HSC Exam Practice

Sample response. Luminosity is the total power output of a star, measured in watts or solar luminosities (L⊙); it is an intrinsic property independent of distance. Apparent brightness is how bright a star looks from Earth and depends on both its luminosity and its distance from the observer. Two stars can have the same apparent brightness if a nearby, dim star emits the same flux at Earth as a distant, luminous star — the inverse-square law means a star 10 times further away must be 100 times more luminous to appear equally bright.

Marking notes. 1 mark for correct definition of luminosity (total power/intrinsic property); 1 mark for correct definition of apparent brightness (depends on distance and luminosity); 1 mark for explaining why two stars can appear equally bright (inverse-square law / distance compensates for luminosity difference).

Sample response. Main sequence: stars fusing hydrogen in their cores; radius ranges from <0.1 R⊙ (low-mass) to ~10 R⊙ (high-mass). Red giants/supergiants: above and right of main sequence; very large radii (10–1 000 R⊙) because, from L = 4πR²σT&sup4;, low T requires very large R to maintain high L. White dwarfs: below and left of main sequence; tiny radii (~0.01 R⊙, Earth-sized) because, although T is high, L is very low, implying small R. (Fourth region: upper main sequence / hot supergiants are part of the main sequence or giant branch, so three main regions plus the full main sequence span suffices.)

Marking notes. 1 mark per correctly identified region with correct relative radius AND a correct reference to Stefan-Boltzmann (accept implicit reasoning from L, T, R relationship). Accept any four of: (1) main sequence, (2) red giant branch, (3) supergiant region, (4) white dwarfs.

Sample response. The HR diagram is plotted with temperature decreasing to the right (a historical convention reflecting how spectra were first classified); equivalently, temperature increases to the left. The spectral sequence OBAFGKM runs from the hottest (O, T > 30 000 K, blue-white) to coolest (M, T < 3 700 K, red). Each type corresponds to characteristic absorption lines: O stars show ionised helium; G stars (like the Sun) show strong calcium lines; M stars show molecular bands. Colour is determined by Wien’s law: hotter stars have λ_max at shorter (bluer) wavelengths.

Marking notes. 1 mark for explaining the temperature axis convention (increases left, historical/classification reason accepted); 1 mark for correctly ordering OBAFGKM hottest to coolest with approximate temperatures or colours; 1 mark for linking spectral type to observational property (absorption lines, colour, Wien’s law — accept any one).

Sample response. T = 2.898 × 10⁻³ / (290 × 10⁻⁹) = 2.898 × 10⁻³ / 2.90 × 10⁻⁷ = 9 993 K ≈ 10 000 K [1 mark]. Spectral type: T ≈ 10 000 K falls at the boundary of A/B; accept A-type (7 500–10 000 K) or B-type depending on convention [1 mark]. Radius: R/R⊙ = √(L/L⊙) / (T/T⊙)² = √20 000 / (10 000/5 778)² = 141.4 / 2.998 ≈ 47 R⊙ [1 mark for correct substitution; 1 mark for correct answer].

Marking notes. 1 mark for correct T calculation (~10 000 K, working shown); 1 mark for spectral type A or B (accept borderline); 1 mark for correct R/R⊙ setup using R ∝ √L/T²; 1 mark for correct numerical answer (accept 45–50 R⊙).

Sample response. (1) Measure the star’s spectral type from its absorption line pattern; this identifies the star’s position on the HR diagram. (2) Use the HR main-sequence calibration to look up the star’s absolute magnitude M (intrinsic luminosity). (3) Compare M to the measured apparent magnitude m using the distance modulus m − M = 5 log(d/10), solving for d in parsecs. This works because spectral type (and luminosity class) uniquely determine absolute magnitude, bypassing the need for a measurable parallax angle.

Marking notes. 1 mark for measuring spectral type and using HR diagram to find absolute magnitude; 1 mark for using distance modulus (m − M = 5 log d/10) to find distance; 1 mark for explaining why this bypasses parallax (spectral type determines intrinsic luminosity; parallax angle too small / unmeasurable at large distances).

Sample response. L ∝ M^3.5 means massive stars are far more luminous; a 20 M⊙ star is ~20^3.5 ≈ 36 000 times more luminous than the Sun. Since main sequence lifetime t ∝ M/L ∝ M^−2.5, this same star lives ~20^−2.5 ≈ 0.00056 times as long as the Sun — only ~5.6 Myr compared to 10 Gyr. In a cluster, all stars formed simultaneously; the most massive stars exhaust their hydrogen first and leave the main sequence first. The main sequence turn-off point (the most massive stars still on the main sequence) indicates when the cluster formed and hence its age.

Marking notes. 1 mark for correctly using L ∝ M^3.5 to explain why massive stars are more luminous and consume fuel faster; 1 mark for correctly deriving/stating t ∝ M^−2.5 and explaining the consequence (massive stars leave MS first); 1 mark for naming the main sequence turn-off point as the HR diagram feature used to determine cluster age.

