iCosmos › Angular diameter distance calculator
Angular diameter distance calculator
How far away an object is, judged by how big it looks: DA = physical size ÷ angular size.
The interactive calculator loads here: change the redshift and the cosmological parameters (Ωm, ΩΛ, Ωr, w0, wa, H0) and the result and plot update instantly.
What it is
The angular diameter distance DA relates an object’s physical size ℓ to the angle θ it covers on the sky: ℓ = DA θ. It is the distance you need to turn an image into physical sizes, for galaxies, clusters or the sound horizon in the cosmic microwave background.
Unlike everyday distances, DA does not keep growing with redshift. In universes like ours it reaches a maximum and then decreases, so galaxies beyond that redshift look larger the farther away they are, because we see them as they were when the universe was smaller.
Formula
- DA = DM / (1 + z)
- DL = (1 + z)2 DA (Etherington’s distance-duality relation)
- ℓ [kpc] = θ [arcsec] × (DA × 1 arcsec) [kpc per arcsec]
Worked example
In the classic model (Ωm = 0.3, ΩΛ = 0.7, H0 = 70 km s−1 Mpc−1), a galaxy at z = 1 has DA = 1,651.91 Mpc (5.388 billion light-years), so 1 arcsecond on the sky corresponds to 8.009 kpc.
DA reaches its maximum of 1,747.6 Mpc at z ≈ 1.61. At z = 3 it is back down to 1,588.9 Mpc, and at z = 1100, where the cosmic microwave background was released, only about 12.4 Mpc.
Common questions
Why does the angular diameter distance decrease at high redshift?
Light from very distant objects left them when the universe was much smaller, so the objects were physically closer to us at emission. The angle they cover is set when the light leaves and does not change on the way to us, so beyond the maximum of DA a galaxy of fixed size looks bigger the farther away it is.
What is the difference between angular diameter distance and luminosity distance?
DA converts angular sizes into physical sizes; DL converts observed fluxes into luminosities. They are always related by DL = (1 + z)2 DA, so at z = 1 the luminosity distance is four times the angular diameter distance.
How do I convert an angular size to a physical size?
Multiply the angular size in arcseconds by the angular scale in kpc per arcsec shown above. For example, 2″ at z = 1 in the classic model is about 16 kpc.
How does curvature change DA?
In a curved universe DM = R0 sinh(DC/R0) (open) or R0 sin(DC/R0) (closed), where R0 is the curvature radius. Positive curvature (a closed universe) focuses light rays: at a given comoving distance DM is smaller than DC, so objects look larger than they would in flat space. Changing the densities also changes DC itself, so the net effect at a given redshift depends on the model. The angular sizes of features in the cosmic microwave background are how the geometry of space is measured.
Related calculators
- Luminosity distance calculator
- Comoving distance calculator
- Redshift to distance calculator
- Curvature of the universe calculator
- The full iCosmos cosmology calculator
About iCosmos
iCosmos was created by Mihran Vardanyan. Mihran Vardanyan is a cosmologist who earned his DPhil in cosmology at the University of Oxford, supervised by Joseph Silk and Roberto Trotta. He developed iCosmos during his research years at Oxford Astrophysics, where his work applied Bayesian model selection to the curvature and size of the Universe.
How to cite: Vardanyan, Mihran. iCosmos V4: Cosmology Calculator. https://icosmos.co.uk/. Include your access date, model parameters and exported data when reporting calculations.