Calculator Inputs
Formula Used
The calculator uses the standard Friedmann expansion rate: E(z) = √[ Ωm(1+z)³ + Ωr(1+z)⁴ + Ωk(1+z)² + Ωde(1+z)3(1+w) ].
The line-of-sight comoving distance is DC = (c/H₀) ∫0z dz′ / E(z′). The transverse comoving distance adjusts for curvature: DM = DC for Ωk = 0, otherwise it uses sine or hyperbolic sine forms.
The angular diameter distance is DA = DM / (1+z). A small-angle scale is kpc/arcsec = DA(Mpc) × 1000 × (π/180)/3600.
How to Use This Calculator
- Enter the redshift z for your source.
- Set H₀ and density parameters (Ωm, Ωde, Ωr).
- Choose auto Ωk for a consistent sum, or set Ωk manually.
- Optionally set w to explore dark energy models.
- Pick an output unit, then click Calculate.
- Use the size tools to convert angles to physical sizes.
Example Data Table
Examples use H₀=70, Ωm=0.30, Ωde=0.70, Ωk=0, w=−1.
| z | H₀ | Ωm | ΩΛ | Ωk | DA (Mpc) | Scale (kpc/arcsec) |
|---|---|---|---|---|---|---|
| 0.5 | 70 | 0.3 | 0.7 | 0 | 1,259.0836 | 6.1042 |
| 1 | 70 | 0.3 | 0.7 | 0 | 1,651.9144 | 8.0087 |
| 2 | 70 | 0.3 | 0.7 | 0 | 1,726.6207 | 8.3709 |
Angular Diameter Distance in Observations
Angular diameter distance links what you see on the sky to a physical transverse size. If an object spans an angle θ and has physical size S, the small-angle relation is S ≈ DA × θ. This calculator estimates DA from redshift and cosmology. It supports planning for imaging, lensing, and size measurements.
Why DA Peaks with Redshift
In an expanding universe, DA first increases with z, then can decline beyond a few in redshift. That means very distant galaxies may appear larger again for the same physical size. The peak location depends on Ωm, Ωde, curvature, and w.
Choosing H₀ and Density Parameters
H₀ sets the overall distance scale through the Hubble distance c/H₀. Ωm controls the matter-driven deceleration, while Ωde and w shape late-time acceleration. Ωr is usually negligible at low redshift, but it matters more for early-universe studies. When comparing papers, watch for h = H₀/100 rescalings in reported distances.
Curvature and Transverse Geometry
The transverse comoving distance DM accounts for spatial curvature. For Ωk = 0, DM equals the line-of-sight comoving distance DC. If Ωk is nonzero, the calculator applies sine or hyperbolic-sine geometry, changing DA accordingly.
Interpreting the kpc per Arcsecond Scale
The “kpc/arcsec” output turns an observed angular size into a transverse size quickly. For example, if the scale is 8 kpc/arcsec, a 0.5 arcsec feature corresponds to about 4 kpc. This is ideal for galaxy structure, lensing arcs, and instrument resolution checks. High-resolution imaging makes these conversions especially useful for compact sources.
Using the Size ↔ Angle Converter
Enter an angular size to estimate physical size at the chosen redshift, or enter a physical size to estimate its angular extent. This helps plan observations: you can compare predicted angular sizes against seeing limits or pixel scales. Use consistent units when cross-checking inputs.
Comparing DA with DL and DC
DC tracks how far light traveled in comoving coordinates, while DL relates to flux and luminosity. The calculator shows DL for reference, because many catalogs provide luminosity distances. For standard cosmology, DL = (1+z)2 DA.
Practical Data Quality Tips
Redshift uncertainty propagates into distance uncertainty, especially at higher z. If you compare different surveys, ensure they use compatible H₀ and parameter sets. For precision work, keep Ωk and w explicit, and document your chosen cosmology alongside reported sizes.
FAQs
1) What is angular diameter distance?
It converts an observed angular size into a transverse physical size at a given redshift, using a cosmological expansion model and the small-angle relation.
2) Why does DA sometimes decrease at high redshift?
Because cosmic expansion and geometry cause the apparent size–distance relation to peak and then reverse beyond a certain redshift, depending on the cosmological parameters.
3) Should I enable auto Ωk?
Use it when you want a consistent parameter sum, setting Ωk = 1 − (Ωm + Ωr + Ωde). Disable it if you are testing a specific curvature value intentionally.
4) What does w represent?
w describes the dark energy equation of state. w = −1 corresponds to a cosmological constant, while other values explore alternative dark energy behaviors that change distances.
5) How do I use kpc per arcsecond?
Multiply the scale by your measured angle in arcseconds. For example, 7 kpc/arcsec × 0.3 arcsec ≈ 2.1 kpc transverse size.
6) Are these distances proper or comoving?
DA is a proper transverse distance per unit angle at emission. DC and DM are comoving distances. The calculator labels each output clearly.
7) What redshift range is supported?
The calculator accepts z from 0 to 50. Extremely large z can be sensitive to parameter choices, so verify inputs and interpret results cautiously for early-universe cases.