Exercises — Chapter 23

Core-Collapse Supernovae

Problem 23.1 ⭐ The iron core of a massive star has electron fraction $Y_e = 0.42$ at the onset of collapse.

(a) Calculate the effective Chandrasekhar mass for this electron fraction using $M_{\text{Ch}} = 1.44(Y_e/0.5)^2\,M_\odot$.

(b) Explain physically why $Y_e < 0.5$ in the iron core (Hint: which iron-peak nuclei dominate in NSE?).

(c) If electron capture during collapse reduces $Y_e$ to $0.36$, by what factor does the Chandrasekhar mass change?

Problem 23.2 ⭐ The photodisintegration of ${}^{56}\text{Fe}$ absorbs $124.4\,\text{MeV}$ per nucleus.

(a) An iron core of mass $1.4\,M_\odot$ is composed entirely of ${}^{56}\text{Fe}$. Calculate the total number of iron nuclei.

(b) Calculate the total energy absorbed if the entire core is photodisintegrated to alpha particles and neutrons.

(c) Compare this energy to the gravitational binding energy of a neutron star ($\sim 3 \times 10^{53}\,\text{erg}$). What fraction is it?

Problem 23.3 ⭐⭐ The neutrino-driven explosion mechanism relies on charged-current reactions $\nu_e + n \to p + e^-$.

(a) The cross section for this reaction at neutrino energy $E_\nu$ is approximately $\sigma \approx 9.5 \times 10^{-44}(E_\nu / \text{MeV})^2\,\text{cm}^2$. Calculate $\sigma$ for $E_\nu = 12\,\text{MeV}$ (typical mean energy for $\nu_e$ from a proto-neutron star).

(b) The material behind the stalled shock has density $\rho \approx 10^{10}\,\text{g/cm}^3$ and is approximately half neutrons by mass. Estimate the mean free path of a $12\,\text{MeV}$ electron neutrino in this material.

(c) The gain region extends from $r_1 \approx 80\,\text{km}$ to $r_2 \approx 200\,\text{km}$. Is the material optically thin or thick to neutrinos? What fraction of neutrino energy is deposited?

Problem 23.4 ⭐⭐ A core-collapse supernova releases $3 \times 10^{53}\,\text{erg}$ in neutrinos over approximately 10 seconds.

(a) Calculate the neutrino luminosity in watts and compare it to the solar luminosity ($L_\odot = 3.83 \times 10^{26}\,\text{W}$).

(b) The observable universe contains approximately $2 \times 10^{11}$ galaxies, each with a luminosity of roughly $10^{10}\,L_\odot$. Calculate the total electromagnetic luminosity of the observable universe and compare it to the supernova neutrino luminosity.

(c) Explain why, despite this enormous luminosity, the supernova neutrino signal from SN 1987A (at 51.4 kpc) yielded only 24 detected neutrino events.

Problem 23.5 ⭐⭐ The ${}^{56}\text{Ni}$ decay chain powers the light curve of a supernova.

(a) A supernova produces $0.07\,M_\odot$ of ${}^{56}\text{Ni}$. Calculate the initial radioactive power (in $\text{erg/s}$) from ${}^{56}\text{Ni}$ decay at $t = 0$, using $Q_{\text{Ni}} = 1.72\,\text{MeV}$ per decay and $t_{1/2} = 6.08\,\text{d}$.

(b) At $t = 30\,\text{d}$, most ${}^{56}\text{Ni}$ has decayed to ${}^{56}\text{Co}$. Calculate the radioactive power from ${}^{56}\text{Co}$ decay at this time, using $Q_{\text{Co}} = 3.73\,\text{MeV}$ per decay, $t_{1/2} = 77.2\,\text{d}$, and assuming all the original ${}^{56}\text{Ni}$ is now ${}^{56}\text{Co}$.

(c) At $t = 200\,\text{d}$, the light curve decline rate should follow the ${}^{56}\text{Co}$ half-life. Verify that a decline of $0.98\,\text{mag}$ per 100 days is consistent with $t_{1/2} = 77.2\,\text{d}$.

