Self-Assessment Quiz — Chapter 23

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

Q1. (Multiple Choice) What triggers the collapse of the iron core in a massive star?

(a) The iron core reaches the Chandrasekhar mass and electron degeneracy pressure can no longer support it (b) A nearby supernova shock wave compresses the core (c) The core cools until thermal pressure is insufficient (d) Nuclear fusion of iron produces energy that destabilizes the core

Q2. (Multiple Choice) Approximately what fraction of a core-collapse supernova's gravitational binding energy is carried away by neutrinos?

(a) ~1% (b) ~10% (c) ~50% (d) ~99%

Q3. (True/False) Photodisintegration of iron in the collapsing core is an exothermic process that helps drive the explosion.

Q4. (Short Answer) What is the nuclear compressibility modulus $K_0$, and why does it matter for core-collapse supernovae? How is it measured in the laboratory?

Q5. (Multiple Choice) The light curve of a Type II supernova's radioactive tail is powered primarily by:

(a) Hydrogen recombination in the envelope (b) The decay chain ${}^{56}\text{Ni} \to {}^{56}\text{Co} \to {}^{56}\text{Fe}$ (c) Neutrino heating of the ejecta (d) Gravitational contraction of the remnant

Q6. (Multiple Choice) Type Ia supernovae produce approximately how much ${}^{56}\text{Ni}$?

(a) $0.01\,M_\odot$ (b) $0.07\,M_\odot$ (c) $0.6$–$0.8\,M_\odot$ (d) $1.44\,M_\odot$

Q7. (True/False) Type Ia supernovae are the primary source of iron-peak elements in the Galaxy.

Q8. (Short Answer) What is the defining condition that distinguishes the s-process from the r-process? Express it in terms of $\lambda_\beta$ and $\lambda_n$.

Q9. (Multiple Choice) The s-process operates in:

(a) The neutrino-driven wind of core-collapse supernovae (b) Asymptotic giant branch (AGB) stars (c) Neutron star mergers (d) The Big Bang

Q10. (Multiple Choice) At neutron magic numbers ($N = 50, 82, 126$), the s-process produces abundance peaks because:

(a) Beta-decay half-lives are very short at magic numbers (b) Neutron capture cross sections are very small, creating bottlenecks (c) Photodisintegration $(\gamma,n)$ destroys nuclei with non-magic neutron numbers (d) The nuclear force is repulsive at magic numbers

Q11. (Short Answer) What is the local approximation for the s-process? Write the formula and explain its physical meaning.

Q12. (True/False) The s-process can produce uranium and thorium.

Q13. (Multiple Choice) The r-process path on the chart of nuclides runs:

(a) Along the valley of stability, like the s-process but faster (b) On the proton-rich side of stability (c) Far to the neutron-rich side of stability, near the neutron drip line (d) Through the superheavy element region ($Z > 118$)

Q14. (Short Answer) Explain why the r-process abundance peaks are shifted to lower mass number ($A$) compared to the s-process peaks, even though both are associated with the same magic neutron numbers.

Q15. (Multiple Choice) The event GW170817 was:

(a) A core-collapse supernova observed through gravitational waves (b) A binary black hole merger with an electromagnetic counterpart (c) A binary neutron star merger observed through gravitational waves and across the electromagnetic spectrum (d) A gamma-ray burst from a magnetar

Q16. (True/False) The kilonova AT 2017gfo was brighter in the infrared than in the optical at late times ($t > 3\,\text{days}$) because lanthanide elements have enormous opacities at optical wavelengths.

Q17. (Short Answer) What element was spectroscopically identified in the kilonova AT 2017gfo, and why was this identification significant?

Q18. (Multiple Choice) Cosmochronology using the ${}^{232}\text{Th}/{}^{238}\text{U}$ ratio works because:

(a) Both isotopes are produced by the s-process with a known ratio (b) Both are r-process products with different half-lives, so their ratio changes with time (c) ${}^{232}\text{Th}$ is stable while ${}^{238}\text{U}$ decays (d) Both are primordial isotopes from the Big Bang

Q19. (True/False) The p-nuclei (proton-rich heavy isotopes) are produced by rapid proton capture in the r-process.

Q20. (Short Answer) Name three facilities designed to study the neutron-rich nuclei on the r-process path, and explain why laboratory measurements of these nuclei are important for r-process calculations.

Solutions

Q1. (a) The iron core exceeds the effective Chandrasekhar mass (reduced from $1.44\,M_\odot$ because $Y_e < 0.5$), and electron degeneracy pressure fails. Electron capture and photodisintegration accelerate the collapse.

Q2. (d) Approximately 99% ($\sim 3 \times 10^{53}\,\text{erg}$ of the $\sim 3 \times 10^{53}\,\text{erg}$ gravitational binding energy). The kinetic energy of the ejecta is only $\sim 1$%, and the radiated light is $\sim 0.01$%.

Q3. False. Photodisintegration of iron is endothermic — it absorbs $124.4\,\text{MeV}$ per ${}^{56}\text{Fe}$ nucleus disassembled, draining energy from the core and accelerating the collapse.

