Angular diameter distance

angular scale versus redshift relation
The angular diameter distance is a distance measure used in astronomy.wikipedia
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Angular diameter

apparent diameterangular sizeapparent size
It is defined in terms of an object's physical size, x, and \theta the angular size of the object as viewed from earth.
In that case, the angular diameter formula can be inverted to yield the angular diameter distance to distant objects as

Standard ruler

Standard ruler
The distance measured by a standard ruler is what is known as the angular diameter distance.

Astronomy

astronomicalastronomerastronomers
The angular diameter distance is a distance measure used in astronomy.

Physical cosmology

cosmologycosmologicalcosmologist
The angular diameter distance depends on the assumed cosmology of the universe.

Redshift

red shiftzred-shift
The angular diameter distance to an object at redshift, z, is expressed in terms of the comoving distance, r as: The angular size redshift relation describes the relation between the angular size observed on the sky of an object of given physical size, and the objects redshift from Earth (which is related to its distance, d, from Earth).

Comoving and proper distances

comoving distancecomoving coordinatesproper distance
The angular diameter distance to an object at redshift, z, is expressed in terms of the comoving distance, r as: In the currently favoured geometric model of our Universe, the "angular diameter distance" of an object is a good approximation to the "real distance", i.e. the proper distance when the light left the object.

Hubble's law

Hubble constantcosmological redshiftHubble parameter
Where \Omega_k is the curvature density and H_0 is the value of the Hubble parameter today.

Lambda-CDM model

ΛCDMstandard cosmological modelΛCDM model
In the currently favoured geometric model of our Universe, the "angular diameter distance" of an object is a good approximation to the "real distance", i.e. the proper distance when the light left the object.

Earth

terrestrialworldGlobal
The angular size redshift relation describes the relation between the angular size observed on the sky of an object of given physical size, and the objects redshift from Earth (which is related to its distance, d, from Earth).

Euclidean geometry

plane geometryEuclideanEuclidean plane geometry
In a Euclidean geometry the relation between size on the sky and distance from Earth would simply be given by the equation:

Euclidean space

EuclideanspaceEuclidean 3-space
This is related to the angular diameter distance, which is the distance an object is calculated to be at from \theta and x, assuming the Universe is Euclidean.

Mattig formula

The Mattig relation yields the angular-diameter distance as a function of redshift for a universe with Ω Λ = 0.

Distance measures (cosmology)

light travel distancedistance measurementdistance
Distance measures (cosmology)

Weak gravitational lensing

weak lensingcluster-scale weak lensingcosmic shear
Angular diameter distances to the lenses and background sources are important for converting the lensing observables to physically meaningful quantities.

Gravitational lensing formalism

convergence and shearcritical densityJacobian
For extragalactic lenses, these must be angular diameter distances.

Etherington's reciprocity theorem

The Etherington's distance-duality equation is the relationship between the luminosity distance of standard candles and the angular diameter distance.

Sunyaev–Zeldovich effect

Sunyaev-Zel'dovich-EffectSunyaev–Zel'dovich (SZ) effect
Another factor which facilitates high-redshift cluster detection is the angular scale versus redshift relation: it changes little between redshifts of 0.3 and 2, meaning that clusters between these redshifts have similar sizes on the sky.

NGC 2460

2460
Its redshift of 0.004837 gives an angular diameter distance of 21.501 megaparsecs, or approximately 70 million light-years.

Einstein ring

extra ringsring
d_L is the angular diameter distance to the lens,

Accelerating expansion of the universe

accelerating universeacceleratingcosmic acceleration
So by looking at the distances at which galaxies at different redshifts tend to cluster, it is possible to determine a standard angular diameter distance and use that to compare to the distances predicted by different cosmological models.

Luminosity distance

distanced'' L
and with the angular diameter distance D_A by the Etherington's reciprocity theorem:

Dark Energy Survey

CTIO-DECamDESThe Dark Energy Survey
The angle subtended by a standard ruler as a function of redshift is related to the Hubble parameter, H(z) through the concept of the angular diameter distance.

Baryon acoustic oscillations

baryon acoustic oscillationBAOacoustic peaks
This measures two cosmological distances: the Hubble parameter, H(z), and the angular diameter distance, d_A(z), as a function of redshift (z).