Physics • Year 12 • Module 8 • Lesson 8
The Hertzsprung-Russell Diagram
Apply Stefan-Boltzmann, Wien’s law, and stellar data to classify stars, calculate radii, and interpret star cluster diagrams.
1. Classify stars from observational data
The table below lists six stars with measured luminosity and surface temperature. Complete the missing columns. 10 marks
| Star | L (L⊙) | T (K) | Spectral type (O/B/A/F/G/K/M) | HR region | Relative radius (compare to Sun) |
|---|---|---|---|---|---|
| Alpha Centauri A | 1.5 | 5 790 | |||
| Rigel | 120 000 | 12 100 | |||
| Betelgeuse | 100 000 | 3 500 | |||
| Sirius B | 0.0028 | 25 000 | |||
| Barnard’s Star | 0.0035 | 3 134 | |||
| Canopus | 10 700 | 7 350 |
Sun: L = 1 L⊙, T = 5 778 K. Spectral boundaries (approximate): O > 30 000 K; B 10 000–30 000 K; A 7 500–10 000 K; F 6 000–7 500 K; G 5 200–6 000 K; K 3 700–5 200 K; M < 3 700 K.
1.1 Complete the “Spectral type”, “HR region”, and “Relative radius” columns. For relative radius, use R ∝ √L / T² with the Sun as reference (R⊙ = 1). 6 marks (1 per row)
1.2 Sirius B has a much higher temperature than Betelgeuse, yet is far less luminous. Using L = 4πR²σT&sup4;, explain quantitatively why this is possible. Show your reasoning using the relative radius values from the table. 2 marks
1.3 Identify which star in the table is most similar to the Sun in HR position and justify your answer with reference to two stellar properties. 2 marks
2. Interpret a star cluster HR diagram
The diagram below shows the HR diagram of two star clusters, X (open symbols) and Y (filled symbols), plotted together. 8 marks
Figure 2. Schematic HR diagram for two star clusters X and Y. Illustrative data only.
2.1 Describe the location of the main sequence turn-off point for each cluster and use this to compare their ages. Which cluster is older? Justify your answer. 3 marks
2.2 Cluster Y shows several stars in the upper-right region of the HR diagram (the red giant region). Explain why these stars are present in the older cluster but absent from the younger cluster. 2 marks
2.3 A student claims that Cluster X must be closer to Earth because its stars appear brighter. Identify the flaw in this reasoning and explain the correct method for determining stellar distance using an HR diagram. 3 marks
3. Predict and justify — a mystery star
An astronomer observes a star with λmax = 966 nm (in the infrared) and apparent magnitude m = 0.5. The star’s parallax angle is too small to measure (distance > 500 pc). Given the Sun’s T = 5 778 K, λmax = 501 nm, and L = 1 L⊙: 6 marks
3.1 Calculate the star’s surface temperature using Wien’s displacement law (λmaxT = 2.898 × 10−3 m K). Show your working. 2 marks
3.2 Using only its temperature, classify the star’s spectral type and predict two possible HR diagram locations where a star of this temperature could be found. 2 marks
3.3 The star’s luminosity is later measured as L = 7 500 L⊙. Using L = 4πR²σT&sup4; (or R ∝ √L / T²), estimate its radius relative to the Sun. Identify the HR region more precisely. 2 marks
4. Compare stellar groups across five properties
Complete the table below. For each property, write a concise contrast across the four stellar groups. 10 marks (1 per cell)
| Property | Main sequence (low mass) | Main sequence (high mass) | Red giant | White dwarf |
|---|---|---|---|---|
| Surface temperature | ||||
| Luminosity | ||||
| Radius | ||||
| HR diagram position | ||||
| Evolutionary stage |
Q1.1 — Stellar classification table
Relative radius: R/R⊙ ≈ √(L/L⊙) / (T/T⊙)²
Alpha Centauri A: T = 5 790 K → G type; main sequence; R/R⊙ = √1.5 / (5790/5778)² ≈ 1.22 / 1.004 ≈ 1.22 R⊙. Rigel: T = 12 100 K → B type; main sequence (or supergiant given extreme luminosity); R/R⊙ = √120 000 / (12100/5778)² ≈ 346 / 4.38 ≈ 79 R⊙. Betelgeuse: T = 3 500 K → M type; red supergiant; R/R⊙ = √100 000 / (3500/5778)² ≈ 316 / 0.366 ≈ 863 R⊙. Sirius B: T = 25 000 K → O/B; white dwarf; R/R⊙ = √0.0028 / (25000/5778)² ≈ 0.053 / 18.7 ≈ 0.0028 R⊙ (about Earth-sized). Barnard’s Star: T = 3 134 K → M; main sequence (red dwarf); R ≈ 0.059 / 0.294 ≈ 0.20 R⊙. Canopus: T = 7 350 K → A/F; bright giant/supergiant; R ≈ √10700 / (7350/5778)² ≈ 103 / 1.62 ≈ 64 R⊙.
