Account for the production of emission and absorption spectra and compare these with a continuous black body spectrum; investigate stellar evolution using the Hertzsprung-Russell diagram and account for the synthesis of elements heavier than iron in supernovae

In the cores of stars on the main sequence, hydrogen fuses to helium via the proton-proton chain (low mass stars) or the CNO cycle (higher mass stars). When the core hydrogen is exhausted, the core contracts and heats; in stars massive enough (about 0.5 solar masses and above) helium fuses to carbon via the triple alpha process.

In massive stars (more than about 8 solar masses), further core contractions ignite successive shells of carbon, oxygen, neon, silicon and so on, building elements up to iron-56. The core of such a star resembles an onion of nuclear-burning shells.

Iron-56 has the highest binding energy per nucleon. Both fusion (making something heavier) and fission (making something lighter) of iron consume energy rather than releasing it. Once iron accumulates in the core, no further exothermic fusion is possible and the core collapses, triggering a supernova.

Elements heavier than iron are produced during the supernova explosion itself by rapid neutron capture (the r-process): neutrons released in the collapse are absorbed by nuclei faster than they can beta-decay, building up to uranium and beyond. The resulting heavy elements are blown out into the interstellar medium and become part of later generations of stars and planets.

Markers reward the up-to-iron stellar nucleosynthesis story, the binding energy explanation for why iron is the cutoff, and the supernova r-process for heavier elements.

The star spends about 90% of its life on the main sequence, fusing hydrogen to helium in its core. On the H-R diagram (luminosity vs surface temperature), this places it in the middle of the main sequence band, with the Sun specifically at $T \approx 5800$ K and $L = 1 L_{\odot}$.

When core hydrogen is exhausted, the core contracts and heats, while the outer envelope expands and cools. The star moves up and to the right on the diagram, into the red giant branch: luminosity increases, surface temperature decreases.

Helium fusion ignites in the core (the triple alpha process), producing carbon and oxygen. After helium exhaustion, the core cannot ignite carbon (insufficient mass). The outer envelope is shed as a planetary nebula, leaving the hot dense carbon-oxygen core exposed.

The remnant moves to the lower left of the H-R diagram as a white dwarf: small, hot, with low luminosity because of its small surface area. It then cools slowly over billions of years.

Markers reward main sequence, ascent to red giant, planetary nebula, and final white dwarf, in correct order with H-R direction noted.

(a) The dense photosphere produces a continuous (blackbody) spectrum: a hot, dense body radiates a smooth range of all wavelengths because collisions are frequent enough to thermalise the radiation.

(b) A cool, thin gas in front of a hotter source produces an absorption spectrum: atoms in the gas absorb photons at their characteristic wavelengths, removing them from the continuum and leaving dark lines.

Marks: one for correctly naming each spectrum type, one for correctly identifying the producing process (thermal emission from a dense hot body; selective absorption by a cooler foreground gas).

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

$T = \dfrac{2.898 \times 10^{-3}}{290 \times 10^{-9}} = \dfrac{2.898 \times 10^{-3}}{2.90 \times 10^{-7}} = 9.99 \times 10^{3}\ \text{K} \approx 1.00 \times 10^{4}\ \text{K}$.

This is hotter than the Sun ($5800\ \text{K}$), so the star would appear blue-white rather than yellow.

Marks: one for stating Wien's law and rearranging for $T$, one for the correct substitution and answer ($T \approx 1.00 \times 10^{4}\ \text{K}$), one for the correct comparison with the Sun's temperature.

Main sequence

A diagonal band running from upper left (hot, high luminosity) to lower right (cool, low luminosity).

Red giants

Upper right of the diagram: cool surface temperature but very high luminosity, because a large radius compensates for the low temperature.

White dwarfs

Lower left of the diagram: hot surface temperature but very low luminosity, because the star's radius is tiny (Earth-sized).

Reversed axis

Surface temperature is plotted decreasing to the right (i.e. hot stars are on the left), by historical convention.

