Angular diameter distance

In astronomy, angular diameter distance is a distance (in units of length) defined in terms of an object's physical size (also in units of length), x {\displaystyle x} , and its angular size (necessarily in radians), θ {\displaystyle \theta } , as viewed from Earth: d A = x θ {\displaystyle d_{A}={\frac {x}{\theta }}} == Cosmology dependence == The angular diameter distance depends on the assumed cosmology of the universe. The angular diameter distance to an object at redshift, z {\displaystyle z} , is expressed in terms of the comoving distance, r {\displaystyle r} as: d A = S k ( r ) 1 + z {\displaystyle d_{A}={\frac {S_{k}(r)}{1+z}}} where S k ( r ) {\displaystyle S_{k}(r)} is the FLRW coordinate defined as: S k ( r ) = { sin ⁡ ( H 0 | Ω k | r ) / ( H 0 | Ω k | ) Ω k < 0 r Ω k = 0 sinh ⁡ ( H 0 | Ω k | r ) / ( H 0 | Ω k | ) Ω k > 0 {\displaystyle S_{k}(r)={\begin{cases}\sin \left(H_{0}{\sqrt {|\Omega _{k}|}}r\right)/\left(H_{0}{\sqrt {|\Omega _{k}|}}\right)&\Omega _{k}<0\\r&\Omega _{k}=0\\\sinh \left(H_{0}{\sqrt {|\Omega _{k}|}}r\right)/\left(H_{0}{\sqrt {|\Omega _{k}|}}\right)&\Omega _{k}>0\end{cases}}} where Ω k {\displaystyle \Omega _{k}} is the curvature density and H 0 {\displaystyle H_{0}} is the value of the Hubble parameter today.

Source: Wikipedia — Angular diameter distance (CC BY-SA 4.0)