Physics • Year 12 • Module 7 • Lesson 9
Spectroscopy and Astronomical Applications
Apply Kirchhoff’s laws, the Doppler shift formula, and spectral line analysis to real astronomical data and exam-style scenarios.
1. Interpret Doppler shift data — five stellar observations
The table below shows observed wavelengths of the Hα line (λ0 = 656.3 nm) and the Ca II K line (λ0 = 393.3 nm) for five different stars. 10 marks
| Star | Hα observed (nm) | Ca II K observed (nm) | Δλ for Hα (nm) | Radial velocity v (km s−1) | Motion |
|---|---|---|---|---|---|
| A | 656.3 | 393.3 | |||
| B | 657.9 | 394.9 | |||
| C | 654.7 | 391.7 | |||
| D | 659.2 | 396.2 | |||
| E | 652.1 | 389.1 |
Use c = 3.00 × 105 km s−1. Formula: v = c × Δλ/λ0.
1.1 Complete the “Δλ”, “Radial velocity” and “Motion” columns. Show calculations for stars B and C on the lines below. 6 marks (1 per row)
1.2 Notice that the Δλ values for Hα and Ca II K for the same star are different in nanometres, but confirm that they give the same radial velocity. Explain why both lines give the same velocity. 2 marks
1.3 Star A shows no shift. What does this mean for the component of its motion along the line of sight? Could Star A still be moving through space? Explain. 2 marks
2. Interpret a graph — binary star radial velocity curve
The graph below shows the observed radial velocity of a star over time. Positive velocity = receding; negative = approaching. 7 marks
Figure 2. Radial velocity of star α Ori-B measured via Doppler shifts of the Ca II K line. Illustrative data.
2.1 Read the period of the velocity oscillation from the graph and state what this period represents physically. 2 marks
2.2 At t = 2 years the star has its maximum recession velocity of +45 km s−1. Calculate the observed wavelength of the Ca II K line (λ0 = 393.3 nm) at this moment. 2 marks
2.3 Explain, using the graph, what is occurring in the binary system at t = 0 years and t = 4 years when the radial velocity is zero. 2 marks
2.4 A student claims this pattern could also be explained by a pulsating star alternately expanding and contracting. Describe one additional observation that would distinguish between a pulsating star and a binary system. 1 mark
3. Compare emission and absorption spectra across five features
Complete the two-column comparison table. 10 marks (1 per cell)
| Feature | Emission spectrum | Absorption spectrum |
|---|---|---|
| Source condition | ||
| Appearance | ||
| Wavelengths of lines | ||
| Energy process | ||
| Astronomical example |
4. Predict and justify — a galaxy cluster observation
The Virgo Cluster is a group of about 1,300 galaxies located approximately 16.5 Mpc from Earth. An astronomer measures the Hβ line (λ0 = 486.1 nm) in the spectra of three member galaxies:
- Galaxy M87: observed at 488.6 nm
- Galaxy NGC 4552: observed at 487.5 nm
- Galaxy M49: observed at 484.8 nm
5 marks
4.1 Calculate the radial velocity of each galaxy relative to Earth. State whether each galaxy is approaching or receding, and identify which of the three has the greatest speed relative to our line of sight. Show working. 3 marks
4.2 Galaxy NGC 4552 is close to the cluster centre while M87 and M49 are further out. Predict whether the velocity spread within the cluster tells us anything about the cluster’s mass, and justify your reasoning. 2 marks
5. Spot the errors — a student’s summary of Kirchhoff’s laws
A Year 12 student wrote the following summary. It contains three errors. Identify each error and write the correction. 6 marks (2 per error: 1 identify, 1 correct)
“Kirchhoff’s First Law states that a hot thin gas produces a continuous spectrum. His Second Law says that a hot solid, liquid or dense gas produces an emission spectrum with bright lines. His Third Law says that a cool gas in front of a continuous source produces an absorption spectrum with bright lines at the same wavelengths as the emission spectrum.”
5.1 Error 1: What is wrong?
Correction:
5.2 Error 2: What is wrong?
Correction:
5.3 Error 3: What is wrong?
Correction:
Q1.1 — Doppler shift table
Star A: Δλ = 0 nm; v = 0 km s−1; No radial motion (stationary along line of sight).
Star B: Δλ = 657.9 − 656.3 = +1.6 nm; v = 3.00×105 × 1.6/656.3 = +732 km s−1; Receding.
Star C: Δλ = 654.7 − 656.3 = −1.6 nm; v = 3.00×105 × (−1.6)/656.3 = −732 km s−1; Approaching.
Star D: Δλ = +2.9 nm; v = +1326 km s−1; Receding.
Star E: Δλ = −4.2 nm; v = −1920 km s−1; Approaching.
Q1.2 — Why both lines give the same velocity (2 marks)
The Doppler formula is v = c × Δλ/λ0. The fractional shift Δλ/λ0 is the same for all spectral lines of the same source because it depends only on the velocity, not on the rest wavelength. Even though the absolute Δλ in nm is larger for Hα than for Ca II K, when divided by their respective rest wavelengths the ratio is identical, giving the same v [1]. This is a consistency check — if the two lines gave different velocities it would indicate a measurement error [1].
