Spectroscopy and Stellar Classification

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

Spectral type and evidence: The star is a K-type star [1 — correct type]. Evidence: (1) Dominant ionised calcium H and K lines are the hallmark of K-type stars in the 3 700–5 200 K range, where Ca⁺ ions are preferentially present [evidence 1]. (2) Hydrogen Balmer lines are present but weak — at K-type temperatures, hydrogen is mostly neutral and the \(n=2\) level is less populated than in A-type stars, reducing absorption [evidence 2]. (3) Neutral iron and magnesium lines indicate temperatures cool enough for neutral metal atoms to survive without being ionised [evidence 3]. (4) Yellow-orange colour is consistent with a surface temperature of ~4 000–5 000 K, confirming K-type classification [supporting evidence — accept in lieu of one of above]. Award 1 mark per clearly explained piece of evidence (3 required).

Full designation and implications: The star is KIII — a K-type giant [1]. Luminosity class III indicates the star has left the main sequence and is now a red giant: it has exhausted hydrogen in its core and has expanded to a much larger radius (tens to hundreds of times the Sun’s radius), with a correspondingly lower surface gravity and higher luminosity than a main-sequence K dwarf [1].

Radial velocity calculation: \(\Delta\lambda = 394.7 - 393.3 = +1.4\) nm (positive = redshift). \(v_r = c \cdot \Delta\lambda/\lambda_\text{rest} = (3.0 \times 10^5 \text{ km s}^{-1}) \times (1.4/393.3) \approx +1\thinsp;068\) km s⁻¹ ≈ 1 070 km s⁻¹ [1 — correct calculation with working shown].

Direction and reasoning: The star is receding from Earth [1]. \(\Delta\lambda\) is positive (\(\lambda_\text{obs} > \lambda_\text{rest}\)), indicating a redshift. By the Doppler formula \(v_r = c \cdot \Delta\lambda/\lambda_\text{rest}\), a positive \(v_r\) corresponds to recession.

Limitation and improvement: Using a single spectral line risks error from instrumental artefacts, line blending with nearby features, or misidentification of the line [1]. Reliability can be improved by measuring the shifts of multiple independent spectral lines (e.g. Ca II K, Hα, neutral iron) and computing a weighted mean radial velocity, as random measurement errors are reduced [1].

Marking criteria summary (8 marks): 1 = correctly identifies K-type; 1 = three pieces of spectroscopic evidence linked to physical cause; 1 = correct full designation KIII; 1 = correct implication of luminosity class III (giant/post-main-sequence); 1 = correct radial velocity calculation with working; 1 = correctly identifies receding and explains using sign of \(\Delta\lambda\); 1 = valid limitation of single-line method; 1 = valid improvement (multiple lines / mean velocity).

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

Hypothesis: If the “star” is a spectroscopic binary, then repeated high-resolution spectra taken over months will show periodic splitting of spectral lines into two components and oscillating radial velocities for each component. Independent variable: time (observation epoch / orbital phase). Dependent variable: measured wavelength position of spectral lines and derived radial velocity. [1 — testable hypothesis with IV and DV]

Procedure: (1) Take a high-resolution spectrum of the target star at observation epoch 1 (night 1). Identify key spectral lines (e.g. Ca II K at 393.3 nm, Hα at 656.3 nm). Record the observed wavelengths and calculate any initial radial velocity. (2) Repeat the observation at regular intervals (e.g. every 2 weeks) for 6 months to sample a full potential orbital period. Maintain identical instrument settings and wavelength calibration using an arc-lamp reference spectrum. (3) For each epoch, measure the centroid wavelength of each target line. If the system is a binary, at some epochs the single line will split into two components (one from each star); at other epochs (when both stars move perpendicular to the line of sight) the two components will overlap. (4) Plot the measured radial velocities of each component versus time. A spectroscopic binary will show two anti-phase sinusoidal radial velocity curves; a single star will show a constant radial velocity (or a single sinusoid if the star itself has a companion planet). [1 — four clear steps with measurement strategy and binary diagnostic]

Spectral feature unique to binary: When the two stars are separating along the line of sight, their individual spectral lines split into two components at slightly different wavelengths (one blueshifted, one redshifted). This line doubling does not occur in single-star spectra and is the definitive spectroscopic signature of a double-lined spectroscopic binary (SB2) [1].

Limitations: (1) If the orbital plane is nearly perpendicular to the line of sight (face-on orbit), the radial velocity amplitude will be very small and the line splitting may be below the spectrograph’s resolution [1]. (2) If the two stars have very different luminosities, the dimmer star’s contribution to the spectrum may be too faint to detect, making it appear as a single-lined binary (SB1) rather than SB2, and the full orbital solution cannot be recovered [1].

Improvement: Use a spectrograph with higher resolving power (larger telescope or echelle spectrograph) to detect smaller velocity separations, and schedule observations more frequently near predicted phases of maximum separation to capture the line splitting clearly [1].

Falsification: If all spectra over 6 months show a single, unshifted spectral line (constant radial velocity, no splitting), the hypothesis that the object is a spectroscopic binary is falsified and it is consistent with a single star [1].

Marking criteria summary (7 marks): 1 = testable hypothesis with IV and DV; 1 = four steps with measurement of wavelength shifts and binary-diagnostic strategy; 1 = correctly identifies line-doubling / anti-phase velocity curves as the binary signature; 1 = first valid limitation; 1 = second valid limitation; 1 = specific improvement; 1 = correct falsification criterion (constant/unshifted lines).