Self-Assessment Quiz — Chapter 22

Test your understanding of stellar nucleosynthesis before moving on. Try to answer each question before checking the solutions at the end.

Q1. (Multiple Choice) The rate-limiting step of the pp chain is:

(a) $d + p \to {}^3\text{He} + \gamma$ (b) ${}^3\text{He} + {}^3\text{He} \to {}^4\text{He} + 2p$ (c) $p + p \to d + e^+ + \nu_e$ (d) ${}^7\text{Be} + e^- \to {}^7\text{Li} + \nu_e$

Q2. (True/False) The $p + p \to d + e^+ + \nu_e$ reaction has been directly measured in the laboratory.

Q3. (Multiple Choice) The pp chain dominates over the CNO cycle in stars with core temperatures below approximately:

(a) $5 \times 10^6$ K (b) $17 \times 10^6$ K (c) $10^8$ K (d) $10^9$ K

Q4. (Short Answer) Why does the CNO cycle convert most of the original C, N, and O into ${}^{14}$N?

Q5. (Multiple Choice) The energy generation rate for the CNO cycle scales approximately as $T^n$ where $n$ is:

(a) 4 (b) 8 (c) 16 (d) 41

Q6. (True/False) In the pp chain, the Sun converts approximately 600 million tonnes of hydrogen into helium every second.

Q7. (Multiple Choice) The triple-alpha process requires a resonance in ${}^{12}$C because:

(a) ${}^8$Be is stable and accumulates in the core (b) ${}^8$Be is unbound and the three-body reaction would otherwise be too slow (c) The Coulomb barrier between two ${}^4$He nuclei is too high (d) The gamma-ray emission probability is too low

Q8. (Short Answer) What are the quantum numbers ($J^\pi$) of the Hoyle state? Why are these important for the decay properties of the state?

Q9. (True/False) Fred Hoyle predicted the existence of the Hoyle state before it was experimentally discovered, based on the observed abundance of carbon in the universe.

Q10. (Multiple Choice) The most important unsolved reaction rate problem in nuclear astrophysics is:

(a) $p + p \to d + e^+ + \nu_e$ (b) ${}^{14}\text{N}(p,\gamma){}^{15}\text{O}$ (c) ${}^{12}\text{C}(\alpha,\gamma){}^{16}\text{O}$ (d) ${}^{12}\text{C} + {}^{12}\text{C}$

Q11. (Short Answer) Why does the C/O ratio at the end of helium burning matter so much for the subsequent evolution of the star?

Q12. (Multiple Choice) Neon burning is driven by:

(a) Direct fusion of two ${}^{20}$Ne nuclei (b) Photodisintegration of ${}^{20}$Ne followed by alpha capture (c) Proton capture on ${}^{20}$Ne (d) Neutron capture on ${}^{20}$Ne

Q13. (True/False) Silicon burning proceeds by direct fusion: ${}^{28}\text{Si} + {}^{28}\text{Si} \to {}^{56}\text{Ni}$.

Q14. (Short Answer) Define nuclear statistical equilibrium (NSE). Under what conditions does it apply?

Q15. (Multiple Choice) The immediate product of silicon burning in conditions with $Y_e \approx 0.50$ is:

(a) ${}^{56}$Fe (b) ${}^{54}$Fe (c) ${}^{56}$Ni (d) ${}^{62}$Ni

Q16. (Short Answer) The burning timescales for a $25 \, M_\odot$ star are: H: $7 \times 10^6$ yr, He: $5 \times 10^5$ yr, C: 600 yr, Si: 1 day. Give two physical reasons for this dramatic acceleration.

Q17. (True/False) Iron-peak elements are the most abundant metals in the universe because iron has the highest binding energy per nucleon.

Q18. (Multiple Choice) In the onion-shell structure of a pre-supernova star, the outermost shell that is actively burning nuclear fuel is:

(a) The hydrogen-burning shell (b) The helium-burning shell (c) The carbon-burning shell (d) The silicon-burning shell

Q19. (Short Answer) The decay chain ${}^{56}\text{Ni} \to {}^{56}\text{Co} \to {}^{56}\text{Fe}$ has what observable consequence in supernovae?

Q20. (Multiple Choice) Fusion beyond iron is endothermic because:

(a) The Coulomb barrier is too high (b) The nuclear force becomes repulsive for $A > 56$ (c) The binding energy per nucleon decreases for $A > 56$–$62$, so the products are less tightly bound than the reactants (d) Iron nuclei are radioactive

Solutions

Q1. (c) The $p + p \to d + e^+ + \nu_e$ reaction is rate-limiting because it requires a weak-interaction process (proton-to-neutron conversion) at the moment of nuclear contact, giving it a cross section $\sim 10^{-25}$ times smaller than typical strong-interaction cross sections.

Q2. False. The S-factor $S_{pp}(0) = 4.01 \times 10^{-25}$ MeV b is too small to be measured in any laboratory. It is calculated from weak-interaction theory, calibrated against the neutron half-life.

Q3. (b) The crossover temperature is $\sim 17 \times 10^6$ K for solar metallicity. Below this temperature, the pp chain dominates; above it, the CNO cycle dominates.

Q4. Because the slowest reaction in the CNO-I cycle is ${}^{14}\text{N}(p,\gamma){}^{15}\text{O}$. Since the cycle rate is limited by this bottleneck, material accumulates as ${}^{14}$N — the nucleus that precedes the slowest step. It is the "traffic jam" nuclide of the CNO cycle.

Q5. (c) The CNO energy generation rate scales as $\sim T^{16}$ near $17 \times 10^6$ K, reflecting the high Coulomb barriers ($Z = 6$–$7$) of the CNO target nuclei.

