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	~~The '''angular diameter distance''' is a distance measure used in [[astronomy]]~~. ~~The angular diameter distance to an object is defined in terms of the object~~'s ~~actual size, <math>x</math>, and <math>\theta</math> the angular size of the object as viewed from earth. <br />~~		Greetings! I am Myrtle Shroyer. For years he's been living in North Dakota and his family members loves it. My working day job is a meter reader. Body building is one of the things I love most.<br><br>Feel free to surf to my page: [http://www.adosphere.com/poyocum http://www.adosphere.com/poyocum]
	~~<math>~~
	~~d_A= \frac{x}{\theta}~~
	~~</math>~~
	~~<br />~~
	~~The angular diameter distance depends on the assumed [[physical cosmology\|cosmology]] of the universe. The angular diameter distance to an object at [[redshift]], <math> z </math>, is expressed~~ in ~~terms of the [[comoving distance]], <math> \chi </math> as:~~
	~~<br />~~
	~~<math>~~
	~~d_A = \frac{r(\chi)}{1+z}~~
	~~</math>~~
	~~<br />~~
	~~Where <math> r(\chi)</math> is defined as:<br />~~
	~~<math>~~
	~~r(\chi) = \begin{cases}~~
	~~\sin \left( \sqrt{-\Omega_k} H_0 \chi \right)/\left(H_0\sqrt{\|\Omega_k\|}\right) & \Omega_k < 0\\~~
	~~\chi & \Omega_k=0 \\~~
	~~\sinh \left( \sqrt{\Omega_k} H_0 \chi \right)/\left(H_0\sqrt{\|\Omega_k\|}\right) & \Omega_k >0~~
	~~\end{cases}~~

	~~</math><br />~~
	~~Where <math> \Omega_k </math> is the curvature density~~ and ~~<math> H_0 </math> is the value of the [[Hubble constant\|Hubble parameter]] today~~.

	~~In the [[Lambda-CDM model\|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 [[comoving distance\|proper distance]] when the light left the object. Note that beyond a certain [[redshift]], the angular diameter distance gets smaller with increasing [[redshift]]. In other words an object "behind" another of the ~~same size, beyond a certain redshift (roughly z=1~~.~~5), appears larger on the sky, and would therefore have a ''smaller'' "angular diameter distance".~~

	~~==Angular size redshift relation==~~
	[[Image:Angular-size-redshift-relation.png\|thumb\|The angular size redshift relation for a [[Lambda-CDM model\|Lambda]] [[physical cosmology\|cosmology]], with on the vertical scale kiloparsecs per arcsecond.]]

	~~[[File:Distanza di diametro angolare.gif\|thumb\|The angular size redshift relation for a [[Lambda-CDM model\|Lambda]] [[physical cosmology\|cosmology]], with on the vertical scale megaparsecs.]]~~

	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, <~~math~~> d <~~/math>, from Earth). In a [[Euclidean geometry]] the relation between size on the sky and distance from Earth would simply be given by the equation:~~

	~~<!-- Image with unknown copyright status removed: [[File:Distanza di diametro angolare.gif]] -->~~

	~~<math> \tan\left ( \theta \right )= \frac{x}{d} </math>~~

	~~where <math> \theta </math> is the angular size of the object on the sky, <math> x </math> is the size of the object and <math> d </math~~> ~~is the distance~~ to ~~the object. Where <math> \theta </math> is small this approximates~~ to:

	~~<math> \theta \approx \frac{x}{d} </math>.~~

	However, in the [[Lambda-CDM model\|currently favoured geometric model of our Universe]], the relation is more complicated. In this model, objects at [[redshift]]s 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 <math> \theta </math> and <math> x </math>, assuming the Universe is [[Euclidean]].~~

	The actual relation between the angular-diameter distance, <math>d_a</math>, and redshift is given below. <math>q_0</math> is called the deceleration parameter and measures the deceleration of the expansion rate of the Universe; in the simplest models, <math>q_0<0.5</math> corresponds to the case where the Universe will expand for ever, <math>q_0>0.5</math> to closed models which will ultimately stop expanding and contract <math>q_0=0.5</math> corresponds to the critical case – Universes which will just be able to expand to infinity without re-contracting.

	~~<math>d_a=\cfrac{c}{H_0 q^2_0} \cfrac{(zq_0+(q_0 -1)(\sqrt{2q_0 z+1}-1))}{(1+z)^2}</math>~~

	~~The [[Mattig relation]] yields the angular-diameter distance as a function of redshift for a universe with ΩΛ = 0.<ref>~~[http://~~books~~.google.se/books?id=RK8qDGKSTPwC&pg=PA102&lpg=PA102&dq=%22The+Mattig+relation%22&source=bl&ots=_WFDJvBzDl&sig=dXymoGGOi92YwRpz_9E5TXHIzbI&hl=sv&ei=8NApTNXGLY2tOI6P7LID&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAYQ6AEwAA#v=onepage&q=%22The%20Mattig%20relation%22&f=false An introduction to the science of cosmology, Chapter 6:2] by Derek J. ~~Raine & Edwin George Thomas (2001)</ref>~~

