Physics • Year 12 • Module 8 • Lesson 3
Hubble's Law and the Expanding Universe
Build HSC Band 5–6 extended-response technique on evaluating evidence for expansion, analysing Hubble's original data, and designing investigations to measure H0.
1. Analyse Hubble's original data (Band 5–6)
9 marks Band 5–6
Scenario. In 1929, Edwin Hubble published the recession velocities and distances for 24 nebulae (galaxies). The table below shows a simplified selection of six data points from his dataset. Distances were estimated using Cepheid variable stars as standard candles; velocities were measured from Doppler shifts of known spectral lines. H0 = 70 km/s/Mpc (modern accepted value; Hubble's own estimate was ~500 km/s/Mpc due to calibration errors in Cepheid period–luminosity relations).
| Nebula | Distance (Mpc) | Observed recession velocity (km/s) | v / d (km/s/Mpc) |
|---|---|---|---|
| NGC 221 | 0.76 | −300 | |
| NGC 598 | 0.87 | −79 | |
| NGC 1023 | 10.5 | 300 | |
| NGC 4736 | 4.7 | 290 | |
| NGC 5055 | 7.4 | 450 | |
| NGC 7331 | 14.7 | 500 |
Simplified and illustrative data based on Hubble (1929). Negative velocities indicate approach (blueshifted); these are Local Group galaxies with peculiar velocities dominating over cosmological expansion at small distances.
Q1. Analyse and evaluate Hubble's 1929 data to assess whether it conclusively supports a linear relationship between recession velocity and distance. In your response you must:
- Calculate v/d for each nebula and complete the table. Comment on the variation you observe.
- Identify which two nebulae are inconsistent with a simple Hubble law and explain the physical reason (hint: consider their distances and what might dominate their velocities).
- Excluding those anomalous nebulae, estimate H0 from the remaining data and compare your estimate with the modern value of 70 km/s/Mpc.
- Explain why Hubble's original estimate of H0 ≈ 500 km/s/Mpc was so much larger than the modern value, and why this would have led to an incorrect estimate of the age of the universe.
- State one specific improvement to Hubble's method that was made in the 20th century that led to the revised, lower value of H0.
2. Experimental design — measuring H0 with Type Ia supernovae (Band 5–6)
8 marks Band 5–6
Research question. A team of astronomers wants to measure the Hubble constant H0 by observing Type Ia supernovae in distant galaxies. Type Ia supernovae are “standard candles” — they all reach approximately the same peak absolute luminosity (L ≈ 1043 W). Design the investigation, including how you would use both spectroscopy and photometry to determine both recession velocity and distance for each supernova host galaxy.
Q2. Design the investigation and present it in the format below.
- State the aim of the investigation as a specific, testable research question.
- Describe how you would measure the recession velocity of each host galaxy (include the equation and the type of data required).
- Describe how you would use the supernova peak apparent brightness b and known luminosity L to calculate the distance d (include the inverse square law formula).
- Explain how you would then determine H0 from these measurements and how many galaxies you would need to obtain a reliable result.
- Identify two sources of systematic error in this method and propose one way to minimise each.
Q1 — Sample Band 6 response (9 marks), annotated
Calculated v/d values: NGC 221: −300/0.76 = −395 km/s/Mpc. NGC 598: −79/0.87 = −91 km/s/Mpc. NGC 1023: 300/10.5 = 28.6 km/s/Mpc. NGC 4736: 290/4.7 = 61.7 km/s/Mpc. NGC 5055: 450/7.4 = 60.8 km/s/Mpc. NGC 7331: 500/14.7 = 34.0 km/s/Mpc. [1 mark — all six values calculated correctly within rounding; accept ±2 km/s/Mpc. Award mark for partially correct table if method is shown.]
Variation and anomalous nebulae: The v/d ratios vary greatly: two (NGC 221 and NGC 598) are negative, indicating the nebulae are approaching rather than receding. These are inconsistent with Hubble's law [1]. The physical reason is that both NGC 221 (M32) and NGC 598 (M33) are members of the Local Group, at distances <1 Mpc. At such small distances, the Hubble flow recession velocity (~d × 70 = ~53–61 km/s) is much smaller than the galaxies’ peculiar velocities (—random gravitational motions of hundreds of km/s within the Local Group). Peculiar motion therefore dominates over cosmological expansion, producing net blueshifts. Hubble’s law only predicts the large-scale average expansion; it is overwhelmed by local gravitational dynamics at small distances [1].
Estimated H0 from remaining data: Excluding NGC 221 and NGC 598, average of NGC 1023, NGC 4736, NGC 5055, NGC 7331: (28.6 + 61.7 + 60.8 + 34.0) / 4 = 185.1 / 4 = 46.3 km/s/Mpc [1 for correct calculation]. This is lower than the modern value of 70 km/s/Mpc [1 for comparison and note on remaining scatter]. The scatter in the individual values (28.6 to 61.7) reflects small-scale peculiar motions even for these more distant objects — a reliable determination requires many more galaxies at larger distances where the Hubble flow dominates [1].
