Doppler Effect, Binary Stars & Exoplanets — Quick Summary

Mr Toogood's Physics · Astrophysics

AQA 3.9.3.1 / 3.9.3.4
z = Δλ/λ = v/c
Redshift
Δf/f = −Δλ/λ
= z (AQA form)
Δλ=λ_lab−λ_obs
AQA sign convention
z>0 recede, z<0 approach
Interpreting z

The Doppler effect & redshift

Diagram of the Doppler effect showing compressed and stretched wavefronts around a moving source

Motion towards the observer compresses wavelengths (blue shift); away, it stretches them (red shift).

A source moving relative to an observer changes the observed wavelength: towards → shorter λ (blue shift); away → longer λ (red shift). Only valid for v≪c.

Δλ = λ_lab − λ_obs    Δf = f_lab − f_obs
Δf/f = −Δλ/λ = v/c = z
AQA's sign convention: both Δλ and Δf are calculated as (lab − observed), then linked by a minus sign — this keeps z>0 for receding objects and z<0 for approaching ones.

Worked example: the Sun's rotation

Comparing the wavelength shift on opposite limbs of the Sun gives its rotational speed via the Doppler equation; combining with ω=v/r and T=2π/ω then gives its rotational period.

Binary stars

Radial velocity method showing spectral line shifts as two binary stars orbit each other

As the stars orbit, spectral lines oscillate between red- and blue-shifted.

Two stars orbiting a common centre of mass. Radial velocity method: as one approaches (blue-shift) and the other recedes (red-shift), spectral lines split and oscillate — this is the only reliable way to measure stellar masses.

  • Visual binaries: individually resolved by telescope.
  • Spectroscopic binaries: too close to resolve — identified only via spectra/light curves.
  • Transit (eclipsing) method: dips in total apparent magnitude as one star eclipses the other — a bigger dip occurs when the brighter star is eclipsed.

Exoplanets

Light curve showing the brightness dip as an exoplanet transits in front of its star

Transit method: periodic dip in brightness as the planet crosses in front of the star.

  • Transit method (~78% of discoveries): periodic brightness dip; the planet contributes no light of its own, so the curve shape differs slightly from an eclipsing binary. Best for short orbital periods; can detect Earth-sized planets.
  • Radial velocity method: tiny periodic Doppler shift as the star wobbles under the planet's gravity — needs very sensitive spectrometers.
  • Direct imaging: only feasible for large planets far from a bright parent star.

Exam essentials

Key equations

  • z=Δλ/λ=v/c
  • Δf/f=−Δλ/λ=z
  • Δλ=λ_lab−λ_obs (AQA)

Detecting binaries/exoplanets

  • Radial velocity: periodic Doppler shift in spectral lines.
  • Transit: periodic dip in brightness.
  • Combining both gives mass and radius.

Common slips

  • Use AQA's specific sign convention consistently — don't mix it with the "observed minus lab" convention used elsewhere.
  • z is only valid for v≪c — no relativistic correction expected.
  • Transit method needs the orbit viewed edge-on (in Earth's line of sight) to work at all.