	~~== References ==~~
	~~<references~~/>

	~~==See also==~~
	* [[Distance measures (cosmology)]]
	* [[Standard ruler]]
	~~==External links==~~
	* [http://~~icosmos~~.co.uk/ ~~iCosmos: Cosmology Calculator (With Graph Generation )]~~


	~~[[Category:Physical quantities]~~]

en>Arno Matthias: wl

2011-08-28T13:12:03Z

New page

The '''angular diameter distance''' is a distance measure used in [[astronomy]]. The angular diameter distance to an object is defined in terms of the object's actual size, <math>x</math>, and <math>\theta</math> the angular size of the object as viewed from earth. 
<math>
d_A= \frac{x}{\theta}
</math>
 
The angular diameter distance depends on the assumed [[physical cosmology|cosmology]] of the universe. The angular diameter distance to an object at [[redshift]], <math> z </math>, is expressed in terms of the [[comoving distance]], <math> \chi </math> as:
 
<math>
d_A = \frac{r(\chi)}{1+z}
</math>
 
Where <math> r(\chi)</math> is defined as: 
<math>
r(\chi) = \begin{cases}
\sin \left( \sqrt{-\Omega_k} H_0 \chi \right)/\left(H_0\sqrt{|\Omega_k|}\right) & \Omega_k < 0\\
\chi & \Omega_k=0 \\
\sinh \left( \sqrt{\Omega_k} H_0 \chi \right)/\left(H_0\sqrt{|\Omega_k|}\right) & \Omega_k >0
\end{cases}

</math> 
Where <math> \Omega_k </math> is the curvature density and <math> H_0 </math> is the value of the [[Hubble constant|Hubble parameter]] today.

In the [[Lambda-CDM model|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 [[comoving distance|proper distance]] when the light left the object. Note that beyond a certain [[redshift]], the angular diameter distance gets smaller with increasing [[redshift]]. In other words an object "behind" another of the same size, beyond a certain redshift (roughly z=1.5), appears larger on the sky, and would therefore have a ''smaller'' "angular diameter distance".

==Angular size redshift relation==
[[Image:Angular-size-redshift-relation.png|thumb|The angular size redshift relation for a [[Lambda-CDM model|Lambda]] [[physical cosmology|cosmology]], with on the vertical scale kiloparsecs per arcsecond.]]

[[File:Distanza di diametro angolare.gif|thumb|The angular size redshift relation for a [[Lambda-CDM model|Lambda]] [[physical cosmology|cosmology]], with on the vertical scale megaparsecs.]]

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, <math> d </math>, from Earth). In a [[Euclidean geometry]] the relation between size on the sky and distance from Earth would simply be given by the equation:



<math> \tan\left ( \theta \right )= \frac{x}{d} </math>

where <math> \theta </math> is the angular size of the object on the sky, <math> x </math> is the size of the object and <math> d </math> is the distance to the object. Where <math> \theta </math> is small this approximates to:

<math> \theta \approx \frac{x}{d} </math>.

However, in the [[Lambda-CDM model|currently favoured geometric model of our Universe]], the relation is more complicated. In this model, objects at [[redshift]]s 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 <math> \theta </math> and <math> x </math>, assuming the Universe is [[Euclidean]].

The actual relation between the angular-diameter distance, <math>d_a</math>, and redshift is given below. <math>q_0</math> is called the deceleration parameter and measures the deceleration of the expansion rate of the Universe; in the simplest models, <math>q_0<0.5</math> corresponds to the case where the Universe will expand for ever, <math>q_0>0.5</math> to closed models which will ultimately stop expanding and contract <math>q_0=0.5</math> corresponds to the critical case – Universes which will just be able to expand to infinity without re-contracting.

<math>d_a=\cfrac{c}{H_0 q^2_0} \cfrac{(zq_0+(q_0 -1)(\sqrt{2q_0 z+1}-1))}{(1+z)^2}</math>

The [[Mattig relation]] yields the angular-diameter distance as a function of redshift for a universe with ΩΛ = 0.<ref>[http://books.google.se/books?id=RK8qDGKSTPwC&pg=PA102&lpg=PA102&dq=%22The+Mattig+relation%22&source=bl&ots=_WFDJvBzDl&sig=dXymoGGOi92YwRpz_9E5TXHIzbI&hl=sv&ei=8NApTNXGLY2tOI6P7LID&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAYQ6AEwAA#v=onepage&q=%22The%20Mattig%20relation%22&f=false An introduction to the science of cosmology, Chapter 6:2] by Derek J. Raine & Edwin George Thomas (2001)</ref>

== References ==
<references/>

==See also==
* [[Distance measures (cosmology)]]
* [[Standard ruler]]
==External links==
* [http://icosmos.co.uk/ iCosmos: Cosmology Calculator (With Graph Generation )]

[[Category:Physical quantities]]