Q6. True. The Sun converts $\sim 4 \times 10^9$ kg/s ($\sim 600$ million tonnes/s) of hydrogen to helium. Of this, $\sim 4.3 \times 10^6$ kg/s is converted to energy ($E = \Delta m \, c^2$).

Q7. (b) ${}^8$Be is unbound ($\tau \sim 10^{-16}$ s), so the equilibrium abundance of ${}^8$Be is tiny. Without the Hoyle state resonance in ${}^{12}$C to enormously enhance the ${}^8$Be$+\alpha$ capture rate, the triple-alpha process would be far too slow to produce the observed cosmic abundance of carbon.

Q8. $J^\pi = 0^+$. The Hoyle state has the same quantum numbers as the ${}^{12}$C ground state ($0^+$), which means single-photon $0^+ \to 0^+$ transitions (E0) are forbidden — the state cannot decay by single gamma emission to the ground state. It must decay by pair emission ($e^+e^-$) to the ground state or by E2 gamma to the $2^+$ first excited state, which suppresses the radiative width and gives the tiny branching ratio $\Gamma_\gamma/\Gamma \approx 4 \times 10^{-4}$.

Q9. True. Hoyle's argument (1953) was that carbon exists in the universe in substantial abundance, which requires an efficient production mechanism in stars. The triple-alpha process can only be efficient enough if ${}^{12}$C has a resonance near the ${}^8$Be$+\alpha$ threshold. This is one of the most famous predictions in nuclear astrophysics.

Q10. (c) ${}^{12}\text{C}(\alpha,\gamma){}^{16}\text{O}$ — the rate is uncertain by $\sim 24\%$ at the astrophysically relevant energy ($E \approx 300$ keV), and this uncertainty propagates into models of white dwarf composition, Type Ia supernova yields, and the pre-supernova structure of massive stars. (d) is also a serious unsolved problem, but (c) has broader astrophysical impact.

Q11. The C/O ratio determines: (i) the composition and cooling properties of white dwarfs; (ii) the ignition conditions and nucleosynthesis yields of Type Ia supernovae; (iii) the fuel mix for subsequent burning stages in massive stars, affecting pre-supernova structure and whether the core produces a neutron star or black hole; (iv) the cosmic abundance of oxygen, the third most abundant element.

Q12. (b) Neon burning is photodisintegration-driven: $\gamma + {}^{20}\text{Ne} \to {}^{16}\text{O} + \alpha$ (endothermic, powered by thermal photons), followed by ${}^{20}\text{Ne} + \alpha \to {}^{24}\text{Mg} + \gamma$ (exothermic). The net reaction is exothermic.

Q13. False. The Coulomb barrier for ${}^{28}\text{Si} + {}^{28}\text{Si}$ ($\sim 39$ MeV) is far too high for fusion at the relevant temperatures. Silicon burning proceeds by photodisintegration rearrangement: photodisintegration breaks down silicon into lighter nuclei and free particles, which are then recaptured to build up heavier nuclei toward the iron peak.

Q14. Nuclear statistical equilibrium (NSE) is a state in which the rates of all nuclear reactions and their reverses are in balance, so the abundance of every nuclear species is determined by its binding energy, the temperature, and the density — not by individual reaction rates. NSE applies at temperatures $T \gtrsim 4 \times 10^9$ K, where photodisintegration and capture reactions are fast enough to maintain equilibrium. The NSE abundance is given by a Saha-like equation with the exponential factor $\exp(B/k_BT)$.

Q15. (c) ${}^{56}$Ni ($Z = N = 28$, doubly magic) is the dominant product of silicon burning at $Y_e \approx 0.50$. It subsequently decays to ${}^{56}$Fe via ${}^{56}$Ni $\to$ ${}^{56}$Co $\to$ ${}^{56}$Fe on a timescale of weeks to months.

Q16. (1) Each successive burning stage releases less energy per reaction (the $B/A$ curve flattens near the iron peak), requiring faster fuel consumption. (2) Neutrino losses from thermal processes ($e^+e^-$ annihilation, plasmon decay, etc.) increase steeply with temperature ($\sim T^9$), draining energy far faster than photon radiation, and accelerating the rate at which the star must burn fuel.

Q17. True (with a caveat). Iron (${}^{56}$Fe) is indeed the most abundant metal because NSE drives material toward the maximum of $B/A$. However, the nucleus with the absolute highest $B/A$ is ${}^{62}$Ni ($B/A = 8.795$ MeV), not ${}^{56}$Fe ($B/A = 8.790$ MeV). Iron dominates because (i) ${}^{56}$Ni (which decays to ${}^{56}$Fe) is the favored NSE product at $Y_e \approx 0.50$, and (ii) the difference in $B/A$ between ${}^{56}$Fe and ${}^{62}$Ni is only $\sim 0.005$ MeV/nucleon.

Q18. (a) The hydrogen-burning shell is the outermost active burning shell. All other burning shells (He, C, Ne, O, Si) are nested inside it. The hydrogen envelope outside the H-burning shell is unprocessed.

Q19. The radioactive decay of ${}^{56}$Co (half-life 77.24 days) powers the optical light curve of supernovae during the nebular phase (weeks to months after explosion). The exponential decline in luminosity directly traces the ${}^{56}$Co half-life, and the amount of ${}^{56}$Ni produced can be inferred from the peak luminosity.

Q20. (c) For nuclei heavier than the iron peak ($A \gtrsim 62$), $B/A$ decreases with increasing $A$. Fusing two iron-peak nuclei would produce a nucleus with lower $B/A$ than the reactants, meaning the products are less tightly bound. The reaction therefore requires energy input (endothermic) rather than releasing energy. This is the fundamental reason iron is the end of the line for stellar nucleosynthesis.