Physics • Year 12 • Module 7 • Lesson 9

Spectroscopy and Astronomical Applications

Build HSC Band 5–6 extended-response technique: synthesise Kirchhoff’s laws, quantum mechanics, Doppler shift calculations and scientific method in complex astronomical contexts.

Master · Extended Response

1. Multi-step calculation — exoplanet detection via the stellar wobble (Band 5–6)

8 marks   Band 5–6

Scenario. A sun-like star is observed to show periodic Doppler shifts in its sodium D-line spectrum (λ0 = 589.0 nm). Analysis reveals that the observed wavelength oscillates between 589.0588 nm (maximum redshift) and 589.0000 nm − 0.0588 nm (maximum blueshift) with a period of 365 days, suggesting an orbiting exoplanet is inducing a stellar wobble. Use c = 3.00 × 108 m s−1.

Q1. Analyse the spectroscopic data to characterise the stellar wobble and discuss what it implies about the exoplanet. In your response you must:

  • Calculate the maximum radial velocity of the star from the maximum redshift measurement. Show full working including units at each step.
  • Using vstar from part (a) and the orbital period T = 365 days, estimate the orbital circumference of the star around the system’s centre of mass and hence the orbital radius of the star. (You may assume a circular orbit.)
  • Explain qualitatively what a larger stellar wobble velocity would imply about the mass of the orbiting exoplanet, with reference to Newton’s third law and conservation of momentum.
  • Discuss one reason why the radial velocity method can only provide a minimum estimate of the exoplanet’s mass.
  • State one technological requirement for spectrographs used in exoplanet searches, given the tiny wavelength shifts involved.
Stuck? Step-by-step: (a) Δλ = 589.0588 − 589.0 = 0.0588 nm; v = c × Δλ/λ0. (b) distance = v × T, then r = distance/(2π). (c) By Newton 3 the exoplanet pulls the star with an equal and opposite force; larger wobble = larger planet mass. (d) If the orbit is inclined to the line of sight, only the radial component of velocity is measured — the true velocity (and hence mass) is larger. (e) High-resolution, high-stability spectrographs (e.g. HARPS) with wavelength stability < 1 m s−1.

2. Experimental design — identifying elements in an unknown star spectrum (Band 5–6)

7 marks   Band 5–6

Research question. An amateur astronomer obtains a digital spectrum of a nearby star and wants to determine: (1) which elements are present in the star’s atmosphere, and (2) whether the star is moving toward or away from Earth. They have access to: a telescope with a diffraction grating spectrograph, a CCD detector, a laptop with spectrum-analysis software, and a database of laboratory reference spectra for common elements (H, He, Na, Ca, Fe, Mg).

Q2. Design the investigation and present it in the format below.

  • State a clear hypothesis for what type of spectrum will be observed and why, using Kirchhoff’s laws.
  • Describe the procedure in at least four numbered steps, including how to calibrate the spectrograph and how to identify elements.
  • Explain how the astronomer would determine whether the star is moving toward or away from Earth.
  • State two limitations of this investigation and one improvement to increase reliability.
  • Predict what result would indicate the star is a member of a binary system.
Stuck? Hypothesis: a cool stellar atmosphere in front of a hot interior (continuous source) will produce an absorption spectrum (Kirchhoff’s Third Law). Calibrate by obtaining the spectrum of a reference lamp of known wavelengths. Identify by matching dark-line positions to database. Doppler: compare line positions to rest wavelengths; shift tells direction and speed. Binary: spectral line positions shift periodically over months to years.
Answers — Do not peek before attempting

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

(a) Maximum radial velocity of the star:

Δλ = 589.0588 − 589.0000 = 0.0588 nm = 0.0588 × 10−9 m [1].

v = c × Δλ/λ0 = 3.00×108 × (0.0588×10−9) / (589.0×10−9) = 3.00×108 × 9.98×10−5 = 29.9 m s−1 ≈ 30 m s−1 [1]. This is consistent with velocities induced by a Jupiter-mass planet. The positive shift at maximum means the star is moving away at that instant.

(b) Orbital radius of the star about the centre of mass:

T = 365 days × 24 × 3600 = 3.15×107 s [1].

Orbital circumference C = v × T = 30 × 3.15×107 = 9.45×108 m [1].

Orbital radius r = C / (2π) = 9.45×108 / (2π) = 1.50×108 m [1]. This is 0.001 AU — the star moves only ~1.5×105 km in a circle; the planet is much further from the star (at ~1 AU for a 365-day period).

