The Doppler effect

When a source of waves travels towards or away from us, the frequency and wavelength that we measure is different to that which would be measured if it was stationary. This is known as the Doppler effect and is essential to the measurement of speeds, and, as we shall see, masses in the Universe.

Consider the situation sketched in Fig. 3.4 in which a source of waves (S, S') moves away from an observer O at speed V_R.

Figure 3.4: A source of waves moves away from an observer O. When at S', it emitted wavefront A', and now it is at S, it is just about to emit the next one.

Assume that when the source was at S' it emitted the wavefront labelled A', and that it is just about to emit the next one having reached position S. By definition the distance from S' to A' is one wavelength, as seen had the source been stationary. This is known as the ``rest'' wavelength and denoted $\lambda_0$, i.e.

$$S'A' = \lambda_0,$$

as indicated in the figure. Also by definition, the time between the emission of one wavefront and the next is $1/f = \lambda/c$, where f is the frequency and c is the wave's speed. Therefore the distance from S to S' is given by

$$SS' = \frac{\lambda_0}{c} V_R$$

where V_R is the speed at which the source moves away from O. At S a new wavefront is just about to be launched and therefore the distance from S to A is the wavelength measured at O, which will be called $\lambda$. Therefore

$$\lambda = SA = SS' + S'A = \frac{\lambda_0}{c} V_R + \lambda_0 .$$

Rearranging we obtain:

(3.2)
This is the equation describing the Doppler effect which you should understand and remember.

The speed V_R is the component of speed away from O along the line-of-sight. This is usually called the ``radial velocity'' which is why the R subscript has been added.

To use the Doppler effect we need to know the rest wavelength $\lambda_0$. This is possible because an atom on the surface of a star is very much the same as it would be on Earth, and crucially, the wavelengths characteristic of its various transitions are the same as they are on Earth. Thus for example, the transition from the n=2 to n=3 level of hydrogen produces emission at $656.276\,\mathrm{nm}$ (red in colour) whether it is on Earth, the Sun or a distant galaxy. We need only measure its wavelength as we observe it to deduce V_R, given that we know c the speed of light.

It is easily possible to measure changes of wavelength of 1 part in 10⁵ or less, giving precision in velocity of about one kilometre per second. On bright stars this can be lowered to around $10\,\mathrm{m}\,\mathrm{s}^{-1}$, which is enough to detect the small wobble induced by Jupiter-mass planets.