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Doppler shifts of light

Waves, sound and light · level 8, lesson 6 of 6

A question to start

Engineers track a spacecraft on its way to Mars by the tiny change in the frequency of its radio signal. Astronomers find planets round other stars by the same trick. How can a frequency measure a speed?

The idea

Light shows the Doppler effect too. For a source moving at a speed \(v\) along the line of sight, much less than \(c\):

\[\frac{\Delta\lambda}{\lambda} = \frac{\Delta f}{f} \approx \frac{v}{c}.\]
  • Moving away: wavelengths get longer, a redshift.
  • Moving towards you: shorter, a blueshift.

Since spectral lines have exactly known wavelengths, measuring how far a line has moved gives the speed. For a radio signal from a spacecraft, the same rule turns a frequency shift into a speed.

Moving through space Mars Earth
Average speed round the Sun 24.08 km/s 29.78 km/s
Speed of a point on the equator, from spin 241 m/s 465 m/s
towardsstillaway

Worked examples

1. A spacecraft's radio. A signal at 8.4 GHz from a craft moving away at 10 km/s: \(\Delta f = 8.4\times10^9 \times 10^4/3.00\times10^8 = 280\) kHz lower.

2. A star. Hydrogen's 656.3 nm line from a star moving towards us at 30 km/s: \(\Delta\lambda = 656.3 \times 3\times10^4/3.00\times10^8 = 0.066\) nm shorter.

3. Mars's spin. The equator moves at \(2\pi \times 3396 \text{ km}/24.62 \text{ h} = 241\) m/s. Light from the edge of Mars turning towards us is shifted by only \(v/c = 8\times10^{-7}\) of its wavelength: tiny, but measurable.

Try it

  1. A line normally at 500.0 nm is seen at 500.1 nm. Is the source moving towards or away from us, and how fast?
  2. A spacecraft's 8.4 GHz signal arrives 140 kHz high. What is its speed along the line of sight?
  3. Why does the rule \(\Delta\lambda/\lambda = v/c\) only work for speeds much less than \(c\)?
  4. A planet makes its star wobble towards and away from us at 12 m/s. Find the shift of a 600 nm line.
Answers
  1. Away (redshift), at \(3.00\times10^8 \times 0.1/500 = 60\) km/s.
  2. \(v = c\,\Delta f/f = 3.00\times10^8 \times 1.4\times10^5/8.4\times10^9 = 5.0\) km/s, towards us.
  3. Near the speed of light, relativity changes the formula.
  4. \(600 \times 12/3.00\times10^8 = 2.4\times10^{-5}\) nm.

Common mistake

"Redshift means the light turns red." The whole spectrum moves a little towards longer wavelengths. A blue line stays blue to the eye; it is just measured slightly longer.

Where it leads