In
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, angular diameter distance is a distance defined in terms of an object's physical size,
, and its
angular size
The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is ...
,
, as viewed from Earth:
Cosmology dependence
The angular diameter distance depends on the assumed
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...
of the universe. The angular diameter distance to an object at
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
,
, is expressed in terms of the
comoving distance
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects. ''Proper distance'' roughly corresponds to where a distant object would be at a spec ...
,
as:
where
is the FLRW coordinate defined as:
where
is the curvature density and
is the value of the
Hubble parameter
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
today.
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.
Angular size redshift relation
![Distanza di diametro angolare](https://upload.wikimedia.org/wikipedia/commons/c/c8/Distanza_di_diametro_angolare.png)
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 object's
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
from
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
(which is related to its distance,
, from Earth). In a
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
the relation between size on the sky and distance from Earth would simply be given by the equation:
where
is the angular size of the object on the sky,
is the size of the object and
is the distance to the object. Where
is small this approximates to:
However, in the
ΛCDM model
The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a Parameter#Modelization, parameterization of the Big Bang physical cosmology, cosmological model in which the universe contains three major components: first, a cosmological constant de ...
, the relation is more complicated. In this model, objects at
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
s greater than about 1.5 appear larger on the sky with increasing
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
.
This is related to the angular diameter distance, which is the distance an object is calculated to be at from
and
, assuming the Universe is
Euclidean.
The
Mattig relation yields the angular-diameter distance,
, as a function of redshift ''z'' for a universe with Ω
Λ = 0.
is the present-day value of the
deceleration parameter
The deceleration parameter ''q'' in cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in a Friedmann–Lemaître–Robertson–Walker universe. It is defined by:
:q \ \stackrel\ -\frac
where a is the s ...
, which measures the deceleration of the expansion rate of the Universe; in the simplest models,
corresponds to the case where the Universe will expand forever,
to closed models which will ultimately stop expanding and contract,
corresponds to the critical case – Universes which will just be able to expand to infinity without re-contracting.
Angular diameter turnover point
The angular diameter distance
reaches a maximum at a
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
(in the ΛCDM model, this occurs at
), such that the slope of
changes sign at
, or