Marks: one for the main sequence description, one for correctly placing both the red giants and white dwarfs, one for correctly identifying the reversed temperature axis.

(a) Reading the marked Sun position: $T \approx 5800\ \text{K}$ and $L \approx 1\ L_{\odot}$, in the middle of the main sequence band.

(b) A cool star ($T \approx 3500\ \text{K}$) with very high luminosity ($L \approx 10^{4}\ L_{\odot}$) plots in the upper-right region: this is a red giant/supergiant. Luminosity depends on both temperature and surface area ($L \propto R^2 T^4$), so a much larger radius can outweigh a lower temperature and still give an enormous luminosity.

(c) After core hydrogen is exhausted, the core contracts and heats while the envelope expands and cools, so the star moves up and to the right on the diagram, off the main sequence and into the red giant region.

Marks: one for reading the Sun's $T$ and $L$ from the figure, one for correctly identifying the red giant region, one for the $L \propto R^2T^4$ (large radius) explanation, one for the correct up-and-right direction of post-main-sequence evolution.

Binding energy per nucleon rises from hydrogen to a maximum at iron-56 (and the closely competing nickel-62), then falls slowly for heavier nuclei. Fusing two light nuclei into a heavier one below iron produces a nucleus with higher binding energy per nucleon than the reactants, so mass is converted to released kinetic energy ($E = \Delta mc^2$).

Beyond iron, fusing two nuclei produces a product with lower binding energy per nucleon, so the reaction would require an energy input rather than releasing one - it is endothermic, not exothermic.

Once the core is entirely iron, there is no further exothermic fusion reaction available to generate outward pressure. The core can no longer support itself against gravity, so it contracts and (in a massive star) collapses catastrophically, triggering a core-collapse supernova.

Marks: one for stating binding energy per nucleon peaks at iron, one for the below-iron exothermic (mass converts to energy) explanation, one for the above-iron endothermic explanation, one for correctly linking the iron core to gravitational collapse/supernova.

Ordinary stellar fusion combines charged nuclei against their mutual Coulomb repulsion and only releases energy up to iron-56; it proceeds on timescales of millions to billions of years inside a stable stellar core.

The r-process captures free neutrons (which carry no charge, so face no Coulomb barrier) onto existing heavy "seed" nuclei in rapid succession, faster than the nuclei can beta-decay, building neutron-rich isotopes up to and beyond uranium in seconds during an explosive event; the products then beta-decay to stable heavy elements.

Another r-process site. A neutron star merger (detected via its gravitational-wave and kilonova electromagnetic signal) is now known to be a major r-process site alongside core-collapse supernovae.

Marks: one for correctly characterising ordinary fusion (charged nuclei, up to iron, slow), one for correctly characterising the r-process (neutron capture faster than beta decay, builds beyond iron, explosive/fast), one for naming a valid second site (neutron star merger).

Band-6 plan. (1) Place the star on the H-R diagram while on the main sequence (hot, luminous, upper left) and state its fuel (hydrogen, CNO cycle). (2) Trace successive core contractions and shell-burning stages up to iron (name at least three fuels). (3) Explain why the iron core cannot fuse further (binding energy) and collapses. (4) Describe the core-collapse supernova and the r-process. (5) Close the loop: link the ejected heavy elements to later star/planet formation. Sequence every stage in the correct order and use both H-R language and nuclear-physics reasoning.

Model answer. A $20\ M_{\odot}$ star spends its main-sequence lifetime in the hot, highly luminous upper-left region of the H-R diagram, fusing hydrogen to helium in its core via the CNO cycle. Because its core mass and temperature are so high, it burns through hydrogen quickly (a few million years) compared with the Sun's $10$ billion years.

