HSC Exam Practice

Sample response. Hubble's law states that the recession velocity of a galaxy is directly proportional to its distance from the observer: v = H₀d, where v = recession velocity (km/s), H₀ = Hubble constant (km/s/Mpc, approximately 70 km/s/Mpc), and d = distance (Mpc). One Mpc = 3.086 × 10¹⁹ km ≈ 3.26 million light-years.

Marking notes. 1 mark for correct verbal statement (velocity proportional to distance); 1 mark for equation v = H₀d with all symbols defined; 1 mark for correct units of H₀ (km/s/Mpc or equivalent).

Sample response. Redshift (z) is the fractional increase in observed wavelength of light from a receding source, defined as z = (λ_obs − λ_rest) / λ_rest. For the Hβ line: z = (510.4 − 486.1) / 486.1 = 24.3 / 486.1 = 0.0500. Since z > 0 (observed wavelength > rest wavelength), the galaxy is receding.

Marking notes. 1 mark for definition of redshift (fractional wavelength increase, source moving away); 1 mark for correct equation; 1 mark for correct numerical calculation (z = 0.0500 ± 0.0002); 1 mark for concluding galaxy is receding with justification (λ_obs > λ_rest).

Sample response. (a) Scale factor: 1 + z = 1/a_then, so a_then = 1/(1 + 2.5) = 1/3.5 = 0.286. The universe was about 28.6% of its current size when this light was emitted. (b) Recession velocity: v = cz = 3.00 × 10⁵ × 2.5 = 7.50 × 10⁵ km/s (= 2.5c). (c) Distance: d = v/H₀ = 750 000/70 = 10 700 Mpc.

Note: For z = 2.5, the low-z approximation v = cz gives a recession velocity exceeding c. This is physically valid in the expanding-universe framework (superluminal recession is permitted as the separation increases due to space expanding, not motion through space), but it means the calculation is only an approximation. Full cosmological treatment gives a somewhat different comoving distance. Accept the low-z calculation as stated in the question.

Marking notes. 1 mark for correct scale factor a = 0.286; 1 mark for recession velocity 7.50 × 10⁵ km/s; 1 mark for distance 10 700 Mpc; 1 mark for noting the low-z approximation is used and acknowledging its limitation at large z (or noting v > c). Accept full marks if all three values are correct even without the caveat on the approximation.

Sample response. The Hubble time t_H = 1/H₀ assumes the universe has expanded at a constant rate since the Big Bang. In reality, the expansion rate has not been constant: gravity has been decelerating the expansion during the matter-dominated era, making the actual elapsed time shorter than t_H. Additionally, dark energy has been accelerating the expansion in recent cosmological history, partially counteracting gravity. The net effect is that the actual age (~13.8 Gyr) is close to but slightly less than the Hubble time (~14 Gyr). The Hubble time is therefore an approximation, not the exact age.

Marking notes. 1 mark for stating that t_H assumes constant expansion rate; 1 mark for explaining that gravity has decelerated expansion in the past (making actual age shorter than t_H); 1 mark for explaining or acknowledging the interplay of deceleration and dark-energy acceleration. Do not penalise if student omits dark energy — deceleration argument alone earns full marks if stated clearly.

Sample response. A galaxy's peculiar velocity is its individual random motion through space due to local gravitational interactions with nearby galaxies and clusters — typically a few hundred km/s. The Hubble flow recession velocity is the velocity due purely to the cosmological expansion of space, given by v = H₀d. For nearby galaxies at small distances, the Hubble flow velocity is comparable to or smaller than the peculiar velocity, so the measured Doppler shift is dominated by peculiar motion and does not reliably indicate distance. For galaxies at large distances (d ≫ 100 Mpc), the Hubble flow velocity (thousands to tens of thousands of km/s) greatly exceeds typical peculiar velocities (~300 km/s), so the total observed velocity is dominated by cosmological expansion and Hubble's law becomes a reliable distance indicator.

Marking notes. 1 mark for correctly distinguishing peculiar velocity (local gravitational motion) from Hubble flow (expansion); 1 mark for explaining why peculiar velocity dominates at small distances; 1 mark for explaining why Hubble flow dominates at large distances, making the law reliable.

Sample response. Convert H₀ to s⁻¹: H₀ = 500 km/s/Mpc = 500 × 10³ m s⁻¹ / (3.086 × 10²² m) = 1.621 × 10⁻¹⁷ s⁻¹. Hubble time (original): t_H = 1/H₀ = 1/(1.621 × 10⁻¹⁷) = 6.17 × 10¹⁶ s = 6.17 × 10¹⁶ / (3.156 × 10⁷) = 1.95 × 10⁹ yr ≈ 1.95 Gyr. Modern H₀ = 70 km/s/Mpc: t_H = 1/(70 × 10³ / 3.086 × 10²²) = 1/(2.27 × 10⁻¹⁸) = 4.41 × 10¹⁷ s ≈ 13.97 Gyr. Implication: Hubble’s original H₀ implied the universe was only ~2 Gyr old — younger than the Earth (~4.5 Gyr) and many stars (~10–13 Gyr), which is physically impossible. This “age crisis” was resolved by successive revisions to the Cepheid distance scale that reduced H₀.