Q4. The nuclear compressibility modulus $K_0 \approx 230\,\text{MeV}$ measures the stiffness of nuclear matter against compression. It determines how hard the core "bounces" when it reaches nuclear density during collapse. It is measured via the excitation energy of the isoscalar giant monopole resonance (the nuclear "breathing mode") in heavy nuclei.

Q5. (b) The ${}^{56}\text{Ni} \to {}^{56}\text{Co} \to {}^{56}\text{Fe}$ decay chain. ${}^{56}\text{Ni}$ is produced by explosive silicon burning; its decay (and that of ${}^{56}\text{Co}$) heats the ejecta and powers the optical light curve.

Q6. (c) $0.6$–$0.8\,M_\odot$ — approximately ten times more than a core-collapse supernova ($\sim 0.07\,M_\odot$).

Q7. True. Type Ia supernovae produce $\sim 10\times$ more ${}^{56}\text{Ni}$ (which decays to ${}^{56}\text{Fe}$) per event, and they are common enough to dominate iron production in the Galaxy.

Q8. s-process: $\lambda_\beta \gg \lambda_n$ (beta decay is much faster than neutron capture — the nucleus decays before capturing another neutron). r-process: $\lambda_n \gg \lambda_\beta$ (neutron capture is much faster than beta decay — the nucleus captures many neutrons before decaying).

Q9. (b) AGB stars, where thermal pulses in the helium-burning shell provide a neutron source via ${}^{13}\text{C}(\alpha,n){}^{16}\text{O}$ and ${}^{22}\text{Ne}(\alpha,n){}^{25}\text{Mg}$.

Q10. (b) Nuclei with magic neutron numbers have closed neutron shells and very small neutron capture cross sections ($\sigma_n \sim$ few mb). The s-process flow slows down dramatically at these nuclei, causing material to accumulate — creating abundance peaks.

Q11. $\sigma_A N_s(A) \approx \text{constant}$. In steady-state s-process flow, the number of nuclei passing through each mass number per unit time is the same. Since the flow rate is proportional to $\sigma_A N_s(A)$, this product must be approximately constant. Isotopes with small $\sigma$ accumulate large $N_s$, and vice versa.

Q12. False. The s-process ends at bismuth (${}^{209}\text{Bi}$). Neutron capture on ${}^{209}\text{Bi}$ leads through ${}^{210}\text{Bi} \to {}^{210}\text{Po} \to {}^{206}\text{Pb}$ (alpha decay), cycling back to lead. The s-process cannot build elements heavier than Bi.

Q13. (c) Far to the neutron-rich side, where $S_n \approx 2$–$3\,\text{MeV}$. At the extreme neutron densities of the r-process ($n_n > 10^{20}\,\text{cm}^{-3}$), nuclei capture neutrons so rapidly that the path extends 10–30 neutrons beyond the valley of stability.

Q14. The r-process waiting points are at magic $N$ but at much lower $Z$ (more neutron-rich) than the stable nuclei at the same $N$. After the neutron flux ceases, these waiting-point nuclei beta-decay toward stability (increasing $Z$, constant $A$). Since $A = N + Z$ and $N$ is the same (magic) but $Z$ is lower at the waiting point, the r-process peak occurs at lower $A$ than the s-process peak at the same magic $N$.

Q15. (c) A binary neutron star merger. GW170817 was detected on August 17, 2017 by LIGO and Virgo (gravitational waves) with follow-up observations across the entire electromagnetic spectrum, including a short gamma-ray burst and a kilonova.

Q16. True. Lanthanide elements (partially filled $4f$ shells) have millions of spectral lines that produce enormous bound-bound opacities ($\kappa \sim 10$–$30\,\text{cm}^2/\text{g}$) at UV/optical wavelengths. This blocks optical emission and forces the radiation into the infrared, producing the "red" kilonova component.

Q17. Strontium (${}^{88}\text{Sr}$, $Z = 38$) was identified by Watson et al. (2019) through its absorption features in the early spectra. This was the first spectroscopic identification of a specific r-process element in a kilonova, providing direct evidence that the merger synthesized heavy elements.

Q18. (b) Both ${}^{232}\text{Th}$ ($t_{1/2} = 14.05\,\text{Gyr}$) and ${}^{238}\text{U}$ ($t_{1/2} = 4.468\,\text{Gyr}$) are produced exclusively by the r-process. Because ${}^{238}\text{U}$ decays faster, the Th/U ratio increases with time. Knowing the initial production ratio from r-process models, the measured present-day ratio gives the elapsed time.

Q19. False. The p-nuclei are produced primarily by the $\gamma$-process — photodisintegration $(\gamma,n)$, $(\gamma,p)$, $(\gamma,\alpha)$ of pre-existing s-process and r-process seeds during explosive supernova burning. Some may also be produced by the $\nu p$-process in the neutrino-driven wind.

Q20. FRIB (Facility for Rare Isotope Beams, Michigan State University, USA), RIKEN RIBF (Radioactive Isotope Beam Factory, Japan), and FAIR (Facility for Antiproton and Ion Research, Germany). These facilities produce neutron-rich nuclei far from stability that lie on the r-process path. Laboratory measurements of their masses, beta-decay half-lives, and neutron capture cross sections are essential because r-process calculations require these nuclear properties as inputs, and for most r-process nuclei, the properties are currently known only from theoretical models with significant uncertainties.