Why Hubble’s original H0 ≈ 500 km/s/Mpc was too large: Hubble used Cepheid variable stars as distance indicators, applying the period–luminosity relation established by Henrietta Leavitt. However, he used an incorrect calibration: he did not distinguish between two types of Cepheid (Population I and Population II), which obey different period–luminosity relations. Population II Cepheids (used for calibration in the Milky Way) are intrinsically fainter than Population I Cepheids (found in distant spiral galaxies). Using the brighter Population I Cepheids with the fainter Population II calibration caused Hubble to underestimate the true distance to the galaxies by a factor of ~2. An underestimated distance with the same measured velocity gives an overestimated H0 = v/d [1]. This led to a Hubble time tH = 1/H0 ≈ 1/(500 km/s/Mpc) ≈ 2 Gyr — embarrassingly shorter than the known age of the Earth (~4.5 Gyr). The modern value of 70 km/s/Mpc gives tH ≈ 14 Gyr, consistent with stellar ages [1].
Key 20th-century improvement: Walter Baade’s 1952 revision of the Cepheid distance scale (distinguishing Population I and II Cepheids using the 200-inch Palomar telescope) doubled estimated extragalactic distances and halved H0. Accept also: use of Type Ia supernovae as standard candles, or the Hubble Space Telescope Key Project (2001) which measured H0 = 72 ± 8 km/s/Mpc using multiple distance-ladder methods. [1]
Marking criteria (9 marks): 1 = all v/d calculated correctly (table complete). 1 = two anomalous nebulae identified (NGC 221 and NGC 598). 1 = physical explanation (peculiar motions dominate over Hubble flow at <1 Mpc for Local Group members). 1 = H0 correctly estimated from remaining 4 nebulae. 1 = comparison with 70 km/s/Mpc and comment on remaining scatter. 1 = explains why Hubble’s H0 was too large (Cepheid calibration / distance underestimated). 1 = links incorrect H0 to incorrect (too short) age of universe. 1 = names a specific 20th-century improvement. 1 = uses precise terminology throughout (peculiar velocity, standard candle, period–luminosity relation, Hubble flow).
Q2 — Sample Band 6 response (8 marks), annotated
Aim: To determine the Hubble constant H0 by measuring both the recession velocities and distances of at least 50 galaxies hosting Type Ia supernovae, and fitting the linear relationship v = H0d to the combined dataset. [1 — specific, testable, names the method]
Recession velocity (spectroscopy): Obtain a spectrum of the supernova or its host galaxy using a high-resolution spectrograph. Identify a known spectral line (e.g. Hα rest wavelength 656.3 nm). Measure the observed wavelength λobs. Calculate redshift: z = (λobs − λrest) / λrest. For z < 0.1, recession velocity v = cz. For larger z, use the relativistic Doppler formula. [1 — correct equation and data type described]
Distance (photometry): Measure the peak apparent brightness b of the supernova using a calibrated photometer or CCD detector (in watts per square metre or flux units). Use the inverse square law: b = L / (4πd2), rearranged as d = √(L / 4πb), where L ≈ 1043 W (the known intrinsic peak luminosity of a Type Ia supernova). This gives distance in metres; convert to Mpc (1 Mpc = 3.086 × 1022 m). [1 — correct formula d = √(L/4πb) with L defined]
Determining H0: For each supernova, calculate v and d. Plot a Hubble diagram (v vs d). Fit a straight line through the origin; the gradient equals H0. To obtain a statistically reliable result with less than 5% uncertainty, at least 50–100 supernovae distributed across a wide range of distances (50–500 Mpc) are needed to reduce the influence of peculiar velocities. [1 — gradient method and justification for large sample]
Systematic error 1 — Dust extinction: Interstellar and host-galaxy dust absorbs and scatters light, making supernovae appear fainter (b lower) than their true brightness. This would cause distance to be overestimated (d = √(L/4πb): lower b → larger d) and H0 to be underestimated. Minimisation: observe supernovae in multiple wavelengths (multi-band photometry); dust reddens light preferentially, so comparing optical and infrared brightness allows dust correction. [1 — correct identification and minimisation]
Systematic error 2 — Intrinsic luminosity variation: Not all Type Ia supernovae reach exactly the same peak luminosity — there is ~15% intrinsic scatter. Using a fixed L would cause distance errors. Minimisation: apply the “Phillips relation” (light-curve width–luminosity correction) — broader, slower light curves indicate intrinsically brighter supernovae. Measure the full light curve and apply the standardisation correction before calculating distance. [1 — correct identification and minimisation]
Marking criteria (8 marks): 1 = specific testable aim including method and quantity to be measured. 1 = correct recession velocity method with z equation. 1 = correct distance calculation including d = √(L/4πb) with L defined. 1 = correct H0 determination (gradient of v vs d plot) with sample size justification. 1 = first systematic error identified and correctly explained. 1 = first systematic error minimisation. 1 = second systematic error identified and correctly explained. 1 = second systematic error minimisation.