Q1.3 — Star A’s zero radial velocity (2 marks)
A zero Doppler shift means there is zero component of velocity along the line of sight (zero radial velocity) [1]. Star A could still be moving through space in a direction perpendicular (transverse) to the line of sight — such motion produces no Doppler shift in light [1]. Only the radial component is detectable by Doppler spectroscopy; the transverse component is detected through proper motion measurements over time.
Q2.1 — Period and physical meaning (2 marks)
The velocity oscillates from v = 0, rises to +45 km s−1 at t = 2 yr, returns to 0 at t = 4 yr, falls to −45 km s−1 at t = 6 yr, and returns to 0 at t = 8 yr. Period = 8 years [1]. This is the orbital period of the star around the common centre of mass of the binary system [1].
Q2.2 — Observed Ca II K at maximum recession (2 marks)
Δλ = λ0 × v/c = 393.3 × 45/(3.00×105) = 393.3 × 1.5×10−4 = 0.059 nm [1].
λobs = 393.3 + 0.059 = 393.36 nm (to 5 sig. figs) ≈ 393.4 nm [1].
Q2.3 — What happens at zero radial velocity (2 marks)
At t = 0 and t = 4 yr the radial velocity is zero. This means the star is moving perpendicular to the line of sight — it is at the closest or furthest point from the observer in its orbit (i.e. at conjunction or at the point where its orbital velocity is directed sideways) [1]. There is no component of motion toward or away from Earth at these moments, so no Doppler shift is observed [1].
Q2.4 — Distinguishing binary from pulsating star (1 mark)
One additional observation: look for periodic changes in the brightness of the star. A pulsating star changes in luminosity as it expands and contracts, whereas a binary (spectroscopic) pair where only one star is visible would show periodic brightness dips only if the plane of the orbit is aligned to cause eclipses. Alternatively, a pulsating star would show correlated changes in temperature (colour) and radius, whereas a binary system would not. Accept any one valid distinguishing observation.
Q3 — Compare emission and absorption spectra
Source condition: Emission: hot thin gas. Absorption: cool thin gas in front of a continuous source.
Appearance: Emission: bright coloured lines on a dark background. Absorption: dark lines on a continuous rainbow background.
Wavelengths of lines: Emission: same wavelengths as the dark lines in the absorption spectrum of the same element. Absorption: same wavelengths as the bright lines in the emission spectrum of the same element.
Energy process: Emission: electron falls from a higher to a lower energy level, releasing a photon (E = hf). Absorption: electron absorbs a photon and jumps from a lower to a higher energy level.
Astronomical example: Emission: bright nebula (e.g. Orion Nebula) where hot gas glows. Absorption: stellar atmosphere producing Fraunhofer lines in the star’s spectrum.
Q4.1 — Virgo Cluster galaxy velocities (3 marks)
Formula: v = c × (λobs − λ0)/λ0 with c = 3.00×105 km s−1, λ0 = 486.1 nm.
M87: Δλ = 488.6 − 486.1 = +2.5 nm; v = 3.00×105 × 2.5/486.1 = +1542 km s−1 (receding) [1].
NGC 4552: Δλ = 487.5 − 486.1 = +1.4 nm; v = 3.00×105 × 1.4/486.1 = +864 km s−1 (receding) [1].
M49: Δλ = 484.8 − 486.1 = −1.3 nm; v = 3.00×105 × (−1.3)/486.1 = −802 km s−1 (approaching) [1].
Greatest speed along the line of sight: M87 at 1542 km s−1.
Q4.2 — Cluster velocity spread and mass (2 marks)
Yes, the velocity spread within the cluster tells us about its mass [1]. The galaxies are gravitationally bound within the cluster. Higher orbital velocities require more gravitational force, which comes from greater cluster mass. By applying the virial theorem (or v2 ∝ GM/r), the spread of radial velocities across cluster members can be used to estimate the total mass of the cluster, including any dark matter that cannot be seen directly [1].
Q5 — Kirchhoff’s laws errors (6 marks)
5.1 Error 1 (Law 1 and Law 2 are swapped): Kirchhoff’s First Law states that a hot thin gas produces an emission spectrum — not a continuous spectrum [1]. Correction: Kirchhoff’s First Law: a hot solid, liquid or dense gas produces a continuous spectrum. Kirchhoff’s Second Law: a hot thin gas produces an emission spectrum [1].
5.2 Error 2 (Second Law description is wrong): The student wrote that Law 2 gives a continuous spectrum from a hot solid/dense gas, which is actually Law 1 [1]. Correction: Kirchhoff’s Second Law correctly states that a hot thin gas produces an emission spectrum of bright lines at characteristic wavelengths [1].
5.3 Error 3 (Absorption spectrum has dark, not bright lines): The student wrote that the absorption spectrum has “bright lines” [1]. Correction: an absorption spectrum has dark lines at the same wavelengths as the element’s emission lines. The continuous background is visible at all other wavelengths; only the absorbed wavelengths appear dark [1].