Doppler Redshift (Radial Velocity) Calculator
Solve the classical Doppler relation for radial velocity, observed wavelength, or rest wavelength — with step-by-step math, a redshift-vs-blueshift chart comparing the classical and relativistic forms, and a spectral-line reference.
λobs vs. λrest — sign sets direction
λobs > λrest gives a positive v (redshift, receding); λobs < λrest gives a negative v (blueshift, approaching).
λrest — sets the sensitivity
A smaller λrest means the same absolute wavelength shift Δλ corresponds to a larger fractional shift, and so a larger implied velocity.
Linear, not relativistic
This classical form is linear in Δλ/λrest and has no built-in speed limit — it's only an approximation, accurate for |v| well below c.
Not the same as cosmological redshift
For very distant galaxies, most of the observed redshift comes from the expansion of space itself, not from motion through space — a distinction the classical Doppler formula doesn't capture.
How to Use This Calculator
The classical Doppler relation links three quantities: the rest wavelength λrest, the observed wavelength λobs, and the radial velocity v. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick v, λobs, or λrest — the tool rearranges the formula automatically.
2. Pick a spectral line, or type your own
Use a preset rest wavelength (Hα, Hβ, Na D, Ca II K, Lyman-α) or enter any custom value in nanometers.
3. Read the result, sign, and steps
The result card flags redshift (receding) or blueshift (approaching) automatically, alongside the step-by-step math.
4. Check the classical-vs-relativistic chart
The chart shows where the simple linear classical formula starts to diverge from the speed-of-light-bounded relativistic form.
5. Switch to Advanced for more depth
Advanced mode adds input sensitivity, the redshift parameter z, the wavelength shift Δλ, and the relativistic-corrected velocity.
6. Convert units, or copy the result
Use the wavelength/velocity converter further down the page, or "Copy result" to grab the solved value.
What This Tool Offers
Built around the classical (non-relativistic) Doppler formula v = c·(λobs−λrest)/λrest, the standard low-velocity approximation used to measure radial velocities from spectral line shifts.
Solve any variable
Go from a measured spectral shift to a velocity, or work backwards from a target velocity to the wavelength that would produce it.
Spectral line presets
One-click presets for common rest wavelengths — Hα, Hβ, Na D, Ca II K, and Lyman-α.
Classical vs. relativistic chart
See exactly where the simple linear approximation starts to diverge from the speed-of-light-bounded relativistic form.
Companion redshift metrics
Advanced mode reports the redshift parameter z, the wavelength shift Δλ, and the relativistic-corrected velocity.
Redshift/blueshift auto-detection
The result card automatically labels the result as a redshift (receding) or blueshift (approaching).
Unit conversion & reference
Convert between nm, Å, and μm, and between km/s, m/s, and fraction of c, further down the page.
Reference Tables & Conversions
Unit converters for wavelength and velocity, plus a lookup table of common rest-frame spectral lines.
Wavelength unit converter
Velocity unit converter
Common Rest-Frame Spectral Lines
| Line | λrest (nm) | Typical use |
|---|---|---|
| Lyman-α | 121.57 | UV; high-redshift quasars & galaxies |
| Ca II K | 393.37 | Stellar & galaxy absorption spectra |
| Ca II H | 396.85 | Stellar & galaxy absorption spectra |
| Hβ | 486.13 | Stellar & nebular emission spectra |
| Mg b | 517.27 | Galaxy absorption spectra |
| Na D | 589.00 | Stellar & interstellar absorption |
| Hα | 656.28 | Stellar radial velocity, exoplanet RV method |
Typical Radial Velocity Regimes
| Context | Typical |v| | Notes |
|---|---|---|
| Exoplanet radial-velocity signal | ~1–100 m/s | Star's wobble due to an orbiting planet |
| Typical stellar peculiar motion | ~10–100 km/s | Random motion relative to local standard of rest |
| Galactic rotation | ~100–300 km/s | Orbital speed within the Milky Way |
| Local group / nearby galaxies | ~100s–1,000s km/s | Peculiar velocity remains a meaningful fraction |
| Distant galaxies / quasars (z ≳ 0.1) | N/A — use cosmological redshift | Classical/relativistic Doppler no longer appropriate |
What Does v = c·(λobs−λrest)/λrest Measure?
When a light source moves toward or away from an observer, its spectral lines shift in wavelength — the Doppler effect. This classical (non-relativistic) form estimates the line-of-sight, or "radial," velocity directly from that shift, and is the standard low-velocity approximation used across observational astronomy and spectroscopy.