Type Ia Supernovae

Problem 23.6 ⭐ A Type Ia supernova produces $0.7\,M_\odot$ of ${}^{56}\text{Ni}$.

(a) Calculate the total energy released by the complete ${}^{56}\text{Ni} \to {}^{56}\text{Fe}$ decay chain (using $Q_{\text{total}} = 3.0\,\text{MeV}$ per decay).

(b) Compare this to the typical kinetic energy of the ejecta ($\sim 1.2 \times 10^{51}\,\text{erg}$). What fraction of the kinetic energy is eventually radiated as light powered by radioactive decay?

Problem 23.7 ⭐⭐ The Phillips relation connects Type Ia peak luminosity to the decline rate $\Delta m_{15}$ (magnitude decline in the first 15 days after maximum).

(a) Explain qualitatively why a supernova that produces more ${}^{56}\text{Ni}$ is both brighter and declines more slowly.

(b) If the peak absolute magnitude ranges from $M_B \approx -18.5$ ($\Delta m_{15} = 1.7$) to $M_B \approx -19.8$ ($\Delta m_{15} = 0.9$), estimate the range of ${}^{56}\text{Ni}$ masses produced, assuming $L_{\text{peak}} \propto M_{\text{Ni}}$.

The s-Process

Problem 23.8 ⭐ The neutron source reaction ${}^{13}\text{C}(\alpha, n){}^{16}\text{O}$ operates in AGB stars.

(a) Calculate the Q-value of this reaction. Use: $M({}^{13}\text{C}) = 13.00335\,\text{u}$, $M({}^{4}\text{He}) = 4.00260\,\text{u}$, $M({}^{16}\text{O}) = 15.99491\,\text{u}$, $m_n = 1.00866\,\text{u}$, and $1\,\text{u} = 931.494\,\text{MeV}/c^2$.

(b) Is this reaction exothermic or endothermic? What is the threshold temperature (roughly) for this reaction to proceed at astrophysically significant rates?

Problem 23.9 ⭐⭐ At a neutron density $n_n = 5 \times 10^7\,\text{cm}^{-3}$ and thermal velocity $v_T = 1.38 \times 10^8\,\text{cm/s}$ (appropriate for $kT = 30\,\text{keV}$):

(a) Calculate the neutron capture timescale $\tau_n = 1/(n_n \sigma_n v_T)$ for ${}^{138}\text{Ba}$ ($\sigma_n = 4.0\,\text{mb}$, a magic $N = 82$ nucleus).

(b) Calculate $\tau_n$ for ${}^{139}\text{La}$ ($\sigma_n = 32\,\text{mb}$, one neutron past the $N = 82$ shell closure).

(c) Explain why the ratio of these timescales leads to the s-process abundance peak at $A = 138$.

Problem 23.10 ⭐⭐ The local approximation. In steady-state s-process flow, $\sigma_A N_s(A) \approx \text{constant}$.

(a) If $\sigma_{88}(\text{Sr}) = 6.4\,\text{mb}$ and $\sigma_{89}(\text{Y}) = 19.0\,\text{mb}$, predict the ratio $N_s(88)/N_s(89)$.

(b) Compare your prediction to the solar s-process abundances: $N_s({}^{88}\text{Sr}) = 23.8$ and $N_s({}^{89}\text{Y}) = 10.4$ (in the Si $= 10^6$ scale). How well does the local approximation work?

(c) Why does the local approximation fail near the s-process peaks and at branching points?

Problem 23.11 ⭐⭐⭐ Branching point analysis. The s-process encounters ${}^{85}\text{Kr}$ with $t_{1/2} = 10.76\,\text{yr}$ and $\sigma_n = 60\,\text{mb}$.

(a) Calculate the beta-decay rate $\lambda_\beta$ for ${}^{85}\text{Kr}$.

(b) At what neutron density $n_n$ is $\lambda_n = \lambda_\beta$? (Use $v_T = 1.38 \times 10^8\,\text{cm/s}$.)

(c) In the main component of the s-process ($n_n \approx 3 \times 10^7\,\text{cm}^{-3}$), what fraction of the flow goes through neutron capture versus beta decay? Use $f_n = \lambda_n/(\lambda_n + \lambda_\beta)$.

