Doppler Redshift (Radial Velocity)

Determines a star or galaxy's radial velocity (toward or away from us) from the shift in its spectral lines.

Written by the Vorzep Team

AstronomySpectroscopyPractical

v ⟷λ_obs, λ_rest, c

Radial Velocity v

km/s

Observed Wavelength λ_obs

Rest Wavelength λ_rest

Speed of Light c

km/s

Solve for: v λ_obs λ_rest c

Result —

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Redshift vs. Observed Wavelength z = (λ_obs − λ_rest) / λ_rest

z(λ_obs) for fixed λ_rest Computed point

λ_obs, λ_rest in same units • c = 299792.458 km/s (speed of light)

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

Scenario: An astronomer observes a star and measures the H‑alpha line (rest wavelength 656.28 nm) shifted to 656.5 nm. They compute the radial velocity using v = c × (λ_obs - λ_rest) / λ_rest. With c = 299,792,458 m/s, the velocity is about 100,500 m/s (≈ 100 km/s) receding. This tells them the star is moving away from us, which is vital for understanding the dynamics of our galaxy.

Parameter	Value
λ_observed (nm)	656.5
λ_rest (nm)	656.28

1v = 3e8 × (656.5 - 656.28) / 656.28 ≈ 3e8 × 0.000335 = 100,500 m/s ≈ 100 km/s

Result ~100 km/s ✓ Receding velocity

Scenario: A galaxy's spectrum shows the H‑beta line (rest 486.13 nm) shifted to 486.3 nm. The astronomer calculates the recession speed v ≈ 3e8 × (486.3 - 486.13)/486.13 ≈ 105,000 m/s ≈ 105 km/s. This redshift indicates the galaxy is moving away, and combined with Hubble's law, they can estimate its distance.

Parameter	Value
λ_obs	486.3
λ_rest	486.13

1v = 3e8 × (486.3 - 486.13)/486.13 ≈ 1.05e5 m/s

Result ~105 km/s ✓ Galaxy recession

Insight: Doppler redshift is used to measure radial velocities. Positive shift (redshift) means receding, negative (blueshift) means approaching. This is fundamental in extragalactic astronomy.

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