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Redshift means recession.

Six original Cambridge-style questions on redshift, the Doppler relation Δλ/λ = v/c, Hubble's law v = H₀d, and what the law says about an expanding universe and its age.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
Keep these straight

Shift gives speed, speed gives distance.

01
Calculation
[2 marks]

A hydrogen line that has a wavelength of 656.3 nm in the laboratory is observed at 663.1 nm in the spectrum of a galaxy. Calculate the speed at which the galaxy is receding.

  • Δλ / λ = (663.1 − 656.3) / 656.3 = 6.8 / 656.3 = 1.04 × 10⁻². ✓
  • v = (Δλ / λ)c = 1.04 × 10⁻² × 3.0 × 10⁸ = 3.1 × 10⁶ m s⁻¹. ✓
02
Calculation
[2 marks]

Use Hubble's law with H₀ = 2.2 × 10⁻¹⁸ s⁻¹ to find the distance of the galaxy in the previous question. Give the answer in metres and in light-years (1 light-year = 9.5 × 10¹⁵ m).

  • d = v / H₀ = 3.1 × 10⁶ / 2.2 × 10⁻¹⁸ = 1.4 × 10²⁴ m. ✓
  • d = 1.4 × 10²⁴ / 9.5 × 10¹⁵ = 1.5 × 10⁸ light-years. ✓
03
Analysis
[3 marks]

State Hubble's law, and explain how the observation on which it is based leads to the conclusion that the universe is expanding.

  • The recession speed of a galaxy is directly proportional to its distance from us: v = H₀d. ✓
  • The light from almost every distant galaxy is redshifted, so they are moving away from us, and the further away a galaxy is the greater its redshift. ✓
  • A uniform expansion of space makes every galaxy recede from every other with a speed proportional to separation, so the observation does not place the Earth at a special centre; it means the whole universe is expanding. ✓
04
Calculation
[3 marks]

Galaxy X is at a distance of 6.0 × 10²⁴ m. (a) Calculate its recession speed. (b) A spectral line of laboratory wavelength 500.0 nm is observed in its light. Calculate the observed wavelength.

  • (a) v = H₀d = 2.2 × 10⁻¹⁸ × 6.0 × 10²⁴ = 1.3 × 10⁷ m s⁻¹. ✓
  • (b) Δλ = λv / c = 500.0 × 1.32 × 10⁷ / 3.0 × 10⁸ = 22 nm. ✓
  • Observed wavelength = 500.0 + 22 = 522 nm, shifted toward the red. ✓
05
Analysis
[3 marks]

A galaxy's distance was found from a type Ia supernova and its recession speed from its redshift. Explain why both measurements are needed to test Hubble's law, and suggest why the law is tested with distant galaxies rather than with the nearby stars of our own galaxy.

  • Hubble's law is a relation between speed and distance, so each galaxy must supply an independent value of both: the redshift gives v and the standard candle gives d. ✓
  • Plotting v against d for many galaxies and obtaining a straight line through the origin tests the law and its gradient gives H₀. ✓
  • Stars in our own galaxy are bound to it by gravity and move with their own orbital velocities, so they do not take part in the expansion of space; the law only appears for objects far enough away for cosmological recession to dominate. ✓
06
Calculation
[4 marks]

Assume that every galaxy has been receding at a constant speed since the Big Bang.

(a) Show that the time since the Big Bang is 1 / H₀. (b) Calculate this time in seconds and in years, using H₀ = 2.2 × 10⁻¹⁸ s⁻¹. (c) State one reason why the true age differs from this estimate.

  • (a) A galaxy now at distance d receding at v = H₀d has taken t = d / v = d / H₀d = 1 / H₀ to get there, the same for every galaxy. ✓
  • (b) t = 1 / 2.2 × 10⁻¹⁸ = 4.5 × 10¹⁷ s. ✓
  • t = 4.5 × 10¹⁷ / 3.15 × 10⁷ = 1.4 × 10¹⁰ years. ✓
  • (c) The expansion rate has not been constant: gravity slowed it in the early universe and dark energy is now speeding it up, so 1/H₀ is only an estimate of the age. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.