(d) Explain how this branching ratio affects the production of the s-only isotope ${}^{86}\text{Kr}$.

Problem 23.12 ⭐ Explain why the s-process cannot produce elements heavier than bismuth (${}^{209}\text{Bi}$). What happens when ${}^{209}\text{Bi}$ captures a neutron? Trace the subsequent decay path back to lead.

The r-Process

Problem 23.13 ⭐ The r-process operates at neutron densities $n_n \sim 10^{20}$–$10^{24}\,\text{cm}^{-3}$.

(a) At $n_n = 10^{22}\,\text{cm}^{-3}$ and $\sigma_n = 100\,\text{mb}$ (typical for neutron-rich nuclei), calculate the neutron capture timescale $\tau_n$.

(b) Compare $\tau_n$ to a typical beta-decay half-life of $\sim 100\,\text{ms}$ for nuclei far from stability. Verify that $\lambda_n \gg \lambda_\beta$.

(c) A seed nucleus of ${}^{56}\text{Fe}$ needs to capture approximately $140$ neutrons to reach the actinide region ($A \sim 196$ after beta-decay). Estimate how long this takes at the capture rate calculated in (a).

Problem 23.14 ⭐⭐ Waiting points and the r-process peaks.

(a) Explain why nuclei at magic neutron numbers ($N = 50, 82, 126$) are waiting points on the r-process path. What two physical effects cause the r-process flow to stall there?

(b) After the neutron flux ceases ("freeze-out"), the r-process waiting-point nuclei beta-decay toward stability. If a waiting-point nucleus at $N = 82$ has $Z \approx 40$ (extremely neutron-rich), and the stable nucleus at $N = 82$ has $Z \approx 56$ (barium), how many beta decays occur? What is the final mass number if $A_{\text{initial}} \approx 122$?

(c) Explain why the r-process peak at $A \approx 130$ is shifted to lower mass relative to the s-process peak at $A \approx 138$.

Problem 23.15 ⭐⭐⭐ The $(n,\gamma) \rightleftharpoons (\gamma,n)$ equilibrium. In the r-process, at temperature $T$ and neutron density $n_n$, photodisintegration equilibrium determines the path.

(a) For the reaction $(A,Z) + n \rightleftharpoons (A+1,Z) + \gamma$, the Saha equation gives:

$$\frac{n(A+1,Z)}{n(A,Z)} = n_n \frac{G(A+1,Z)}{2 G(A,Z)} \left(\frac{2\pi\hbar^2}{m_n k_B T}\right)^{3/2} \exp\left(\frac{S_n(A+1,Z)}{k_B T}\right)$$

Show that the condition $n(A+1)/n(A) = 1$ (equal populations) defines a critical neutron separation energy $S_n^{\text{crit}}$ that depends on $T$ and $n_n$.

(b) For $T_9 = 1.5$ ($T = 1.5 \times 10^9\,\text{K}$) and $n_n = 10^{22}\,\text{cm}^{-3}$, neglecting the partition function ratio, estimate $S_n^{\text{crit}}$.

(c) Explain why the r-process path follows a contour of approximately constant $S_n$ on the chart of nuclides.

Problem 23.16 ⭐⭐⭐ Fission recycling. The r-process reaches $A \sim 260$ where nuclei undergo fission.

(a) If a nucleus with $A = 260$, $Z = 100$ fissions into two fragments with $A_1 = 130$, $A_2 = 130$, what are the approximate $Z$ values of the fragments, assuming $Z/A$ is preserved?

(b) These fragments are very neutron-rich ($Z/A \approx 0.385$ vs. $Z/A \approx 0.44$ for stable nuclei at $A = 130$). How many beta decays will each fragment undergo before reaching stability?

(c) Explain how fission recycling can establish a steady-state r-process abundance pattern and contribute to the rare-earth peak at $A \sim 160$.

GW170817 and the Kilonova

Problem 23.17 ⭐ The chirp mass of GW170817 was measured to be $\mathcal{M} = 1.188\,M_\odot$.

