HSC Exam Practice

Sample response. An absorption spectrum consists of a continuous rainbow background with dark lines at specific wavelengths; it is produced when light from a hot, dense source passes through a cooler, low-density gas that absorbs photons at its electron transition wavelengths. An emission spectrum consists of bright coloured lines on a dark background; it is produced by a hot, low-density gas in which excited atoms spontaneously emit photons at specific wavelengths as electrons drop to lower energy levels.

Marking notes. 1 mark for correct definition of absorption spectrum (continuous background + dark lines); 1 mark for correct physical conditions (hot dense source behind cool gas); 1 mark for correct definition of emission spectrum (bright lines on dark background); 1 mark for correct physical conditions (hot low-density gas, excited atoms).

Sample response. (a) M-type: molecular TiO bands appear only in very cool stars (below ~3 700 K) because at higher temperatures TiO molecules dissociate. (b) A-type: strong Balmer lines indicate the surface temperature (~7 500–10 000 K) that maximises the population of the hydrogen \(n=2\) level for absorption. (c) O-type: ionised helium (He II) requires temperatures above ~30 000 K to strip an electron from neutral helium.

Marking notes. 1 mark per correctly identified spectral type with correct justification referencing the dominant spectral feature and its temperature implication.

Sample response. Kirchhoff’s second law states that an absorption spectrum is produced when a continuous-spectrum source is viewed through a cooler gas. In a star, the hot, dense photosphere acts as the continuous source (Kirchhoff law 1), and the cooler outer atmosphere of the star lies in front of this source. The atmospheric gas absorbs photons at wavelengths matching its electron transition energies, removing them from the outgoing beam and creating dark absorption lines in the observed spectrum. There is no geometry in an isolated main-sequence star that would produce a pure emission spectrum as seen from Earth.

Marking notes. 1 mark for identifying the photosphere as the continuous source; 1 mark for identifying the cooler outer atmosphere as the absorbing gas; 1 mark for linking the dark lines specifically to photon absorption at electron transition wavelengths (correct atomic mechanism).

Sample response. A blueshift is a shift of spectral lines to shorter wavelengths (\(\lambda_\text{obs} < \lambda_\text{rest}\), \(\Delta\lambda < 0\)); it occurs when the source is moving toward the observer, compressing the wavefronts. A redshift is a shift to longer wavelengths (\(\lambda_\text{obs} > \lambda_\text{rest}\), \(\Delta\lambda > 0\)); it occurs when the source is moving away from the observer, stretching the wavefronts. Both are described by the Doppler formula: \(\dfrac{\Delta\lambda}{\lambda_\text{rest}} = \dfrac{v_r}{c}\), where a negative \(v_r\) (by convention) indicates approach (blueshift) and a positive \(v_r\) indicates recession (redshift).

Marking notes. 1 mark for correct definition of blueshift (\(\lambda_\text{obs}\) shorter, source approaching); 1 mark for correct definition of redshift (\(\lambda_\text{obs}\) longer, source receding); 1 mark for correctly stating the Doppler formula with all symbols defined; 1 mark for correctly linking the sign of \(\Delta\lambda\) (or \(v_r\)) to the direction of motion.

Sample response. The statement is only partially correct because spectral type primarily depends on surface temperature, not just composition [1]. Temperature determines which atoms are ionised and which electron energy levels are populated; different ions dominate at different temperatures even when composition is the same [1]. For example, both an O-type and a G-type star contain hydrogen, but the O-type star shows no hydrogen Balmer lines because all hydrogen is ionised at >30 000 K, while the G-type Sun shows calcium lines because at ~5 800 K calcium ions are preferentially stable [1 — named example]. Composition becomes apparent only once temperature is accounted for.

Marking notes. 1 mark for correctly identifying surface temperature as the primary determinant; 1 mark for explaining the mechanism (ionisation state / energy level population depends on temperature); 1 mark for a named or specific example contrasting two spectral types to demonstrate why temperature, not composition alone, drives classification.

Sample response. An orbiting planet gravitationally pulls its host star into a small circular orbit around the common centre of mass. As the star moves toward Earth in this orbit, its spectral lines are blueshifted; as it moves away, they are redshifted. The astronomer measures the Doppler shift of the star’s spectral lines over time using \(\Delta\lambda/\lambda_\text{rest} = v_r/c\) [1]. If an exoplanet is present, the radial velocity will oscillate sinusoidally with a period equal to the planet’s orbital period [1]. The amplitude of the oscillation is related to the planet’s mass — a more massive planet in a closer orbit produces a larger velocity amplitude, allowing the planet’s mass and orbital radius to be estimated [1].

Marking notes. 1 mark for correctly describing what is measured (periodic Doppler shifts in spectral line wavelengths); 1 mark for explaining the sinusoidal radial velocity pattern and its period representing the orbital period; 1 mark for explaining how amplitude relates to planet mass or orbit.

Sample response (a). Hα: \(z = (682.6 - 656.3)/656.3 = 26.3/656.3 = 0.0401\). Hβ: \(z = (505.6 - 486.1)/486.1 = 19.5/486.1 = 0.0401\). Ca II K: \(z = (409.2 - 393.3)/393.3 = 15.9/393.3 = 0.0404\). The three values are consistent (all \(\approx 0.040\)), confirming that the entire spectrum has been uniformly shifted by the same factor and that the identification of each line is correct [1 mark per z-calculation × 3; 1 mark for consistency comment].