(c) Larger wobble → more massive exoplanet:

By Newton’s Third Law, the star and planet exert equal and opposite gravitational forces on each other. For momentum conservation in the two-body system, mstarvstar = mplanetvplanet. A larger stellar wobble velocity means the planet must have a larger mass to produce the required momentum (since the star’s mass is fixed). [1] More massive planets exert a stronger gravitational pull, inducing larger-amplitude oscillations in the star’s motion.

(d) Why radial velocity gives only minimum mass:

The radial velocity method measures only the component of the star’s velocity along the line of sight to Earth. If the orbit is inclined at angle i to the plane of the sky, the measured velocity is vobs = vtrue sin i. For i < 90°, vobs < vtrue, so the calculated planet mass mplanet sin i is less than the true mass. The inclination angle i is generally unknown, so only mplanet sin i (a lower bound) is determined [1].

(e) Technological requirement:

Spectrographs must achieve wavelength stability and precision better than approximately 1 m s−1 in velocity terms (corresponding to sub-picometer wavelength precision). Instruments such as HARPS (High Accuracy Radial velocity Planet Searcher) use temperature-stabilised vacuum-enclosed gratings, simultaneous calibration with a reference lamp or laser frequency comb, and fibre-fed spectrographs to reach this precision [1].

Marking criteria summary (8 marks): 1 = correct Δλ with units; 1 = correct v calculation (≈30 m s−1); 1 = correct T in seconds; 1 = correct orbital circumference; 1 = correct orbital radius with correct formula; 1 = correct Newton 3 / momentum reasoning linking wobble amplitude to planet mass; 1 = correct orbital inclination explanation for minimum mass; 1 = valid technological requirement with justification.

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

Hypothesis: The star has a hot, dense interior that produces a continuous spectrum, with a cooler outer atmospheric layer that produces dark absorption lines at wavelengths characteristic of elements present (Kirchhoff’s Third Law). The spectrum observed will be an absorption spectrum. If the star is moving relative to Earth, the absorption line positions will be shifted from their laboratory rest wavelengths. [1 — hypothesis uses Kirchhoff’s Third Law correctly]

Procedure: (1) Point the telescope at the star, ensuring sufficient exposure time for the CCD to record a high signal-to-noise spectrum. First obtain a wavelength calibration spectrum using a known reference lamp (e.g. neon or thorium-argon lamp) immediately before and after the stellar observation to correct for any spectrograph drift [1 — calibration step]. (2) Record the CCD image of the stellar spectrum and apply the wavelength calibration to convert pixel positions to wavelengths in nanometres. Identify the positions of dark absorption lines in the stellar spectrum. (3) Compare the observed dark-line positions to the reference database of emission line wavelengths for H, He, Na, Ca, Fe, and Mg. A match (within measurement uncertainty) identifies that element as present in the star’s atmosphere [1 — identification method]. (4) To determine stellar motion, for each identified line calculate Δλ = λobs − λ0,ref and compute v = c × Δλ/λ0. A positive v indicates recession (redshift); a negative v indicates approach (blueshift). Average velocities from multiple lines to reduce measurement uncertainty [1 — Doppler interpretation]. [1 — minimum 4 steps each clearly numbered and executed]

Limitations: (1) Atmospheric dispersion and scintillation (twinkling) in Earth’s atmosphere can broaden and blur spectral lines, reducing the precision of wavelength measurements [1]. (2) A single observation cannot distinguish between a constant radial velocity and a velocity that is slowly changing; a single epoch does not reveal orbital motion [1].

Improvement: Repeat the observation on multiple nights over several months to detect any periodic variation in the line positions (indicating a binary partner) and to average out atmospheric effects. Cross-correlating with a mask (cross-correlation radial velocity technique) further improves precision [1].

Prediction for binary system: If the star is a binary, the wavelength positions of all absorption lines will shift periodically between redshifted and blueshifted positions over a timescale equal to the orbital period. The amplitude of the shift gives the radial velocity amplitude, and the period of repetition gives the orbital period.

Marking criteria summary (7 marks): 1 = correct hypothesis invoking Kirchhoff’s Third Law; 1 = calibration step included (reference lamp); 1 = element identification by matching line positions to database; 1 = Doppler shift calculated and interpreted for direction of motion; 1 = four clearly numbered steps; 1 = two valid limitations; 1 = one specific improvement to reliability.

HSCScience • Physics • Year 12 • Module 7 • Lesson 9 • Worksheet 3 (Master)