To understand galactic redshift, you first need the Doppler effect, which is the apparent change in the wavelength (and frequency) of a wave when its source is moving relative to the observer.
What the Doppler effect is
- Imagine a stationary source emitting waves. The waves spread out in a series of evenly spaced spherical wavefronts in every direction
- Now make the source move. The wavefronts are still produced at the same rate, but the source is catching up with the wavefronts ahead of it and moving away from the wavefronts behind it
- In front of the moving source:
- The wavefronts are bunched closer together
- The wavelength is shorter (λ − Δλ)
- The frequency is higher
- Behind the moving source:
- The wavefronts are stretched apart
- The wavelength is longer (λ + Δλ)
- The frequency is lower
An everyday example: ambulance sirens
- An ambulance siren has a higher pitch when the ambulance is coming towards you (wavelengths bunched up in front, higher frequency, higher pitch)
- The pitch then suddenly drops as the ambulance passes and starts moving away (wavelengths stretched behind, lower frequency, lower pitch)
- The change in pitch is obvious to the ear. Light waves do exactly the same thing, but the change is invisible to your eyes; you need a spectrometer to detect it
The Doppler equation for light
- The fractional change in wavelength is equal to the speed of the source divided by the speed of light:
Δλ / λ₀ = v / c
- Where:
- λ₀ = the reference wavelength, the wavelength the source would emit if it were not moving (often measured in a laboratory on Earth)
- λ = the observed wavelength, what arrives at your detector
- Δλ = λ − λ₀, the change in wavelength
- v = the speed of the source away from (or towards) the observer (m/s). Positive for moving away (redshift), negative for moving towards (blueshift)
- c = the speed of light = 3 × 10⁸ m/s
- Both sides of the equation are dimensionless, since it is wavelength divided by wavelength, speed divided by speed. So the Doppler shift itself has no units
- Rearranging for v:
v = c × (λ − λ₀) / λ₀
Example — In a laboratory, a hot hydrogen lamp produces a green emission line at a wavelength of 500 nm. The same emission line, observed from a distant galaxy, arrives at the Earth with a wavelength of 510 nm. Taking the speed of light as 3 × 10⁸ m/s, work out how fast the galaxy is receding from us.
- Step 1 — List the known quantities
- Reference wavelength λ₀ = 500 nm = 500 × 10⁻⁹ m
- Observed wavelength λ = 510 nm = 510 × 10⁻⁹ m
- Speed of light c = 3 × 10⁸ m/s
- Step 2 — Find the change in wavelength
- Δλ = λ − λ₀ = (510 − 500) × 10⁻⁹ = 10 × 10⁻⁹ m
- Step 3 — Apply v = c × Δλ / λ₀
- v = (3 × 10⁸) × (10 × 10⁻⁹) / (500 × 10⁻⁹)
- v = (3 × 10⁸) × (10 / 500)
- v = (3 × 10⁸) × 0.02
- v = 6 × 10⁶ m/s (6000 km/s)
- The galaxy is moving away from us (because the wavelength has increased) at 6000 km/s, about 2% of the speed of light