Sample response (a). Star P: T = 10 000 K → A-type (or B-type boundary); L = 80 L⊙ is moderately high but not supergiant level; combined with T = 10 000 K, Star P lies on the upper main sequence [1]. Star Q: T = 3 500 K → M-type; L = 500 L⊙ is high for a 3 500 K star — a main-sequence M star has L << 1 L⊙, so 500 L⊙ indicates Star Q is a red giant or supergiant in the upper-right of the HR diagram [1]. Spectral types and regions earn 1 mark each = 4 marks total for part (a).

Sample response (b). R/R⊙ = √(L/L⊙) / (T/T⊙)². Star P: R = √80 / (10 000/5 778)² = 8.94 / 2.998 ≈ 2.98 R⊙ [1 mark working; 1 mark answer]. Star Q: R = √500 / (3 500/5 778)² = 22.36 / 0.366 ≈ 61 R⊙ [1 mark working; 1 mark answer]. Physical meaning: Star Q’s enormous radius (~61 R⊙) accounts for its high luminosity despite low temperature; its surface area (proportional to R²) is ~420 times that of Star P, vastly outweighing Star Q’s lower T&sup4; factor.

Sample response (c). The conclusion is incorrect [1]. Although Star Q is more luminous (500 vs 80 L⊙), its temperature (3 500 K, M-type) and radius (~61 R⊙) are inconsistent with a main-sequence M star, which would have L < 0.01 L⊙ and R < 0.3 R⊙. Star Q’s luminosity can only be explained by its very large radius, placing it in the red giant region, not the main sequence [1].

Marking notes (10 marks). Part (a): 1 mark per star classification with HR region justified by T and L = 4 marks. Part (b): 1 mark per star for correct substitution, 1 mark per star for correct answer (accept R within ±5%); plus 0 additional marks if physical meaning not commented on but the marks above cover it = 4 marks. Part (c): 1 mark for identifying conclusion as incorrect; 1 mark for quantitative justification using radius/temperature data = 2 marks.

Sample response. The HR diagram is one of the most powerful tools in stellar astrophysics because it organises observational data (luminosity and temperature) into meaningful groupings that reveal stellar physics. Its primary strength is that it instantly classifies stars into evolutionary groups — main sequence, giants, supergiants, white dwarfs — using only two observables. For example, Betelgeuse (T ≈ 3 500 K, L ≈ 100 000 L⊙) is unambiguously a red supergiant in the upper-right, and its location immediately implies an enormous radius from L = 4πR²σT&sup4; (~860 R⊙ by calculation). The Stefan-Boltzmann law is essential here: it transforms a star’s (L, T) position into a physical radius, providing a quantitative connection between observation and physical structure that cannot be obtained from a single datum alone. The turn-off method for star clusters is another major strength: in clusters such as the Pleiades (turn-off ≈ 6 M⊙, B-type), the mass of the most massive main-sequence star directly gives the cluster age via t ∝ M^−2.5 × 10 Gyr, without requiring knowledge of individual star distances. However, the HR diagram has significant limitations as a standalone tool. It reveals nothing about a star’s distance directly; spectroscopic parallax is required, and this technique introduces systematic errors through interstellar dust absorption and luminosity class misidentification. The diagram also does not distinguish evolutionary state unambiguously for all stars: a G-type star at L ≈ 5 L⊙ could be a slightly evolved main-sequence star or a subgiant without additional spectral analysis. Furthermore, the diagram describes only current evolutionary stage; it cannot directly reveal past history. In summary, the HR diagram is most powerful when combined with the Stefan-Boltzmann law (for radii), the mass-luminosity and lifetime relations (for cluster ages), and spectroscopic data (for distance and evolutionary classification). Used in isolation with only (L, T) data, it is a strong but incomplete tool.

Marking criteria (7 marks). 1 = identifies and explains at least two strengths of the HR diagram (classification into regions; turn-off age determination; Stefan-Boltzmann radius calculation; spectroscopic parallax). 1 = identifies and explains at least two limitations (no direct distance; ambiguous evolutionary stage at some positions; no past-history information; dust absorption affecting spectroscopic parallax). 1 = uses Stefan-Boltzmann law explicitly to explain how HR position yields stellar radius (with formula or clear reasoning). 1 = first named stellar example correctly placed on HR diagram and used to illustrate a strength or limitation (Betelgeuse, Sirius B, Pleiades turn-off, or similar). 1 = second named example. 1 = reaches an explicit evaluative judgement about the usefulness of the HR diagram (not just descriptive; must weigh strengths and limitations). 1 = uses precise physics terminology throughout (luminosity, spectral type, Stefan-Boltzmann, main sequence turn-off, spectroscopic parallax, red giant, white dwarf — at least four distinct technical terms used correctly).