The Doppler effect shifts observed frequency and wavelength when source and observer move relative to each other

Astrophysics | AQA A-Level Physics

Key Definition

Doppler effect (for light): The change in observed frequency (or wavelength) of electromagnetic radiation caused by relative motion between the source and the observer.

The Doppler shift equation

For a light-emitting source moving at non-relativistic speeds ($v \ll c$), the Doppler shift is described by:

$$\frac{\Delta f}{f} = \frac{\Delta \lambda}{\lambda} \approx \frac{v}{c}$$

$\Delta f$ = change in frequency (Hz)
$f$ = reference frequency of the source (Hz)
$\Delta \lambda$ = change in wavelength (m)
$\lambda$ = reference wavelength of the source (m)
$v$ = relative velocity of the source and observer along the line joining them (m s$^{-1}$)
$c$ = the speed of light (m s$^{-1}$)

Working with wavelength changes

The change in wavelength $\Delta \lambda$ is defined as the difference between the observed wavelength $\lambda'$ and the emitted (reference) wavelength $\lambda$:

$$\Delta \lambda = \lambda' - \lambda$$

The relative speed between source and observer along the line joining them is $\Delta v = v_s - v_o$, where $v_s$ is the velocity of the source and $v_o$ is the velocity of the observer.
Crucially, the observer (on Earth) can usually be assumed stationary, so $v_o = 0$ and $\Delta v = v_s = v$.
The full Doppler equation then becomes:

$$\frac{\Delta f}{f} = \frac{\lambda' - \lambda}{\lambda} = \frac{\Delta \lambda}{\lambda} = \frac{v}{c}$$

Redshift and blueshift

Key Definition

Redshift ($z$): The fractional change in wavelength of light from a source moving away from the observer. It is defined as $z = \dfrac{\Delta \lambda}{\lambda} = \dfrac{v}{c}$.

If the source moves away from the observer: the observed wavelength increases, the observed frequency decreases, and redshift ($z > 0$) is observed.
If the source moves towards the observer: the observed wavelength decreases, the observed frequency increases, and blueshift ($z < 0$) is observed.
In terms of frequency, redshift is given by $z = -\dfrac{\Delta f}{f} = \dfrac{v}{c}$. The minus sign arises because a decrease in frequency corresponds to a positive recession velocity.

Common Mistake

The sign of $z$ can trip you up. Redshift is defined as positive for objects moving away from us. If you calculate $\Delta f / f$ and get a negative number, the source is receding and $z$ is positive. Keep track of minus signs carefully: if the observed frequency is lower than the emitted frequency, the object is moving away.