Once core hydrogen is exhausted, the core contracts and heats, igniting helium fusion (the triple-alpha process, producing carbon and oxygen) while the envelope expands into a red supergiant, moving the star up and to the right on the H-R diagram. Because the star is massive enough, each subsequent core exhaustion triggers another contraction and ignition: carbon burns to neon, oxygen and magnesium; neon burns to oxygen and magnesium; oxygen burns to silicon; and finally silicon burns to iron-56. Each successive fuel burns for a shorter time, leaving an "onion-skin" core of concentric burning shells around an iron centre.

Iron-56 sits at the peak of the binding-energy-per-nucleon curve, so fusing it further would require energy input rather than releasing it. With no further exothermic fusion available, the iron core cannot generate the outward pressure to resist gravity. When its mass exceeds the Chandrasekhar limit ($\approx 1.4\ M_{\odot}$), it collapses in under a second; protons capture electrons to form neutrons and release a burst of neutrinos, and the in-falling outer layers rebound off the dense core, driving a core-collapse supernova that blows the star's layers into space, leaving a neutron star or black hole.

The same collapse floods the surrounding material with free neutrons. Because neutrons carry no charge, existing heavy seed nuclei (built up to iron in the preceding stages) can capture them in rapid succession, faster than the unstable products can beta-decay - the r-process. This builds neutron-rich nuclei up to and beyond uranium within seconds, which then beta-decay to stable heavy elements such as gold, platinum and uranium. These elements are ejected into the interstellar medium and are ultimately incorporated into the gas clouds that collapse to form the next generation of stars and their planets - which is why elements heavier than iron on Earth were forged in a supernova (or neutron star merger) from an earlier stellar generation.

Marker's note: the top band sequences every stage correctly (main sequence to supergiant to successive shell-burning to iron core to collapse to r-process to enrichment), explicitly uses the binding-energy argument to explain the iron cutoff, and explicitly explains why neutrons (not fusion) build elements beyond iron. A response that lists fusion stages without the binding-energy reasoning, or that never explains the r-process mechanism, caps in the middle band.

Band-6 plan. (1) Explain the continuous background as blackbody radiation from the dense photosphere. (2) Explain the absorption lines as selective absorption by cooler, thinner outer-atmosphere gas, and link the wavelengths to composition. (3) Explain the nebula's emission-only spectrum as hot, low-density gas with nothing behind it to absorb. (4) Explain why the same wavelengths appear in both, and evaluate what this confirms about shared composition/physics. (5) Reach an explicit judgement.

Model answer. The continuous part of the star's spectrum is thermal (blackbody) radiation: the photosphere is dense enough that photons undergo many collisions before escaping, producing a smooth Planck-like curve of all wavelengths whose peak wavelength depends only on temperature (Wien's law). Superimposed dark lines form because the cooler, less dense gas in the star's outer atmosphere, lying in front of this continuum, absorbs photons at the precise wavelengths corresponding to electron transitions in its constituent atoms, removing those photons from the light reaching us: this is an absorption spectrum, and the missing wavelengths identify the elements present.

The nebula, by contrast, is a hot, very low-density gas with no dense photosphere behind it to supply a continuum. Its atoms are excited (for example by nearby hot stars) and re-emit photons at their own characteristic wavelengths as electrons fall back to lower energy levels, producing bright lines on an otherwise dark background: an emission spectrum.

That the star's absorption lines and the nebula's emission lines occur at the same wavelengths is strong evidence that both processes involve the same atomic transitions in the same elements - the wavelength of a spectral line is a fixed property of the atom, not of whether it is absorbing or emitting. This confirms that spectroscopy reliably identifies chemical composition regardless of whether the gas is seen in absorption or emission, and that the star's atmosphere and the nebula plausibly share a common origin or composition (consistent with both having condensed from, or been enriched by, the same interstellar material).

Marker's note: the top band correctly attributes the continuum to blackbody radiation from a dense source, distinguishes absorption (cooler foreground gas) from emission (hot low-density gas with nothing behind it), and explicitly uses the matching-wavelength evidence to draw a composition/physical-conditions conclusion rather than simply describing the two spectra side by side.