Classical Doppler Formula
The defining relationship is:
Rearranged to solve for the wavelengths:
λrest = λobs / (1 + v/c)
The redshift parameter z, and the relativistic form used for larger velocities:
v_rel = c · [ (1+z)² − 1 ] / [ (1+z)² + 1 ] (relativistic)
Model Variables
| Symbol | Meaning | Notes |
|---|---|---|
| v | Radial (line-of-sight) velocity | km/s or m/s; positive = receding, negative = approaching |
| λobs | Observed wavelength | Wavelength of the line as measured, nm |
| λrest | Rest (laboratory) wavelength | Known unshifted wavelength of the line, nm |
| c | Speed of light | 299,792.458 km/s (exact, by definition) |
| z | Redshift parameter | z = Δλ/λrest; dimensionless |
Redshift vs. Blueshift
By convention, a positive shift (λobs > λrest, so v > 0) is called a redshift — light is shifted toward longer, "redder" wavelengths as the source recedes. A negative shift (λobs < λrest, v < 0) is a blueshift, as the source approaches.
Calculation Examples
v = 299,792.458 × (656.35 − 656.28) / 656.28 = 299,792.458 × 0.0001066
v = ≈31.97 km/s (redshift)
λobs = 589.00 × (1 + (−60/299,792.458)) = 589.00 × 0.999800
λobs = ≈588.88 nm (blueshift)
Frequently Asked Questions
The line-of-sight (radial) velocity of a light source relative to the observer, inferred from how much a known spectral line's wavelength has shifted from its rest value.
Not exactly. This classical Doppler formula describes redshift from genuine relative motion through space. For very distant galaxies, most of the observed redshift instead comes from the expansion of space itself (cosmological redshift), which follows different physics.
The classical formula is a fine approximation for |v| up to a few percent of c. Above roughly 10% of c, the relativistic form (shown in Advanced mode) should be used instead, since the two start to diverge noticeably.
These are strong, well-characterized spectral lines with precisely known rest wavelengths, which makes them reliable references for measuring a shift against.
A longer observed wavelength than the rest wavelength (λobs > λrest) gives a positive v — a redshift, meaning the source is receding. A shorter observed wavelength gives a negative v — a blueshift, meaning the source is approaching.
Mathematically yes, since the classical formula is unbounded — but that's a sign the classical approximation has broken down, not a physical result. Check the relativistic-corrected velocity in Advanced mode instead.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| v | Radial velocity | m/s |
| c | Speed of light | m/s |
| λ_observed | Observed wavelength | nm |
| λ_rest | Rest wavelength | nm |
What it means
The Doppler redshift formula relates the observed shift in wavelength (λ_observed) of a spectral line to the radial velocity v of the source relative to the observer. It assumes non‑relativistic speeds (v << c). This is fundamental in astronomy for measuring the motions of stars, galaxies, and even the expansion of the Universe. It is used in the radial velocity method for exoplanet detection and in cosmology to measure redshifts. Understanding this equation is essential for interpreting spectra and for studying the dynamics of celestial objects.
Worked example
Doppler Redshift (Radial Velocity) – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| λ_observed (nm) | 656.5 |
| λ_rest (nm) | 656.28 |
| Parameter | Value |
|---|---|
| λ_obs | 486.3 |
| λ_rest | 486.13 |
Common mistakes
- Doppler redshift: v = c × (λ_observed − λ_rest) / λ_rest – for radial velocity.
- λ_rest: The laboratory (rest) wavelength of a spectral line.
- λ_observed: The observed wavelength (redshifted or blueshifted).
- Sign: Redshift (positive v) – object receding; blueshift (negative) – approaching.
- Relativistic: For high speeds (v > 0.1c), use the relativistic Doppler formula.
Applications
The Doppler redshift formula, v = c (λ_observed − λ_rest) / λ_rest, relates the radial velocity of an object to the shift in wavelength of its spectral lines. This is one of the most powerful tools in astronomy, used to measure the motion of stars, galaxies, and quasars. By observing redshift, astronomers can determine the recession velocity of galaxies, study galactic rotation curves, and detect exoplanets via radial velocity. It also underpins the discovery of dark matter and dark energy. Understanding redshift is essential for cosmology and for interpreting the motion of celestial bodies.
- Measurement of galaxy redshifts and cosmic expansion
- Detection of exoplanets via radial velocity method
- Study of stellar radial velocities and binary star orbits
- Investigation of galactic dynamics and dark matter halos
- Determination of the universe's expansion history
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