galaxy distances and the D_n–σ relation at the NASA/IPAC Extragalactic Database
• Original 1987 distance-estimator paper at the Astrophysics Data System
• Application to the bulges of disk galaxies at the Astrophysics Data System
• ENEAR cluster template study at the Astrophysics Data System