Physics › Astrophysics › The Doppler effect and Hubble's law
The Doppler effect and Hubble's law
Motion leaves fingerprints on light: lines shift red when a source recedes and blue when it approaches, and they wobble to the beat of unseen companions. Apply the same shift to whole galaxies and a straight-line graph rewrites cosmic history, right back to a beginning.
Builds on Progressive waves and Black-body radiation and spectral classes.
IN THIS TOPIC
- Use Δf/f = v/c and z = Δλ/λ for sources moving much slower than light.
- Interpret the periodic Doppler shift of a spectroscopic binary seen in the plane of its orbit.
- Use Hubble's law to find distances and estimate the age of the universe, and state the Big Bang evidence.
WHAT YOU PROBABLY THINK
The Big Bang was an explosion at a point in space, and the galaxies are its shrapnel.
Motion, printed on light
The pitch of a siren rises as it approaches and falls as it passes: the waves ahead of a moving source are bunched, the waves behind stretched. Light does the same. A source moving away stretches every wavelength it emits toward the red; a source approaching squeezes them toward the blue. For speeds much less than c, the fractional shift equals the speed as a fraction of light's:
The quantity z is the red shift. Recession stretches wavelengths, so a receding source gives positive z; sign conventions vary between books, and the safe working habit is to reason physically: away means red, toward means blue, and the size of the shift gives the speed. The measurement is possible at all because spectra carry absorption lines at exactly known laboratory wavelengths: the whole line pattern slides together, and the slide is unmistakable.
WORKED EXAMPLE
A galaxy's speed from one line
Hydrogen's 656.3 nm line arrives from a galaxy at 662.9 nm. Find the galaxy's velocity.
Δλ = 662.9 − 656.3 = 6.6 nm, so z = 6.6 / 656.3 = 0.010.
v = zc = 0.010 × 3.0 × 108 = 3.0 × 106 m s−1, receding, since the shift is toward longer wavelength.
The nanometres never needed converting: z is a ratio, so any wavelength unit cancels. State the direction as part of the answer; the shift's sign is half the physics.
Binary stars: the wobble in the lines
Many stars come in orbiting pairs too close for any telescope to split. The spectrum betrays them. Watch a binary in the plane of its orbit and each star alternately approaches and recedes along the line of sight, so its spectral lines swing blue, then red, then blue again, once per orbit. The result is a wavelength that oscillates about the rest value.
Two readings come free. The period of the oscillation is the orbital period of the pair, and the maximum shift gives the orbital speed through z = Δλ/λ. Tilt the orbit out of the line of sight and the measured shifts shrink, which is why the clean textbook case specifies the plane of the orbit.
Hubble's law, and what it implies
Apply the red shift measurement to galaxies and a pattern with no exceptions appears: apart from a handful of close neighbours, every galaxy is receding, and the further away it is, the faster it goes. Speed is proportional to distance:
with H the Hubble constant, around 65 km s−1 Mpc−1: every megaparsec of extra distance adds 65 kilometres per second of recession. The law reads naturally as evidence that the universe is expanding, and here the shrapnel picture needs dismantling carefully. The galaxies are not flying through space away from one privileged spot: space itself is stretching, carrying the galaxies apart, and an observer in any galaxy sees the same law with themselves at the apparent centre. There is no shrapnel and no crater.
Run the expansion backwards and everything was once together: the Big Bang. Two further observations support it qualitatively. The cosmic microwave background is a faint glow arriving from every direction, a black-body spectrum at 2.7 K, the cooled and red-shifted afterglow of the hot early universe; Wien's law puts its peak near one millimetre. And the universe's relative abundance of hydrogen and helium, about three to one by mass, matches what fusion in the first few minutes of a hot dense universe would cook up, and no ordinary stellar history explains it.
YOUR TURN
From red shift to distance
A galaxy shows z = 0.020. Taking H = 65 km s−1 Mpc−1, find its recession speed and its distance, before opening the working.
Show the working
v = zc = 0.020 × 3.0 × 108 = 6.0 × 106 m s−1, which is 6000 km s−1.
d = v/H = 6000 / 65 = 92 Mpc. Keeping v in km s−1 and H in km s−1 Mpc−1 lets the units hand you megaparsecs directly.
TRY IT UNSEEN
The age of everything
Assuming the expansion speed of each galaxy has stayed constant, estimate the age of the universe from H = 65 km s−1 Mpc−1, with 1 Mpc = 3.08 × 1022 m.
Show the working
A galaxy now at distance d has travelled for a time t = d/v = d/(Hd) = 1/H, the same for every galaxy.
In SI units H = 65 000 / 3.08 × 1022 = 2.1 × 10−18 s−1, so t = 1/H = 4.7 × 1017 s ≈ 15 billion years.
The estimate leans on H never changing, which is exactly what the supernova measurements of the last lesson called into question; the modern figure is 13.8 billion years. For an assumption that crude, landing within ten per cent is a triumph.
THE EXAM BIT
- z = Δλ/λ uses the laboratory wavelength on the bottom, and the formulas hold only for v much less than c. Quote both facts when the question probes understanding.
- State the direction with every shift: longer wavelength means receding, shorter means approaching. A magnitude without a direction is half marks.
- The binary-star answer is a sentence about geometry: seen in the plane of the orbit, each star alternately approaches and recedes, so its lines oscillate red and blue with the orbital period.
- In v = Hd, match the units: v in km s−1 with d in Mpc, or convert H to SI for the age estimate. The age 1/H needs H in s−1.
- The Big Bang evidence is two items, each with its reason: the 2.7 K microwave background as the cooled afterglow, and the hydrogen-helium ratio matching early-universe fusion. Name both.
CHECK YOURSELF
A quasar shows z = 0.15. Estimate its recession speed and its distance for H = 65 km s−1 Mpc−1, and explain one reason the answer is only an estimate.
Show a hint
v = zc first; then keep v in km per second so H's units do the conversion.
Show the answer
v = zc = 0.15 × 3.0 × 108 = 4.5 × 107 m s−1, or 45 000 km s−1.
d = v/H = 45 000 / 65 = 690 Mpc.
At fifteen per cent of light speed the small-shift approximation z = v/c is beginning to strain, and H itself is only known roughly; both make the distance an estimate rather than a measurement.
Away means red, toward means blue, and the fractional shift is the speed over c.
Hubble: v = Hd, so red shift measures distance, and 1/H clocks the universe.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Use Δf/f = v/c and z = Δλ/λ for sources moving much slower than light.
- Interpret the periodic Doppler shift of a spectroscopic binary seen in the plane of its orbit.
- Use Hubble's law to find distances and estimate the age of the universe, and state the Big Bang evidence.
Open the full revision checklist to track your progress across the whole unit.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.