Sample response (b). Mean \(z \approx 0.0402\). \(v_r = zc = 0.0402 \times 3.0 \times 10^5 \approx 12\thinsp;060\) km s⁻¹ ≈ 12 100 km s⁻¹ [1 mark]. Since \(z \approx 0.04 < 0.1\), the non-relativistic approximation is valid here — the relativistic correction would be less than a few per cent [1 mark]. Award a third mark for showing the assessment of validity clearly referenced to the \(z < 0.1\) criterion [1 mark].

Sample response (c). The galaxy’s redshift is attributed to cosmological expansion rather than peculiar motion because virtually every galaxy beyond the Local Group shows a redshift proportional to its distance (Hubble’s law: \(v = H_0 d\)), consistent with space itself expanding uniformly in all directions [1]. If galaxies were simply moving through static space, we would expect roughly equal numbers of blueshifted and redshifted galaxies; instead, the systematic redshift of all distant galaxies is best explained by the metric expansion of the universe carrying galaxies apart [1].

Marking criteria summary (9 marks): Part (a): 1 mark per correct z-value (3); 1 mark for noting consistency and what it confirms. Part (b): 1 mark for correct mean z and multiplication by c; 1 mark for correct recession velocity; 1 mark for explicit validity assessment (\(z < 0.1\)). Part (c): 1 mark for referencing Hubble’s law or the systematic nature of the redshift; 1 mark for explaining why metric expansion is preferred over peculiar motion.

Sample response. Spectroscopy is arguably the single most powerful investigative tool in astrophysics, transforming starlight into quantitative physical information. A stellar spectrum encodes at minimum five properties: (1) surface temperature — inferred from the dominant absorption lines and Wien’s displacement law (e.g. Sirius, an A-type star at ~9 700 K, shows the strongest hydrogen Balmer lines because this temperature maximises the \(n=2\) hydrogen population); (2) chemical composition — every element has a unique set of spectral lines that serve as a fingerprint (Fraunhofer lines in the solar spectrum identified sodium, calcium, iron, and hydrogen in the Sun long before spacecraft could sample the solar wind); (3) radial velocity — via the Doppler formula \(\Delta\lambda/\lambda_\text{rest} = v_r/c\), allowing stellar motions, galactic rotation curves, and recession velocities of distant galaxies to be measured; (4) stellar classification and evolutionary stage — the OBAFGKM system, combined with the luminosity class (I–V), enables stars to be placed on the H-R diagram, revealing their evolutionary history; (5) magnetic field strength — via Zeeman line splitting, and rotation rate — via rotationally broadened line profiles. The Doppler radial-velocity method has particular strengths: it requires no spatial resolution (it works even on unresolved point sources), it can detect motions as small as 1 m s⁻¹ with modern instrumentation, and the first exoplanet around a sun-like star (51 Pegasi b, 1995) was discovered precisely via the 56 m s⁻¹ periodic wobble it induced in its host star. However, the method has important limitations: it detects only the component of velocity along the line of sight, so edge-on orbits are fully detected while face-on orbits show no radial velocity signal. It also cannot uniquely determine a planet’s mass without knowing the orbital inclination. Stellar intrinsic radial-velocity noise (jitter from convection and stellar activity) can mimic or obscure planetary signals. Spectroscopic classification has contributed enormously to stellar evolution theory: the OBAFGKM sequence is ordered by temperature, and when plotted against luminosity it traces the main sequence, giant branch, and supergiant region of the H-R diagram. The spectral types of Betelgeuse (M supergiant, red, cool) and Mintaka (O main-sequence, blue, hot) anchor opposite ends of the sequence and illustrate how stellar mass determines both temperature and evolutionary timescale — massive O stars exhaust their fuel in millions of years while low-mass M dwarfs burn for trillions. In summary, spectroscopy is uniquely powerful because it extracts a rich dataset from a single beam of light, with the Doppler and classification tools being complementary: classification tells us what stage the star has reached, while Doppler measurements reveal its dynamics. Its limitations (line-of-sight restriction, stellar jitter) can be mitigated by multi-line averaging, complementary photometric monitoring, and space-based observation.

Marking criteria (9 marks). 1 = correctly states and explains at least three distinct types of physical information extractable from a spectrum (temp, composition, velocity, class, magnetic field — any three). 1 = explains the mechanism by which spectral lines encode one of these properties at a physical/atomic level (e.g. electron transitions, temperature-dependent ionisation). 1 = names and correctly describes Doppler radial-velocity method (formula, what is measured). 1 = identifies at least one strength of the method (no spatial resolution needed / high sensitivity / exoplanet success). 1 = identifies at least one limitation (line-of-sight only / inclination degeneracy / stellar jitter). 1 = links spectroscopic classification (OBAFGKM) to stellar evolution via the H-R diagram concept. 1 = uses at least two named specific examples correctly (acceptable: Sirius/A-type, Sun/G2V, Betelgeuse/M supergiant, Mintaka/O-type, 51 Peg b exoplanet detection, Fraunhofer lines). 1 = reaches an explicit evaluative judgement about the overall power of spectroscopy relative to its limitations. 1 = sustained use of precise physics terminology throughout (Doppler, redshift, blueshift, radial velocity, OBAFGKM, luminosity class, absorption spectrum, Fraunhofer, H-R diagram — minimum five distinct terms).