Doppler Shift and Radial Velocity: Redshift, Blueshift and Worked Calculations

How do spectral lines give radial velocity?

The measured ratio and the physical interpretation are separate steps. Redshift is dimensionless; velocity has units and a reference frame. At large shifts, an optical or radio “velocity” convention need not equal the physical relative speed of a simple moving-source model.

This guide uses invented spectral lines and c = 299,792.458 km/s. It teaches calculations rather than claiming measurements of named stars. The ESA radial-velocity overview explains how stellar spectra reveal motion along the line of sight.

Redshift and blueshift describe a relative wavelength change

If λobs is greater than λ₀, then z is positive: the feature is redshifted. If λobs is smaller, z is negative and the feature is blueshifted. The label describes a shift relative to the identified rest feature; it does not require the star to look visibly red or blue to the eye.

A cool star can have a red-looking overall spectrum while approaching. Its continuum colour and the displacement of an identified spectral line are different observables. The line's reference wavelength is essential; a bright or dark feature at 500 nm alone is not a velocity measurement.

For this article, both rest and observed wavelengths are vacuum wavelengths. If a dataset uses another wavelength convention, convert consistently before taking a ratio. Matching units such as nanometres is necessary but does not by itself guarantee the same wavelength scale or calibration.

Seven line-position examples around 500 nm

The rest wavelength 500 nm is chosen for easy arithmetic and is not assigned to a real atomic transition. In the row with λobs = 500.050 nm, Δλ = 0.050 nm and z = 0.0001. The low-speed estimate is 299,792.458 × 0.0001 = 29.9792458 km/s.

The relativistic result is slightly smaller, about 29.977747 km/s. Whether that difference matters depends on the required accuracy. “Small compared with the speed of light” is a scale comparison, not permission to ignore an error larger than the precision of the question.

The mirrored wavelength shifts at 499.950 and 500.050 nm produce equal and opposite cz values. Their exact relativistic velocities are not perfectly symmetric because equal positive and negative z are not reciprocal wavelength ratios.

Keep digits through the ratio and round the final result. Subtracting nearly equal wavelengths after rounding them too coarsely can erase the shift entirely. A displayed 500.0 nm value may be insufficient for a tens-of-kilometres-per-second calculation.

Use consistent units and distinguish speed from redshift

NASA JPL's constants page lists c = 299,792,458 m/s, equivalent to 299,792.458 km/s. Use the version whose velocity unit you intend to report. The ratio z does not change when both wavelengths are converted by the same factor.

For example, 500.000 to 500.050 nm is the same fractional shift as 5000.00 to 5000.50 Å, or 0.500000 to 0.500050 micrometres. All give z = 0.0001. Mixing 500 nm with an observed value in Å would create an enormous false shift.

Do not write “z = 30 km/s.” The redshift is 0.0001 in this example; approximately 30 km/s is its low-speed velocity interpretation. Similarly, β = v/c is dimensionless and is not itself a wavelength.

Frequency moves in the opposite direction

For light in vacuum, νλ = c. With a positive wavelength shift, the observed frequency decreases. If R = λobs/λ₀, then νobs/ν₀ = 1/R, so z = ν₀/νobs − 1.

It is tempting to replace wavelength with frequency in z = (observed−rest)/rest, but that would reverse the sign and use the wrong denominator for the same z. The reciprocal relation must be carried through the algebra.

At λobs = 500.5 nm, the wavelength ratio is 1.001 while the frequency ratio is approximately 0.999000999. Their product is one. This provides a useful check when a catalogue supplies frequencies but a worksheet was originally written for wavelengths.

Several lines should agree in fractional shift

These three invented features share z = 0.0001. Their absolute wavelength shifts differ: 0.04, 0.05 and 0.07 nm. A common Doppler factor multiplies wavelengths, so equal physical motion does not require the same additive shift at every wavelength.

