C.5.3—Light Doppler approximation

Use the low-speed model

This approximation applies when the relative source–observer speed $v$ is much smaller than the speed of light $c$. Here $f$ and $\lambda$ are emitted values, while $\Delta f$ and $\Delta\lambda$ are the magnitudes of the observed shifts.

Keep the direction

Motion away produces a redshift: $\Delta\lambda>0$ and the observed frequency decreases. Motion toward produces a blueshift: wavelength decreases and frequency increases. If a question asks for speed, use the absolute shift; if it asks for direction, use the sign of the wavelength or frequency change.

Worked example — spectral line

A line emitted at $4.86\times10^{-7}\,\text{m}$ is observed from a receding galaxy at $5.21\times10^{-7}\,\text{m}$. Then $\Delta\lambda=3.5\times10^{-8}\,\text{m}$. Substitution gives $v=(3.00\times10^8)(3.5\times10^{-8}/4.86\times10^{-7})=2.16\times10^7\,\text{m s}^{-1}$. The longer observed wavelength identifies recession.

Common trap

Do not put $\Delta\lambda$ in the denominator, and do not add $v$ to $c$. The approximation changes the observed frequency or wavelength, not the invariant speed of light.