Q1.2 — Sirius B vs Betelgeuse (2 marks)
From L = 4πR²σT&sup4;: Sirius B has T ≈ 25 000 K but R ≈ 0.003 R⊙ (white dwarf). Betelgeuse has T ≈ 3 500 K but R ≈ 860 R⊙. The T&sup4; term for Sirius B (~3.9 × 1017) is (25 000/3 500)&sup4; ≈ 260 times larger than Betelgeuse’s, yet its R² is ~(0.003/860)² ≈ 1.2 × 10−11 times smaller. The enormous radius of Betelgeuse overwhelms its low temperature, making it far more luminous [1]. The radius (R²) term dominates the luminosity comparison [1].
Q1.3 — Most similar to the Sun (2 marks)
Alpha Centauri A is most similar to the Sun [1]. It has T = 5 790 K (close to T⊙ = 5 778 K, both G-type) and L = 1.5 L⊙ (very close to 1 L⊙), placing it in almost the same position on the main sequence [1].
Q2.1 — Cluster ages (3 marks)
Cluster X has its turn-off at high luminosity and high temperature (approximately B-type, L ∼ 10 000 L⊙), indicating that very massive, short-lived stars are still on the main sequence [1]. Cluster Y has its turn-off at much lower luminosity (approximately G-type, L ∼ 3 L⊙), meaning all stars more massive than ~1.1 M⊙ have already evolved off [1]. Cluster Y is significantly older because only long-lived, low-mass stars remain on its main sequence; massive stars have already exhausted their hydrogen. The age of a cluster ≈ main-sequence lifetime of the turn-off mass star (t ∝ M−2.5 × 10 Gyr) [1].
Q2.2 — Red giants in Cluster Y (2 marks)
Red giants are present in Cluster Y (older) because stars that were originally on the main sequence at moderate masses (~1–2 M⊙) have had sufficient time to exhaust their core hydrogen and evolve off the main sequence [1]. As they do, they expand dramatically and cool at the surface, moving to the upper-right of the HR diagram (red giant branch). In the younger Cluster X, these stars have not yet had enough time to exhaust their hydrogen and remain on the main sequence [1].
Q2.3 — Apparent brightness vs distance (3 marks)
Flaw: apparent brightness depends on both intrinsic luminosity AND distance; a more luminous but distant cluster could appear just as bright as a dim but nearby one [1]. The HR diagram method for distance is spectroscopic parallax: measure the star’s spectral type (from absorption lines) to determine its position on the HR diagram, hence its absolute magnitude (intrinsic luminosity) [1]. Comparing absolute magnitude to apparent magnitude gives the distance modulus: m − M = 5 log(d/10), from which distance d can be calculated [1].
Q3.1 — Wien’s law calculation (2 marks)
T = 2.898 × 10−3 / λmax = 2.898 × 10−3 / (966 × 10−9) [1] = 2.898 × 10−3 / 9.66 × 10−7 ≈ 3 000 K [1].
Q3.2 — Spectral type and possible HR locations (2 marks)
T ≈ 3 000 K → spectral type M [1]. A star with T = 3 000 K could be located: (1) on the lower main sequence as a low-mass red dwarf (dim, L < 0.01 L⊙), or (2) in the red giant/supergiant region (L ≫ 1 L⊙, very large radius) [1]. Temperature alone does not distinguish between the two; luminosity or radius is required.
Q3.3 — Radius estimate (2 marks)
R/R⊙ = √(L/L⊙) / (T/T⊙)² = √7 500 / (3 000/5 778)² = 86.6 / 0.269 ≈ 322 R⊙ [1]. This enormous radius (~300 solar radii) places the star firmly in the red supergiant region of the HR diagram (upper-right), consistent with a high-mass star that has evolved far off the main sequence [1].
Q4 — Compare and contrast table
Surface temperature: Low-mass MS: ~3 000–5 000 K (K–M type) | High-mass MS: 10 000–50 000 K (O–B) | Red giant: 3 000–5 000 K (cool) | White dwarf: 10 000–100 000 K (hot). Luminosity: Low-mass MS: <0.1 L⊙ | High-mass MS: 103–106 L⊙ | Red giant: 10–10 000 L⊙ | White dwarf: <0.01 L⊙. Radius: Low-mass MS: <0.5 R⊙ | High-mass MS: 5–20 R⊙ | Red giant: 10–1 000 R⊙ | White dwarf: ~0.01 R⊙ (Earth-sized). HR position: Low-mass MS: lower-right main sequence | High-mass MS: upper-left main sequence | Red giant: upper-right | White dwarf: lower-left. Evolutionary stage: Low-mass MS: hydrogen-burning in core; very long lifetime (~100 Gyr) | High-mass MS: hydrogen-burning; very short lifetime (~10 Myr) | Red giant: core H exhausted; shell burning; expanded envelope | White dwarf: inert remnant of low/intermediate mass star; no fusion.