Doppler effect
Astrophysics & Cosmology - OCR A-Level Physics
Key Definition
Doppler effect
The change in observed wavelength (or frequency) of a wave due to relative motion between the source and the observer.
The change in observed wavelength (or frequency) of a wave due to relative motion between the source and the observer.
$$\frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}$$
$$\frac{\Delta\lambda}{\lambda} \approx \frac{\Delta f}{f} \approx \frac{v}{c}$$
- If a source moves away from the observer, wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m). increases (redshiftAn increase in the observed wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m). of light compared to the emitted wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m)., caused by the source moving away from the observer or by the expansion of space.): $\Delta\lambda$ is positive.
- If a source moves towards the observer, wavelength decreases (blueshiftA decrease in the observed wavelength of light from a source moving towards the observer (Doppler effect).): $\Delta\lambda$ is negative.
- This equation is valid for v << c (non-relativistic speeds).
- Astronomers measure redshiftAn increase in the observed wavelength of light compared to the emitted wavelength, caused by the source moving away from the observer or by the expansion of space. by observing the absorption linesDark lines in a continuous spectrum caused by atoms absorbing specific wavelengths; used to identify elements in stars. in a star's or galaxy's spectrum. The pattern is recognisable (e.g. hydrogen Balmer series) but shifted to longer wavelengths.
- The cosmological redshiftAn increase in the observed wavelength of light compared to the emitted wavelength, caused by the source moving away from the observer or by the expansion of space. of distant galaxies is due to the expansion of space itself, not just the motion of galaxies through space.
Worked Example
A hydrogen absorption line at 656.3 nm is observed at 659.2 nm in the spectrum of a distant galaxy. Calculate the galaxy's recessional speed.
Show Solution
1
$\Delta\lambda = 659.2 - 656.3 = 2.9 nm = 2.9 \times 10^{-9} m.$
2
$v/c = \Delta\lambda / \lambda.$
3
$v = c \times \Delta\lambda / \lambda = 3.0 \times 10^8 \times 2.9 \times 10^{-9} / (656.3 \times 10^{-9}).$
4
$v = 3.0 \times 10^8 \times (2.9 / 656.3).$
5
$v = 3.0 \times 10^8 \times 4.42 \times 10^{-3} = 1.33 \times 10^6 m s^{-1}.$
6
$$v \approx 1300$ km $s^{-1}$.$
Answer
$$v \approx 1.3$ $\times 10$^6 $m s^{-1}$$
Common Mistake
MEDIUM
Wrong: Using the observed wavelength as \lambda in the denominator instead of the emitted (rest) wavelength.
Right: \lambda in \Delta\lambda/\lambda is the EMITTED (rest) wavelength, not the observed wavelength. The difference \Delta\lambda = \lambda_{observed} - \lambda_{emitted}.
Right: \lambda in \Delta\lambda/\lambda is the EMITTED (rest) wavelength, not the observed wavelength. The difference \Delta\lambda = \lambda_{observed} - \lambda_{emitted}.