If correctly identified, similarly calibrated lines disagree strongly in z, investigate the identification, blending, calibration and the physical region producing the feature. Averaging unexplained mismatches is not automatically an improvement.

A line's width and its centre also convey different information. The calculation here uses a centre position. A broad feature is not evidence that the entire star has the centre velocity at every emitting location, and a width should not be substituted for Δλ in the centre-shift equation.

The radial relativistic formula and its inverse

For a single relative velocity directed along the line of sight, set β = v/c with −1 < β < 1 and positive β for recession. The wavelength ratio is R = √((1+β)/(1−β)). Squaring and rearranging gives β = (R²−1)/(R²+1).

The Astropy relativistic convention gives the equivalent relation in frequency. Converting frequency to wavelength produces the equations used here. The conditions matter: this is not a general instruction to turn every cosmological redshift into a unique local speed.

At β = 0.6, R = √(1.6/0.4) = 2 and z = 1. At β = −0.6, R = 0.5 and z = −0.5. Reversing the velocity replaces R by its reciprocal, not z by its negative.

For every positive finite R, the inverse formula remains between −1 and +1. Near zero, its leading term is β ≈ z. That limiting behaviour explains why the simpler expression works at low speed and why its validity should be checked against the accuracy required.

See where the straight-line approximation separates

The curves touch at the origin and are close for small magnitudes of β. On the receding side, the exact redshift grows more rapidly. At 0.8c it is 2, corresponding to three times the rest wavelength.

The vertical axis is redshift, not kilometres per second. A value of z = 2 therefore does not mean a physical speed of 2c in this model. Using the exact inverse returns β = 0.8.

Optical, radio and relativistic velocities are different conventions

The optical convention defines vopt/c = z. The radio convention defines vradio/c = (ν₀−νobs)/ν₀ = z/(1+z). The relativistic convention uses the inverse ratio above.

At small shifts the three are close, but at z = 1 they give 1, 0.5 and 0.6 in units of c. These are three ways to label the same measured spectral ratio. Calling all three the exact physical speed would be inconsistent.

The optical convention can produce a value above c, and the radio convention can give a sufficiently negative value below −c. That does not demonstrate superluminal local motion; it shows why a convention-defined velocity must retain its label. A catalogue's column name and reference frame should be read together.

Estimate the effect of a small wavelength uncertainty

With an exact rest wavelength and a small independent line-centre uncertainty σλ, the low-speed expression gives σv ≈ cσλ/λ₀. The next table isolates that one contribution for the invented 500 nm line.

For σλ = 0.005 nm, the velocity uncertainty from that term is about 2.998 km/s. The central shift of 0.050 nm would then be ten times that line-centre uncertainty. This ratio is an arithmetic illustration, not a claim about a real instrument's detection significance.

Real uncertainty also depends on the rest reference, calibration and correlated effects. If λ₀ is uncertain, its derivative in z is −λobs/λ₀², so its uncertainty must also be propagated. A table of calculator digits does not establish equivalent measurement precision.

Instrumental or observer corrections should follow the data product's documented convention. A topocentric measurement and a barycentric value can refer to different observers; copying one into a column labelled with the other does not perform a correction.

Keep radial motion, total motion and cosmic expansion separate

A radial velocity measures the line-of-sight component in the stated frame. It does not require a visible change in angular position. The proper-motion guide explains the transverse component and why distance is needed to turn angular drift into physical speed.

For low-speed stellar kinematics, transverse and radial components can be combined in one consistent frame. That does not make the purely radial relativistic formula a universal description of arbitrary three-dimensional relativistic motion.

NASA's spectroscopy guide distinguishes Doppler motion from cosmological redshift caused by the expansion of space. Inferring distance or a cosmological recession rate requires the appropriate cosmological model, not only the algebra for a locally moving source.

A reliable worksheet records the line identification, wavelength convention, rest and observed values, units, sign convention, velocity definition, reference frame and uncertainty. Those details make the answer reproducible and explain why two correct catalogues may display different velocity numbers for the same spectral shift.

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