Redshift to Radial Velocity Calculator
The Redshift to Radial Velocity Calculator converts a measured redshift value (z) into the recession speed of an astronomical object in kilometers per second (km/s). Redshift is defined as z = (λ_observed − λ_emitted) / λ_emitted. For z < 0.1 the classical approximation v ≈ z × c is used (error under ~5%); for z ≥ 0.1 the relativistic Doppler formula v = c × [(1+z)² − 1] / [(1+z)² + 1] is required. This tool applies to stars, galaxies, and quasars observed via spectroscopy.
To convert redshift z to radial velocity: for small redshifts (z < 0.1) use v = z × 299,792 km/s. For larger redshifts use the relativistic formula v = c × [(1+z)² − 1] / [(1+z)² + 1]. Example: z = 0.05 → v ≈ 14,990 km/s; z = 1.0 → v ≈ 179,875 km/s (not 299,792 km/s as the classical formula would predict — a 67% overestimate).
When to use this calculator
- Determining the recession velocity of a distant galaxy from its spectral emission lines (e.g., the Hα line shifted from 656.3 nm to 689 nm yields z ≈ 0.05, v ≈ 14,990 km/s).
- Estimating the distance to a quasar using Hubble's Law (d = v / H₀) once the radial velocity is computed from its observed redshift.
- Verifying whether a stellar object is gravitationally bound to the Milky Way or receding cosmologically, using the sign and magnitude of z.
- Converting published redshift catalog values (e.g., from SDSS or NED) into physical recession speeds for use in cosmological simulations or student lab reports.
Worked Example
- Given: z = 0.5 (a moderately distant galaxy)
- Classical estimate: v = 0.5 × 299,792 = 149,896 km/s — unphysical, approaches c
- Relativistic formula: v = 299,792 × [(1.5)² − 1] / [(1.5)² + 1]
- = 299,792 × 1.25 / 3.25 = 115,305 km/s (≈ 0.38c)
How it works
2 min readHow It Is Calculated
Redshift (z) is a dimensionless quantity measured from spectroscopic observations. Two formulas connect z to radial velocity:
Classical (non-relativistic) approximation — valid for z < 0.1:
v = z × c = z × 299,792.458 km/sRelativistic Doppler formula — required for z ≥ 0.1:
v = c × [(1 + z)² − 1] / [(1 + z)² + 1]Where:
z = observed redshift (dimensionless)c = 299,792.458 km/s (NIST CODATA defined value)v = radial recession velocity in km/s> Note: For z > ~2, recession velocities can exceed c due to the expansion of space — this is not a violation of special relativity.
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Reference Table: Redshift → Radial Velocity
| Redshift (z) | Classical v (km/s) | Relativistic v (km/s) | Error if classical | Example Object |
|---|---|---|---|---|
| 0.001 | 300 | 300 | ~0% | Nearby star |
| 0.005 | 1,499 | 1,495 | 0.3% | Nearby galaxy group |
| 0.01 | 2,998 | 2,983 | 0.5% | Local group |
| 0.05 | 14,990 | 14,615 | 2.6% | Typical Seyfert galaxy |
| 0.1 | 29,979 | 28,487 | 5.2% | Medium-distance galaxy |
| 0.3 | 89,938 | 76,898 | 17% | Distant galaxy |
| 0.5 | 149,896 | 115,305 | 30% | Distant galaxy cluster |
| 1.0 | 299,792 | 179,875 | 67% | High-z galaxy |
| 2.0 | 599,584 | 239,834 | 150% | Distant quasar |
| 6.5 | 1,948,648 | 289,319 | 574% | GN-z11 galaxy |
The classical formula becomes dangerously inaccurate above z ≈ 0.1. Always use the relativistic version for extragalactic work.
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Typical Cases
Case 1: Nearby Galaxy (NGC 4889 in the Coma Cluster, z = 0.0266)
Since z = 0.0266 < 0.1, code uses classical approximation:
v = 0.0266 × 299,792 = 7,974 km/s
For reference, the relativistic result: v = 7,880 km/s
Error from using classical: ~1.2% — acceptable at this z.Case 2: Moderate Redshift Galaxy (z = 0.5)
Classical: v = 0.5 × 299,792 = 149,896 km/s ← UNPHYSICAL (approaches c)
Relativistic: v = 299,792 × [1.25] / [3.25] = 115,305 km/s
Error: ~30%. Always use relativistic for z ≥ 0.1.Case 3: Quasar at z = 3.0
Relativistic: v = 299,792 × [(4)² − 1] / [(4)² + 1]
= 299,792 × 15 / 17 = 264,523 km/s ≈ 0.88c
Classical: v = 3.0 × 299,792 = 899,376 km/s — nearly 3× the speed of light (meaningless).---
Common Errors
1. Using v = z × c for large redshifts. At z = 1.0, the classical formula gives v = c, while the relativistic result is v ≈ 0.67c. Use relativistic for z ≥ 0.1.
