Doppler Shift & Radial Velocity Calculator
Measure the radial velocity of stars and galaxies from spectral line shifts using classical and relativistic Doppler formulas.
🌈 What is the Doppler Shift in Astronomy?
Astronomical Doppler shift is the change in the observed wavelength of light caused by the relative motion between a source and an observer. When a star or galaxy moves away from Earth, its light is stretched to longer (redder) wavelengths, producing a redshift. When it moves toward Earth, light is compressed to shorter (bluer) wavelengths, producing a blueshift. Astronomers quantify this shift with the dimensionless redshift parameter z, defined as z = (lambda_obs minus lambda_rest) divided by lambda_rest, where lambda_rest is the known laboratory wavelength and lambda_obs is the measured wavelength in the spectrum.
Radial velocity spectroscopy is one of the oldest and most productive techniques in observational astronomy. By comparing the observed positions of spectral lines to their laboratory rest values, astronomers can measure the line-of-sight (radial) velocity of virtually any luminous object. This technique has been used to discover thousands of exoplanets via the stellar wobble they induce, to map the rotation curves of spiral galaxies that provided the first evidence for dark matter, to confirm the expansion of the universe, and to detect supermassive black holes in galactic nuclei through extreme velocity dispersions in their surrounding gas.
The key distinction between classical and relativistic Doppler is important for high-redshift objects. The classical approximation v = z times c gives the correct answer to within 0.5% for z less than 0.1, but breaks down badly at high redshifts: at z = 2, it predicts a velocity of twice the speed of light, which is physically impossible. The relativistic formula v = c times ((1+z)^2 minus 1) divided by ((1+z)^2 plus 1) is always valid and should be used for any source with z > 0.1. This calculator computes both values side by side so users can see the divergence directly.
Spectral lines commonly used in radial velocity work include H-alpha (656.28 nm, the strongest hydrogen line in the optical), H-beta (486.13 nm), the calcium H and K doublet (396.85 and 393.37 nm), which trace cool stellar atmospheres, and Lyman-alpha (121.567 nm), which dominates quasar spectra at z > 1.7 where it shifts into the visible band. This calculator includes all eight of these standard lines as presets, with a custom input for any other wavelength.
📐 Formulas
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — H-alpha Line from a Receding Galaxy Cluster (z = 0.067)
H-alpha rest: 656.28 nm, observed at 700.00 nm in a galaxy cluster spectrum
Example 2 — Lyman-alpha Line from a Quasar at z = 2.0
Lyman-alpha rest: 121.567 nm, observed at 364.701 nm in a high-redshift quasar spectrum
Example 3 — Predicting Observed H-alpha at 10,000 km/s Recession
H-alpha rest: 656.28 nm, source receding at 10,000 km/s (Velocity to Wavelength mode)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Doppler shift in astronomy?
The Doppler shift is the change in wavelength of light emitted by a source that is moving toward or away from the observer. In astronomy, it is used to measure the radial (line-of-sight) velocity of stars, galaxies, and gas clouds from the shift in their spectral lines. A shift toward longer wavelengths is called a redshift (source receding); a shift toward shorter wavelengths is a blueshift (source approaching). The fractional shift is defined as z = (lambda_obs - lambda_rest) / lambda_rest.
What is redshift z and how is it measured?
Redshift z is the dimensionless ratio of the wavelength shift to the rest wavelength: z = delta-lambda / lambda_rest. For a source with z = 0.066, the observed wavelength is 6.6% longer than the rest wavelength. Astronomers measure z by identifying known spectral lines (such as H-alpha at 656.28 nm) in a spectrum and comparing their observed positions to laboratory values. For cosmological sources, z also encodes the expansion of the universe, not only a Doppler velocity.
What is the difference between classical and relativistic Doppler formulas?
