Physics • Year 12 • Module 8 • Lesson 8

The Hertzsprung-Russell Diagram

Build HSC Band 5–6 extended-response technique on applying Stefan-Boltzmann, analysing cluster data, and evaluating stellar classification methods.

Master · Extended Response

1. Data + scenario: The Pleiades and the age of the Universe (Band 5–6)

8 marks Band 5–6

Scenario. The Pleiades is a young open star cluster visible from New South Wales during summer nights. When astronomers plot the Pleiades on the HR diagram, the main sequence extends to high luminosities (O and B-type stars still present). The most massive star still on the Pleiades main sequence has a mass of approximately 6 M⊙. Using the main sequence lifetime formula t = t⊙ × (M / M⊙)^−2.5, where t⊙ = 10 Gyr:

Star property	Pleiades turn-off star (6 M⊙)	Solar analogue (1 M⊙)
Mass (M⊙)	6	1
Luminosity L ∝ M^3.5 (L⊙)	(calculate)	1
Main sequence lifetime t (Gyr)	(calculate)	10
Surface temperature (K, approximate)	~18 000	~5 778
Spectral type	(identify)	G
HR region	upper main sequence	main sequence

Q1. Analyse and evaluate the data above to determine the age of the Pleiades and assess what this implies about stellar evolution. In your response you must:

Calculate the luminosity of the 6 M⊙ star using L ∝ M^3.5, showing your working.
Calculate its main sequence lifetime t using t = 10 × (6)^−2.5 Gyr. Round to appropriate significant figures.
Explain what the turn-off point tells us about the age of the Pleiades.
Identify the spectral type of the 6 M⊙ star and describe where it sits on the HR diagram relative to the Sun.
Predict what the Pleiades HR diagram will look like in 10 Gyr and justify your answer using the mass-luminosity and lifetime relations.

Stuck? Plan: L = 1 × 6^3.5 → t = 10 × 6^−2.5 Gyr → turn-off age → spectral type from T ≈ 18 000 K → predict future: turn-off moves to lower mass stars, main sequence depletes from the top down.

2. Experimental design — testing spectroscopic parallax (Band 5–6)

7 marks Band 5–6

Research scenario. A student claims that the spectroscopic parallax method is unreliable because “you’re just guessing the star’s absolute magnitude from its spectral class.” Design an analysis procedure using real stellar data to evaluate the accuracy of spectroscopic parallax for main-sequence stars within 500 parsecs.

Available resources: access to a database of stars with known trigonometric parallax (independent distance), measured apparent magnitudes, and classified spectra (OBAFGKM type + luminosity class). You also have the HR diagram main-sequence calibration (spectral type → absolute magnitude).

Q2. Design the analysis procedure and present it in the format below.

State your hypothesis: a testable prediction about spectroscopic parallax accuracy.
Identify the independent variable, dependent variable, and at least two controlled variables.
Describe the analysis procedure in at least four numbered steps, including how you would calculate distances from both methods and compare them.
Explain what result would support the student’s claim that the method is unreliable.
State two sources of systematic error in spectroscopic parallax and suggest one way to improve accuracy.

Stuck? Consider: hypothesis (spectroscopic parallax distances agree within ±20% for main-sequence stars); IV = method used (spectroscopic vs trigonometric); compare using distance modulus; errors: interstellar absorption, misclassification of luminosity class, metallicity differences; improvement: correct for interstellar reddening using colour excess E(B−V).

Answers — Do not peek before attempting

Q1 — Sample Band 6 response (8 marks), annotated

Luminosity calculation: L = L⊙ × (M/M⊙)^3.5 = 1 × 6^3.5 = 6³ × 6^0.5 = 216 × 2.449 ≈ 529 L⊙ [1 mark: correct calculation with working shown].

Lifetime calculation: t = 10 × (6)^−2.5 = 10 / 6^2.5 = 10 / (6² × 6^0.5) = 10 / (36 × 2.449) = 10 / 88.2 ≈ 0.113 Gyr ≈ 113 Myr [1 mark: correct calculation; accept 100–120 Myr].

Turn-off point and cluster age: The Pleiades’ main sequence turn-off at 6 M⊙ indicates that stars more massive than 6 M⊙ have already exhausted their core hydrogen and evolved off the main sequence [1 mark]. Therefore the cluster’s age equals the main sequence lifetime of the turn-off mass star: approximately 113 Myr. All stars above this mass have completed (or exceeded) their main sequence lifetime and have moved to the giant/supergiant branch or beyond [1 mark].

