3.9.3.1 - Doppler Effect

What changes when a source moves

A source approaching an observer emits successive wavefronts from progressively nearer positions, so they arrive closer together in time. The observed frequency rises and the wavelength falls. Recession reverses those changes.

Astronomers first identify a recognisable pattern of spectral lines, then compare each observed line with its rest wavelength or rest frequency, measured in a laboratory. A shorter wavelength is a blueshift; a longer wavelength is a redshift. These names also apply to radio waves: the radiation need not be visible or actually blue or red.

For light in vacuum, $c=f\lambda$. The speed $c$ does not change when a source approaches. Instead, $f$ increases while $\lambda$ decreases. At the small speeds considered here, transverse motion produces no first-order Doppler shift: a large sideways speed can therefore have a very small measured radial velocity.

An optical hydrogen line and the radio line of neutral hydrogen probe the same type of motion. Radio observations are useful for gas that does not emit much visible light, but the comparison must still be with the correct laboratory line.

Equations and sign convention

A sign convention is a definition, not a physical disagreement. Here $v>0$ means motion towards the observer, as in the AQA frequency equation. Define changes as observed minus rest:

The definition of $z$ is exact. The velocity relations are approximations for small shifts. Use the rest value in each denominator. Frequency and wavelength shifts have opposite signs because $f\lambda=c$.

[DIAGRAM: asset_name: A.3.1 - Doppler Effect - Diagram 2; asset_slug: A.3.1 - Doppler Effect - Diagram 2; caption: A small blueshift and redshift around a 600.0 nm rest line; description: Numerical wavelength axis with a rest line at 600.0 nm, approaching line at 599.8 nm and receding line at 600.2 nm.]

For example, an optical line shifts from $500.0\,\mathrm{nm}$ to $500.2\,\mathrm{nm}$. Then $z=0.2/500.0=4.0\times10^{-4}$, so $v=-1.2\times10^5\,\mathrm{m\,s^{-1}}$: recession. Nanometres cancel in the ratio. The result is only $0.0004c$, consistent with the approximation. A speed answer is the positive magnitude together with its direction.

Binary stars viewed in the plane of orbit

Two binary stars orbit their common centre of mass. A spectroscopic binary is recognised from periodic changes in spectral lines even when the stars cannot be resolved in an image. If both spectra are visible, the two line systems move in opposite directions about their mean positions.