The classical Doppler formula v = z x c is valid only when the velocity is much less than the speed of light. For z < 0.1 it is accurate to better than 0.5%. The relativistic formula v_rel = c x ((1+z)^2 - 1) / ((1+z)^2 + 1) is exact at any speed and must be used for high-redshift objects. For a quasar at z = 2, the classical formula gives 599,585 km/s (twice c), which is physically impossible. The relativistic result is 239,834 km/s, or 80% of c, which is the correct recession velocity in the source rest frame.
What is radial velocity and how is it used in astronomy?
Radial velocity is the component of a source's velocity directed toward or away from the observer, measured in km/s. It is derived from the Doppler shift of spectral lines and is the primary tool for detecting stellar companions (spectroscopic binaries), measuring the expansion of the universe, searching for exoplanets (the radial velocity method), and mapping the rotation of galaxies. Modern spectrographs achieve radial velocity precision below 1 m/s, enabling detection of Earth-mass exoplanets.
Which spectral lines are most commonly used to measure radial velocity?
H-alpha (656.28 nm) is the strongest hydrogen emission line and the most widely used line in optical spectroscopy of nearby stars and galaxies. H-beta (486.13 nm) and the Ca H and K lines (396.85 and 393.37 nm) are used for stellar classification and galactic studies. Lyman-alpha (121.567 nm) dominates high-redshift quasar spectra, where it shifts into the optical window for z > 1.7. The Na D doublet (589 nm) traces cool gas and interstellar sodium absorption.
Can the recession velocity of a galaxy exceed the speed of light?
Yes, in cosmology it can. The classical Doppler formula v = z x c can give values larger than c for z > 1, but this is an artifact of applying a non-relativistic formula outside its range of validity. The special-relativistic radial velocity in the source rest frame is always less than c. However, in general relativity, the cosmological recession velocity at large distances can genuinely exceed c because space itself is expanding. Galaxies at z > 1.7 are receding faster than c due to cosmic expansion, not local motion through space.
What is the blueshift of the Andromeda Galaxy?
The Andromeda Galaxy (M31) shows a radial velocity of approximately -300 km/s, meaning it is approaching the Milky Way at about 300 km/s. This corresponds to a blueshift of z = -0.001, or a shift of -0.66 nm in the H-alpha line (from 656.28 nm to about 655.62 nm). The minus sign indicates approach rather than recession. Despite this approach velocity, the two galaxies are expected to merge in roughly 4.5 billion years due to the combination of radial and transverse motions.
What is the highest redshift galaxy ever observed?
As of 2024, the highest confirmed spectroscopic redshift belongs to JADES-GS-z14-0 at z = 14.32, observed by the James Webb Space Telescope. At this redshift, light we see today left the galaxy only about 290 million years after the Big Bang. The Lyman-alpha line (121.567 nm rest frame) appears at roughly 1,864 nm in the infrared, well beyond the optical window. The relativistic recession velocity at z = 14.32 is approximately 97.7% of c.
How is the Doppler shift used to detect exoplanets?
The radial velocity method detects exoplanets by measuring the tiny Doppler wobble a planet induces in its host star as both orbit their common center of mass. As the star moves toward us, its spectral lines blueshift slightly; as it moves away, they redshift. The amplitude of this wobble is proportional to the planet mass and inversely proportional to the orbital period. Jupiter-mass planets cause wobbles of about 12 m/s in solar-type stars; Earth-mass planets cause wobbles of less than 0.1 m/s, at the limit of current instruments.
What is the difference between redshift and blueshift?
A redshift (positive z, positive delta-lambda) means the observed wavelength is longer than the rest wavelength, indicating the source is receding. A blueshift (negative z, negative delta-lambda) means the observed wavelength is shorter, indicating the source is approaching. The terms come from the direction of the shift on the visible spectrum: red light has longer wavelengths, blue light shorter. In practice, redshifts in extragalactic astronomy are overwhelmingly more common because the universe is expanding and most distant galaxies are receding.