Spectral type and HR position: T ≈ 18 000 K falls in the B-type range (10 000–30 000 K). The 6 M⊙ star is a B-type star on the upper main sequence, significantly above and to the left of the Sun on the HR diagram: L ≈ 529 L⊙ vs L⊙ = 1, and T ≈ 18 000 K vs 5 778 K [1 mark for spectral type + HR comparison to Sun].

Future prediction in 10 Gyr: In 10 Gyr, all main sequence stars with t < 10 Gyr (M > 1 M⊙, since t ∝ M^−2.5) will have evolved off the main sequence. The turn-off will have moved down to ~1 M⊙ (G-type), near the Sun’s position. The upper main sequence will be completely absent; instead the HR diagram will show a prominent red giant branch and potentially a white dwarf sequence, while only K and M dwarfs remain on the lower main sequence [1 mark for prediction; 1 mark for linking to mass-lifetime relation; total: 2 marks for prediction part].

Marking criteria summary (8 marks): 1 = correct L calculation (529 L⊙, showing 6^3.5 working); 1 = correct t calculation (~113 Myr); 1 = explains turn-off as indicating cluster age (stars >6 M⊙ already evolved off); 1 = states this means the Pleiades is ~100–120 Myr old; 1 = correct spectral type B and HR position relative to Sun (hotter, more luminous, upper-left); 1 = predicts turn-off moves to lower mass in 10 Gyr; 1 = justifies prediction using t ∝ M^−2.5; 1 = describes qualitative appearance of future HR diagram (no upper MS; red giant branch; white dwarfs).

Q2 — Sample Band 6 response (7 marks), annotated

Hypothesis: If spectroscopic parallax is reliable for main-sequence stars within 500 pc, then distances calculated using spectroscopic parallax (from spectral type and apparent magnitude) will agree with independently-measured trigonometric parallax distances to within ±20%. Independent variable: distance measurement method (spectroscopic vs trigonometric parallax). Dependent variable: calculated distance (parsecs). Controlled variables: (1) same set of stars analysed by both methods; (2) stars restricted to luminosity class V (main sequence only) to ensure HR diagram calibration applies; (3) apparent magnitudes measured in the same photometric band (V-band) for both methods. [1 mark: hypothesis with IV, DV, and two controlled variables]

Procedure: (1) Select a sample of at least 50 main-sequence (luminosity class V) stars within 500 pc that have both published trigonometric parallax distances and classified spectra. (2) For each star, use its spectral type to look up its calibrated absolute magnitude M from the HR diagram main-sequence calibration table. Combine with the measured apparent magnitude m to calculate the spectroscopic parallax distance: d = 10^(m−M+5)/5 parsecs. (3) For each star, calculate the trigonometric parallax distance independently: d = 1/p, where p is the parallax angle in arcseconds (from, e.g., the Hipparcos or Gaia catalogue). (4) Compare the two sets of distances by calculating the percentage difference [(d_spec − d_trig)/d_trig] × 100 for each star and plotting d_spec vs d_trig on a scatter graph. A 1:1 line indicates perfect agreement. [1 mark: four steps including both calculation methods and comparison]

Falsification: The student’s claim would be supported if the scatter plot showed large, systematic deviations (>20%) between the two distance estimates across many stars, with no correlation between d_spec and d_trig. Specifically, if the spectroscopic method consistently over- or under-estimates distance (systematic bias) by >20% for most stars, this would indicate the method is unreliable. [1 mark]

Systematic errors and improvement: (1) Interstellar dust absorption: dust between the star and Earth reddens and dims starlight, making stars appear further away than they are in the spectroscopic method (apparent magnitude is too faint) [1 mark]. (2) Luminosity class misclassification: if a subgiant (class IV) is misidentified as a main-sequence star (class V), its absolute magnitude will be underestimated, biasing the distance calculation [1 mark]. Improvement: apply an interstellar reddening correction using the colour excess E(B−V) (measured by comparing the star’s observed B−V colour to its expected intrinsic colour) to remove the dust absorption effect before calculating the distance modulus [1 mark].

Marking criteria summary (7 marks): 1 = testable hypothesis naming IV, DV, and two controlled variables; 1 = four procedure steps including distance calculations by both methods; 1 = states what result would falsify (support student’s claim); 1 = first systematic error (interstellar absorption / dust reddening); 1 = second systematic error (luminosity class misclassification / metallicity / stellar variability); 1 = specific improvement method (reddening correction / restrict to unreddened nearby stars); 1 = uses precise technical terminology throughout (distance modulus, apparent/absolute magnitude, luminosity class, parallax angle).