The Doppler Effect of Light
- The Doppler shift for a light-emitting non-relativistic (v << c) source can be described using the equation:
- Where:
- = change in frequency (Hz)
- = reference frequency (frequency of the source) (Hz)
- = change in wavelength (m)
- = reference wavelength (wavelength of the source) (m)
- = relative velocity of the source and observer (m s–1)
- = the speed of light (m s–1)
- The change in wavelength is equal to:
- Where:
- = observed wavelength of the source (m)
- The relative speed between the source and observer along the line joining them is given by:
- Where:
- = velocity of the light source (m s–1)
- = velocity of the observer (m s–1)
- The velocity of the observer (usually from Earth) can be assumed to be stationary, i.e.
- The relative speed then simply becomes the speed of the source:
- Hence, the Doppler shift equation can be written as:
Redshift
- The fractional change in the wavelength is called the redshift and is given the symbol
- In terms of wavelength, redshift is given by:
- In terms of frequency, redshift is given by:
- This shows that if the source moves away from the observer then
- The wavelength increases
- The frequency decreases
- Note: the sign of z can cause some confusion, remember to look at the context - z is a measure of redshift so write it as positive for receding objects
Doppler Shift of Light
- Doppler shift can be observed in the spectra of stars and galaxies
- If the star is approaching the Earth, blueshift is observed (negative )
- The relative velocity is positive
- The change in wavelength is negative
- If the star is receding from the Earth, redshift is observed (positive )
- The relative velocity is negative
- The change in wavelength is positive
Worked example
A stationary source of light is found to have a spectral line of wavelength 438 nm. The same line from a distant star that is moving away from the Earth has a wavelength of 608 nm.
Calculate the speed at which the star is travelling away from the Earth.
Answer:
Step 1: List the known quantities
- Unshifted wavelength, λ = 438 nm
- Shifted wavelength, λ' = 608 nm
- Change in wavelength, Δλ = (608 – 438) nm = 170 nm
- Speed of light, c = 3.0 × 108 m s–1
Step 2: Write down the Doppler equation and rearrange for velocity v
Step 3: Substitute values to calculate v
= 1.16 × 108 m s–1
Worked example
The stars in a distant galaxy can be seen to orbit about a galactic centre. The galaxy can be observed 'edge-on' from the Earth.
Light emitted from a star on the left-hand side of the galaxy is measured to have a wavelength of 656.44 nm. The same spectral line from a star on the right-hand side is measured to have a wavelength of 656.12 nm.
The wavelength of the same spectral line measured on Earth is 656.28 nm.
Answer:
(a)
- The light from the right-hand side (656.12 nm) is observed to be at a shorter wavelength than the reference line (656.28 nm)
- Therefore, the right-hand side shows blueshift and must, therefore, be moving towards the Earth
(b)
Step 1: List the known quantities
- Observed wavelength on LHS, = 656.44 nm
- Observed wavelength on RHS, = 656.12 nm
- Reference wavelength, λ = 656.28 nm
- Speed of light, c = 3.0 × 108 m s−1
Step 2: Calculate the average change in wavelength
- The magnitude of redshift or blueshift from each side is different, so you must calculate the average
= 0.16 nm
- Tip: you don't need to change the wavelengths from nm to m, as the units will cancel out later
Step 3: Write down the Doppler equation and rearrange for velocity v
Step 4: Substitute values into the velocity equation
Rotational speed:
Exam Tip
You need to recall that, in the visible light spectrum, red light has a longer wavelength and a lower frequency compared to blue light which has a shorter wavelength and a higher frequency