# Angular size redshift relation

Angular size redshift relation

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

$anleft \left( heta ight \right)= frac\left\{x\right\}\left\{d\right\}$

where $heta$ is the angular size of the object on the sky, $x$ is the size of the object and $d$ is the distance to the object. Where $heta$ is small this approximates to:

$heta = frac\left\{x\right\}\left\{d\right\}$.

However, in the currently favoured geometric model of our Universe, the relation is more complicated. In this model, objects at redshifts greater than about 1.5 appear "larger" on the sky with increasing redshift.

This is related to the angular diameter distance, which is the distance an object is calculated to be at from $heta$ and $x$, assuming the Universe is Euclidean.

The actual relation between theangular-diameter distance, $d_a$, and redshift is given below. $q_0$ is called thedeceleration parameter and measures the deceleration of the expansion rate of theUniverse; in the simplest models, $q_0<0.5$ corresponds to the case where the Universewill expand for ever, $q_0>0.5$ to closed models which will ultimately stop expandingand contract $q_0=0.5$ corresponds to the critical case – Universes which will just beable to expand to infinity without re-contracting.

$d_a=cfrac\left\{c\right\}\left\{H_0 q^2_0\right\} cfrac\left\{\left(zq_0+\left(q_0 -1\right)\left(sqrt\left\{2q_0 z+1\right\}-1\right)\right)\right\}\left\{\left(1+z\right)^2\right\}$

* Angular diameter distance

[http://icosmos.co.uk/ iCosmos: Cosmology Calculator (With Graph Generation )]

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