Investigate emission and absorption spectra, distinguish continuous, line emission and line absorption spectra, and analyse stellar spectra to identify chemical composition, surface temperature and motion

Continuous spectrum: a smooth rainbow of all wavelengths, produced by a hot dense source (an incandescent solid, liquid or compressed gas) where the energy levels are smeared together by close packing.

Line emission spectrum: a series of bright lines on a dark background at specific wavelengths, produced by a hot, low-pressure gas. The lines correspond to photons released as electrons drop from higher to lower discrete energy levels in the atoms, $E_{\text{photon}} = E_i - E_f = h f$.

Line absorption spectrum: a continuous spectrum crossed by dark lines at specific wavelengths, produced when continuum light passes through a cool gas. Electrons in the cool gas absorb only those photons whose energies exactly match the gas atoms' allowed transitions, removing those wavelengths from the transmitted beam.

A star produces continuum light from its hot dense interior, which passes through the cooler stellar atmosphere. Atoms in the atmosphere absorb the wavelengths matching their characteristic transitions, leaving dark lines in the stellar spectrum. Each element has a unique fingerprint, so matching the observed absorption-line wavelengths to laboratory spectra identifies the elements present (this is how helium was discovered in the Sun in 1868, before being found on Earth).

Markers reward all three spectrum types correctly described with examples, plus the absorption-by-atmosphere mechanism and the fingerprint-matching idea.

Redshift fraction:

$z = \Delta \lambda / \lambda_0 = (689.1 - 656.3) / 656.3 = 32.8 / 656.3 = 0.0500$.

For $v \ll c$, the non-relativistic Doppler formula gives:

$v = z c = 0.0500 \times 3.00 \times 10^8 = 1.5 \times 10^7$ m/s.

Sign and direction: the observed wavelength is longer than the laboratory wavelength (redshift), so the galaxy is receding from Earth at about $1.5 \times 10^7$ m/s.

Markers reward the formula, correct value of $z$, conversion to velocity, and identification of "receding" because of the redshift. Some answers will use the relativistic Doppler formula; at $v / c = 0.05$ the correction is small ($\sim 0.1\%$).

Line emission spectrum. Bright lines on a dark background at specific wavelengths, produced by a hot, low-density gas (e.g. a neon discharge tube or a nebula), where excited electrons drop to lower energy levels and release photons.

Line absorption spectrum. A continuous spectrum crossed by dark lines at specific wavelengths, produced when continuum light passes through a cooler gas (e.g. the Sun's photosphere continuum passing through its cooler outer atmosphere), which removes photons matching its atoms' allowed transitions.

Marks: one for a correct description of each spectrum type, one for a correct source example for each.

Wien's law: $\lambda_{\max} = \dfrac{b}{T}$.

$\lambda_{\max} = \dfrac{2.898 \times 10^{-3}}{3000} = 9.66 \times 10^{-7}\ \text{m} = 966\ \text{nm}$.

Marks: one for the correct formula with values substituted, one for the answer to three significant figures with the unit (this falls in the infrared, consistent with a cool red star).

(a) A continuous spectrum comes from a hot, dense source (a glowing solid, liquid or high-pressure gas), where atoms are packed so closely that their energy levels blend into an effectively continuous range of emitted photon energies.

(b) A line emission spectrum comes from a hot, low-density gas, where widely separated atoms emit photons only at the discrete energies corresponding to their allowed transitions.

Why they differ. In a dense source, collisions and pressure broaden and smear the discrete atomic energy levels into a continuum; in a dilute gas the atoms are isolated, so only their sharp, discrete transition energies appear as lines.

Marks: one for the continuous-spectrum source, one for the line-emission-spectrum source, one for the pressure/density explanation of why the same element gives different-looking spectra.

Redshift fraction: $z = \dfrac{\Delta \lambda}{\lambda_0} = \dfrac{657.9 - 656.3}{656.3} = \dfrac{1.6}{656.3} = 2.44 \times 10^{-3}$.

For $v \ll c$: $v = zc = 2.44 \times 10^{-3} \times 3.00 \times 10^8 = 7.31 \times 10^5\ \text{m s}^{-1}$.

Since $\lambda_{\text{obs}} > \lambda_0$ this is a redshift, so the galaxy is receding from Earth at about $7.31 \times 10^5\ \text{m s}^{-1}$ ($731\ \text{km s}^{-1}$).

Marks: one for the correct $z$, one for $v = zc$ correctly evaluated with unit, one for correctly identifying "receding" from the redshift.

(a) Star A's curve peaks at approximately $\lambda_{\max} \approx 480\ \text{nm}$ (reading directly off the graph).

(b) $\lambda_{\max} = \dfrac{b}{T} = \dfrac{2.898 \times 10^{-3}}{6000} = 4.83 \times 10^{-7}\ \text{m} = 483\ \text{nm}$, which matches the graph's peak within reading error.

(c) Star A ($6000\ \text{K}$) is hotter than star B ($4000\ \text{K}$). Its curve peaks at a shorter wavelength (Wien's law: $\lambda_{\max} \propto 1/T$, so a hotter object peaks further toward the blue/violet end), and its curve also sits higher and encloses a larger area (a hotter blackbody radiates more total power per unit area at every wavelength, consistent with the Stefan-Boltzmann law).

