Physics • Year 12 • Module 8 • Lesson 3
Hubble's Law and the Expanding Universe
Apply Hubble's law, redshift calculations, and the scale factor to real spectral data, graphical analysis and multi-step quantitative problems.
1. Interpret spectral data, galaxy recession velocities
Five galaxies have been observed with spectrographs. The table records the rest wavelength and observed wavelength of their Hα emission line (rest λrest = 656.3 nm). Use H0 = 70 km/s/Mpc. 10 marks
| Galaxy | λobs (nm) | Redshift z | Recession velocity v (km/s) | Distance d (Mpc) |
|---|---|---|---|---|
| NGC-A | 659.2 | |||
| NGC-B | 669.4 | |||
| NGC-C | 689.1 | |||
| NGC-D | 722.0 | |||
| NGC-E | 787.6 |
1.1 Complete all three calculated columns in the table above. Show your working for NGC-C below. 6 marks (1 per complete row)
1.2 Galaxy NGC-E has a relatively large redshift. State one reason why using v = cz becomes less reliable as z increases. 2 marks
1.3 Using your calculated distances, identify which galaxy (NGC-A through NGC-E) would appear as the faintest in the night sky, assuming all five galaxies have the same intrinsic luminosity. Justify your answer. 2 marks
2. Interpret a Hubble diagram
The graph below is a simplified Hubble diagram plotting recession velocity (km/s) against distance (Mpc) for a set of Type Ia supernovae. Use H0 = 70 km/s/Mpc. 7 marks
Figure 2.1. Illustrative Hubble diagram for a set of Type Ia supernovae used as standard candles. H0 = 70 km/s/Mpc.
2.1 Describe the relationship shown in the Hubble diagram and identify what the gradient of the best-fit line represents. 2 marks
2.2 Use the graph to estimate the recession velocity of a galaxy at 350 Mpc. Show how you would check your reading using Hubble's law. 2 marks
2.3 The data points do not lie exactly on the best-fit line. State two physical reasons (not experimental error) why individual galaxies might deviate from the ideal Hubble relationship. 3 marks
3. Predict and justify, scale factor and cosmic history
The James Webb Space Telescope has detected a galaxy at redshift z = 12. 5 marks
3.1 Calculate the scale factor of the universe when the light from this galaxy was emitted. Show your working. 2 marks
3.2 Interpret what this scale factor means for the size of the universe at that time compared with today. 1 mark
3.3 Using the low-z approximation, estimate the lookback time for this galaxy in Gyr (use H0 = 70 km/s/Mpc; 1 Mpc = 3.086 × 1019 km). Comment on whether the low-z approximation is appropriate here and why. 2 marks
4. Compare, redshift, recession velocity and distance for four objects
Complete the two-column comparison for each pair of objects. Use H0 = 70 km/s/Mpc and c = 3.00 × 105 km/s. 8 marks (1 per correct cell)
| Feature | Galaxy P (z = 0.03) | Galaxy Q (z = 0.15) |
|---|---|---|
| Recession velocity (km/s) | ||
| Distance (Mpc) | ||
| Scale factor at emission | ||
| Observed Hα wavelength (nm) given λrest = 656.3 nm |
Q1.1, Galaxy data table
Using z = (λobs − 656.3) / 656.3, v = cz (c = 3.00 × 105 km/s), d = v / 70:
NGC-A: z = (659.2 − 656.3)/656.3 = 0.00442; v = 1326 km/s; d = 18.9 Mpc.
NGC-B: z = (669.4 − 656.3)/656.3 = 0.0200; v = 5994 km/s; d = 85.6 Mpc.
NGC-C: z = (689.1 − 656.3)/656.3 = 0.0500; v = 14 990 km/s; d = 214 Mpc. (Working: z = 32.8/656.3 = 0.0500; v = 0.0500 × 300 000 = 15 000 km/s; d = 15 000/70 = 214 Mpc.)
NGC-D: z = (722.0 − 656.3)/656.3 = 0.1001; v = 30 025 km/s; d = 429 Mpc.
NGC-E: z = (787.6 − 656.3)/656.3 = 0.200; v = 60 000 km/s; d = 857 Mpc.
Q1.2, Reliability of v = cz for large z
The relation v = cz is a non-relativistic, linear approximation valid only for z < ~0.1. For larger redshifts the special relativistic Doppler formula (or full cosmological treatment) must be used, otherwise recession velocity is overestimated. At z = 0.20, the low-z formula gives v = 0.20c, but the relativistic formula gives a slightly different value because time dilation and length contraction become non-negligible.
Q1.3, Faintest galaxy
NGC-E is faintest because it is the most distant (~857 Mpc). Apparent brightness decreases with the inverse square of distance, so for equal intrinsic luminosity, the most distant galaxy will appear faintest.
Q2.1, Hubble diagram description and gradient
The graph shows a linear (straight-line) relationship between recession velocity and distance, passing through the origin. The gradient of the best-fit line equals the Hubble constant H0 (units km/s/Mpc), the constant of proportionality in Hubble's law.
Q2.2, Reading from graph
Reading from the graph at d = 350 Mpc: v ≈ 24 500 km/s. Check: v = H0d = 70 × 350 = 24 500 km/s. Consistent.
Q2.3, Physical reasons for scatter
Any two of: (1) Peculiar velocities, galaxies have their own random motions relative to the overall Hubble flow (gravitational attraction to nearby galaxy clusters), which adds or subtracts a few hundred km/s to the measured recession velocity. (2) Gravitational interactions, galaxies in dense clusters are gravitationally bound and may not follow pure Hubble expansion. (3) Non-uniform matter distribution (large-scale structure), voids and filaments cause slight deviations from smooth expansion in different directions.
Q3.1, Scale factor at z = 12
1 + z = 1 / athen, so athen = 1 / (1 + 12) = 1/13 ≈ 0.077.
Q3.2, Interpretation of scale factor
When this light was emitted, the universe was approximately 1/13 (about 7.7%) of its current size, far smaller and denser than today.
Q3.3, Lookback time and validity of approximation
Low-z approximation: t ≈ z / H0. H0 in s−1: 70 km/s/Mpc = 70 × 103 / (3.086 × 1022) = 2.27 × 10−18 s−1. t = 12 / (2.27 × 10−18) = 5.29 × 1018 s ≈ 168 Gyr. This is clearly unphysical (much greater than the age of the universe at ~13.8 Gyr), demonstrating that the low-z approximation is completely invalid for z = 12. Full numerical integration of the Friedmann equations gives a lookback time of ~13.4 Gyr for this galaxy (near the beginning of the universe). Only use t ≈ z/H0 for z < ~0.1.
Q4, Comparison table
Galaxy P (z = 0.03): v = 0.03 × 3 × 105 = 9 000 km/s. d = 9 000 / 70 = 128.6 Mpc. athen = 1/(1 + 0.03) = 0.971. λobs = 656.3 × (1 + 0.03) = 676.0 nm.
Galaxy Q (z = 0.15): v = 0.15 × 3 × 105 = 45 000 km/s. d = 45 000 / 70 = 642.9 Mpc. athen = 1/(1 + 0.15) = 0.870. λobs = 656.3 × 1.15 = 754.7 nm.