2. Confusing cosmological and Doppler redshift. At z > 0.5, most of the redshift is from space expansion, not peculiar motion.
3. Ignoring the sign of z. A negative z (blueshift) means the object is approaching. Andromeda (M31) has z ≈ −0.001, approaching at ~300 km/s.
4. Conflating recession velocity with peculiar velocity. Objects at z > ~1.7 have recession velocities exceeding c — this is the expansion rate of space, not a particle velocity.
5. Rounding c to 300,000 km/s. The exact value is 299,792.458 km/s. For precision spectroscopy, rounding introduces ~70 km/s error per unit of z.
Frequently asked questions
What is redshift (z) and how is it measured?
Redshift (z) is the fractional increase in the wavelength of light from a moving or distant source: z = (λ_obs − λ_emit) / λ_emit. Astronomers measure it by comparing the observed wavelength of a known spectral line (e.g., hydrogen Hα at 656.28 nm) to the lab value. z = 0.05 means the wavelength stretched 5%, implying recession. A negative z (blueshift) means the source is approaching.
When should I use the relativistic formula instead of v = z × c?
Use the relativistic formula v = c × [(1+z)² − 1] / [(1+z)² + 1] whenever z ≥ 0.1. At z = 0.1 the classical formula overestimates by about 0.9%; at z = 0.5 the error is ~30%; at z = 1.0 the classical formula gives the nonsensical result v = c exactly. For any galaxy survey data or quasars, always use the relativistic version.
Can recession velocity exceed the speed of light?
Yes — in the ΛCDM cosmological framework, galaxies at z ≳ 1.7 have recession velocities exceeding c due to the expansion of space itself. This does not violate special relativity, which prohibits objects from moving through space faster than c. The Hubble volume has a current radius of about 4,200 Mpc (~13.7 billion light-years).
What speed of light value does this calculator use?
This calculator uses the exact NIST CODATA defined value: c = 299,792.458 km/s. Many quick estimates use c ≈ 300,000 km/s, which introduces a ~0.07% error (~208 km/s per unit of z). For precision spectroscopy or Hubble constant measurements, the full value matters.
How do I convert radial velocity to distance using Hubble's Law?
Using Hubble's Law: d = v / H₀. The current best estimates are H₀ ≈ 67–73 km/s/Mpc depending on the measurement method. For example, z = 0.05 gives v ≈ 14,990 km/s, so d ≈ 14,990 / 70 ≈ 214 Mpc ≈ 698 million light-years. This approximation is valid only for z ≪ 1.
What is the difference between cosmological and Doppler redshift?
Doppler redshift arises from an object's peculiar velocity (motion through space), while cosmological redshift arises from space itself expanding and stretching photon wavelengths during travel. For nearby objects (z < 0.01), the two are nearly equivalent. For distant galaxies (z > 0.1), cosmological expansion dominates.
What are some famous objects and their redshifts?
Andromeda Galaxy (M31): z ≈ −0.001 (approaching at ~300 km/s). Virgo Cluster: z ≈ 0.004 (~1,200 km/s). 3C 273 (first identified quasar, 1963): z = 0.158, v ≈ 44,000 km/s. GN-z11 galaxy: z = 10.957 — one of the most distant confirmed galaxies, with light emitted ~430 million years after the Big Bang. The CMB: z ≈ 1,100.
Does this formula apply to gravitational redshift?
No. The Doppler/relativistic formula applies to kinematic and cosmological redshift only. Gravitational redshift follows z_grav = 1/√(1 − 2GM/rc²) − 1. For the Sun's surface, z_grav ≈ 2.1 × 10⁻⁶ (~0.6 km/s equivalent). For a neutron star, z_grav can reach 0.3–0.4. Mixing up these types is a common error in stellar spectroscopy.
What is the cutoff z value where the classical and relativistic formulas diverge significantly?
The 5% error threshold for the classical formula occurs around z ≈ 0.1. At z = 0.05 the error is ~2.6%; at z = 0.1 it rises to ~5%; at z = 0.5 it reaches ~30%. This calculator automatically switches to the relativistic formula for z ≥ 0.1. For precise spectroscopic work, use the relativistic version for any z > 0.05.