Marks: one for a peak-wavelength reading close to $480\text{-}485\ \text{nm}$, one for the Wien's-law calculation confirming it, one for correctly identifying star A as hotter, one for citing both the shorter peak wavelength AND the greater area/height as evidence.

Presence confirms. Each element has a unique set of allowed electron transitions, so a set of absorption lines at wavelengths matching an element's known laboratory spectrum can only be produced by that element being present in the light path - it is a unique spectral fingerprint.

Absence does not rule it out. A given transition line only appears strongly if the atmosphere's temperature and density populate the right starting energy level for that transition. An element may be present but at too low a temperature (or too fully ionised, or too rarefied) to produce a detectable line at that particular wavelength, so a missing line can reflect the physical conditions rather than the element's absence.

Marks: one for the fingerprint-matching argument for presence, one for identifying that line strength depends on temperature/level population, one for concluding this means absence of a line is not proof of absence of the element.

Band-6 plan. Structure by the three quantities in the question: (1) composition - line identification against laboratory spectra; (2) temperature - Wien's law on the continuum plus relative line strengths; (3) radial velocity - Doppler shift of the line positions. For each, name the underlying physics (quantised energy levels, blackbody radiation, wave motion of light) and the observable feature used. Finish by noting all three are extracted from the SAME spectrum simultaneously.

Model answer. A star's photosphere produces a near-continuous blackbody spectrum that is then absorbed at specific wavelengths by cooler gas in the star's outer atmosphere, producing a line absorption spectrum. Three independent pieces of information can be read from this one spectrum.

Composition is determined from the wavelengths of the dark absorption lines. Because each element has a unique set of quantised electron energy levels, it absorbs photons only at the specific energies $hf = E_i - E_f$ matching its allowed transitions. Matching the observed line wavelengths against laboratory reference spectra identifies which elements are present in the star's atmosphere (this is how helium was first identified in the Sun in 1868, before being isolated on Earth).

Surface temperature is found in two independent ways from the same spectrum. First, the overall shape of the continuum approximates a blackbody curve, and its peak wavelength obeys Wien's law, $\lambda_{\max} = b/T$; a shorter peak wavelength means a hotter star. Second, the relative strengths of different absorption lines depend on temperature, since each transition requires the lower energy level to be sufficiently populated, so the pattern of strong and weak lines places a star into a spectral class (O, B, A, F, G, K, M).

Radial velocity is found from a uniform shift of every absorption line's wavelength relative to its laboratory rest value. If the star is moving away from Earth along the line of sight, all lines are shifted to longer wavelengths (redshift); if approaching, to shorter wavelengths (blueshift), with $\Delta\lambda/\lambda_0 = v/c$ for $v \ll c$. Because this shift affects every line by the same fractional amount, it can be distinguished cleanly from the fixed line positions used for composition.

All three results, composition, temperature and radial velocity, come from analysing a single captured spectrum: the line wavelengths (shifted by a common factor) give composition and velocity, and the continuum shape plus line-strength pattern gives temperature.

Marker's note: the top band explains the physical origin of each measurement (quantised levels for composition, blackbody/Wien's law and level population for temperature, wave-Doppler shift for velocity) rather than simply naming the three results, and explicitly notes that all three are extracted from one spectrum. A response that lists the three uses without the underlying physics caps in the middle band.

Band-6 plan. Take a genuine evaluative stance: (1) what a line absorption spectrum CAN determine (composition, temperature, radial velocity, rotation) with brief physical justification for each; (2) what it CANNOT determine from itself alone (e.g. distance, luminosity, transverse motion, total mass) and why; (3) a concluding judgement (powerful but partial; needs combining with other data such as parallax or a light curve).

Model answer. A line absorption spectrum is a remarkably information-rich single measurement. The wavelengths of the absorption lines identify the elements present in the star's atmosphere by matching quantised-transition fingerprints against laboratory spectra. The shape of the underlying continuum, via Wien's law $\lambda_{\max} = b/T$, together with which lines are strong or weak, gives the surface temperature and hence spectral type. A uniform shift of all the line wavelengths gives the radial velocity by the Doppler effect, and a symmetric broadening of the lines (from one limb approaching and the other receding) gives the star's rotation rate. In this sense the claim has real force: composition, temperature, radial velocity and rotation are all genuinely obtainable from spectroscopy alone.

However, a spectrum alone cannot determine everything about a star. It gives no direct measure of distance (needed for parallax or standard-candle methods) or of luminosity (total power output), since apparent brightness depends on both luminosity and distance, and a spectrum records neither in isolation. It also cannot determine transverse velocity (motion across the line of sight), because the Doppler effect is sensitive only to the radial component of velocity; a star moving purely sideways produces no line shift at all. Total mass likewise cannot be read from a single spectrum, though a binary system's spectrum can give some mass information via orbital Doppler variations combined with Kepler's laws.

On balance, the claim overstates what a single spectrum provides: it is an unusually powerful tool for atmospheric and kinematic (line-of-sight) properties, but a complete stellar characterisation, including distance, luminosity and full three-dimensional motion, requires combining spectroscopy with other observations such as parallax, proper motion and photometric brightness.

Marker's note: the top band gives specific correct physics for each "can" (fingerprint matching, Wien's law/line strengths, uniform Doppler shift, symmetric broadening) and each "cannot" (no distance/luminosity/transverse velocity from a spectrum alone, with a stated reason), then reaches an explicit, justified evaluative conclusion. A response that only lists what spectra reveal, without addressing the limitation, caps in the middle band.