Marking notes. 1 mark for correct conversion of H₀ = 500 km/s/Mpc to s⁻¹; 1 mark for correct Hubble time calculation using original H₀ (~2 Gyr); 1 mark for correct Hubble time using modern H₀ (~14 Gyr); 1 mark for explaining the implication (universe younger than Earth — physically impossible — age crisis resolved by revised distance scale).

Sample response (a) — redshift calculations.

Galaxy X, Hα: z = (672.5 − 656.3)/656.3 = 16.2/656.3 = 0.0247. Galaxy X, Ca II K: z = (403.1 − 393.4)/393.4 = 9.7/393.4 = 0.0247. Both lines give z = 0.0247 for Galaxy X — highly consistent, confirming the redshift is real and not an instrumental artifact or misidentification of the spectral line. [2 marks: 1 for both correct z values; 1 for comment on consistency]

Galaxy Y, Hα: z = (696.5 − 656.3)/656.3 = 40.2/656.3 = 0.0613. Galaxy Y, Ca II K: z = (417.9 − 393.4)/393.4 = 24.5/393.4 = 0.0623. Galaxy Y: both lines give z ≈ 0.062 — consistent within ~2%, which is acceptable given observational uncertainty. Consistent z values from multiple spectral lines increase confidence that the observed wavelength shift is genuinely cosmological and not caused by an error in line identification. [2 marks: 1 for both correct z values; 1 for comment on consistency / significance]

Sample response (b) — recession velocity and distance (Galaxy Y, showing working).

Average z (Galaxy Y) = (0.0613 + 0.0623)/2 = 0.0618. Recession velocity: v = cz = 3.00 × 10⁵ × 0.0618 = 18 540 km/s. Distance: d = v/H₀ = 18 540/70 = 265 Mpc. Galaxy X (z = 0.0247): v = 7 410 km/s; d = 106 Mpc. [3 marks: 1 for working for Galaxy Y; 1 for correct Galaxy Y answer; 1 for correct Galaxy X answer]

Sample response (c) — which galaxy appears larger.

Galaxy X (at 106 Mpc) would appear larger. Both galaxies are physically the same size, so apparent angular size scales inversely with distance. Galaxy X is approximately 265/106 ≈ 2.5 times closer than Galaxy Y, so it subtends ~2.5 times the angular diameter. Quantitatively, if both have true diameter D, angular size θ = D/d: θ_X/θ_Y = d_Y/d_X = 265/106 = 2.50. [2 marks: 1 for correct identification (Galaxy X) with reason (smaller distance); 1 for quantitative comparison using angular size ∝ 1/d]

Sample response. The primary evidence that the universe is expanding comes from the systematic redshift of galaxies, quantified by Hubble's law. In 1929, Hubble observed that almost all galaxies are receding from us, and that recession velocity v is proportional to distance d: v = H₀d (H₀ ≈ 70 km/s/Mpc). This relationship holds in all directions; a galaxy at 100 Mpc recedes at ~7 000 km/s, while one at 500 Mpc recedes at ~35 000 km/s. The simplest physical interpretation is that space itself is expanding uniformly: galaxies are not moving through space, but are being carried apart by the expansion of the space between them, like raisins in rising bread. This explains why every galaxy appears to be receding from us regardless of direction — there is no special centre to the expansion.

However, Hubble's law alone has limitations as evidence for expansion. It could, in principle, be explained by a large explosion from a central point, with matter simply moving outward through static space at speeds proportional to distance. This alternative (the “tired light” or “steady state” interpretation) was eventually ruled out by independent evidence. Furthermore, Hubble's law breaks down at very small distances (local group galaxies dominated by peculiar velocities) and requires modification at very large redshifts.

The concept of the scale factor a(t) provides a more complete model. The scale factor quantifies the relative size of the universe at time t: a = 1 today by convention, a < 1 in the past. Redshift is directly related to the scale factor by 1 + z = 1/a_then, so a measured redshift directly encodes the size of the universe when the light was emitted. For example, a galaxy at z = 4 emitted its light when the universe was a = 1/5 = 0.20 of its current size — five times smaller and denser than today. This is a much more powerful statement than Hubble's law alone: it maps cosmic history and allows reconstruction of the universe's expansion history at all epochs, not just in the nearby universe. The scale factor model, embedded in the Friedmann equations, predicts how the expansion rate changed over time under the influence of gravity and dark energy — predictions confirmed by supernova observations in 1998 that discovered the accelerating expansion.

In summary, Hubble's law provides the first quantitative evidence for expansion and remains the primary tool for cosmological distance measurement, but the scale factor provides a theoretically complete framework that links observations at all redshifts to the dynamical history of the universe.

Marking criteria (8 marks). 1 = states Hubble's law correctly and links to expansion evidence. 1 = uses a specific numerical example (e.g. v = H₀d with values, or z from a specific galaxy). 1 = explains the physical interpretation (space expanding, not motion through static space). 1 = identifies at least one limitation of Hubble's law as evidence for expansion (e.g. doesn't distinguish expansion from central explosion at low z; fails at very small distances; approximate at large z). 1 = defines scale factor correctly (a = 1 today, a < 1 in the past) and writes 1 + z = 1/a_then. 1 = applies scale factor to a specific numerical example (e.g. z = 4 → a = 0.2 → universe was 1/5 its current size). 1 = explains how scale factor provides a more complete model than Hubble's law alone (maps full expansion history, links to Friedmann equations or dark energy). 1 = reaches an explicit, evaluative conclusion integrating both lines of evidence.