(a) If the two neutron stars have equal mass ($m_1 = m_2 = m$), what is $m$? Use $\mathcal{M} = m \cdot 2^{-1/5}$.

(b) The actual mass ratio was $q = m_2/m_1 \approx 0.7$–$1.0$. For $q = 0.85$, find $m_1$ and $m_2$ from the chirp mass formula $\mathcal{M} = (m_1 m_2)^{3/5}/(m_1 + m_2)^{1/5}$ and $m_2 = 0.85\,m_1$.

Problem 23.18 ⭐⭐ The kilonova AT 2017gfo ejected an estimated $M_{\text{ej}} \approx 0.04\,M_\odot$ of r-process material.

(a) If the ejecta has a solar-system-like r-process abundance pattern, approximately 1% of the mass is in elements with $A > 195$ (third r-process peak and actinides). Estimate the mass of gold ($A = 197$) produced, assuming gold is $\sim 10$% of the third-peak material by mass. Express your answer in Earth masses ($M_\oplus = 5.97 \times 10^{27}\,\text{g}$).

(b) The Milky Way has experienced approximately $10^3$–$10^4$ neutron star mergers over its history (based on merger rate estimates of $\sim 80$–$810\,\text{Gpc}^{-3}\text{yr}^{-1}$). Estimate the total mass of gold produced by all mergers in the Galaxy.

(c) The Earth contains approximately $4.8 \times 10^{22}\,\text{g}$ of gold ($\sim 0.5\,\text{ppm}$ by mass). Is this consistent with your estimate?

Problem 23.19 ⭐⭐ The kilonova heating rate follows $\dot{Q}(t) \approx 2 \times 10^{10}\,\text{erg}\,\text{s}^{-1}\,\text{g}^{-1} \times (t/1\,\text{day})^{-1.3}$.

(a) Calculate the luminosity of the kilonova at $t = 1\,\text{day}$ for $M_{\text{ej}} = 0.04\,M_\odot$, assuming the thermalization efficiency $\epsilon_{\text{th}} = 0.5$ (half the decay energy is deposited as heat).

(b) Convert to solar luminosities. Compare to the peak luminosity of a typical Type Ia supernova ($L_{\text{Ia}} \sim 10^{43}\,\text{erg/s}$). Why is it called a "kilo"-nova?

(c) At what time does the kilonova luminosity equal the solar luminosity?

Problem 23.20 ⭐⭐⭐ Lanthanide opacity. The key diagnostic of heavy r-process elements in a kilonova is the large opacity of lanthanide elements.

(a) The opacity of iron-peak elements is $\kappa \sim 0.2\,\text{cm}^2/\text{g}$. The opacity of lanthanide-rich material is $\kappa \sim 10$–$30\,\text{cm}^2/\text{g}$. Explain qualitatively why lanthanides have such large opacities (Hint: consider the atomic electron structure).

(b) The diffusion timescale for photons in the ejecta is $t_{\text{diff}} \sim (\kappa M_{\text{ej}} / v c)^{1/2}$, where $v \approx 0.1c$ is the expansion velocity. Calculate $t_{\text{diff}}$ for lanthanide-poor ($\kappa = 0.5\,\text{cm}^2/\text{g}$) and lanthanide-rich ($\kappa = 10\,\text{cm}^2/\text{g}$) ejecta with $M_{\text{ej}} = 0.02\,M_\odot$ each.

(c) Use your results to explain the two-component (blue + red) kilonova model.

The p-Process

Problem 23.21 ⭐⭐ Explain why the p-nuclei cannot be produced by either the s-process or the r-process. Use ${}^{92}\text{Mo}$ as an example: show that it is shielded from the r-process beta-decay path by ${}^{92}\text{Zr}$ (stable) and explain why the s-process path does not produce it.

Cosmochronology

Problem 23.22 ⭐⭐ The ${}^{187}\text{Re}$–${}^{187}\text{Os}$ chronometer.

${}^{187}\text{Re}$ beta-decays to ${}^{187}\text{Os}$ with $t_{1/2} = 41.6\,\text{Gyr}$.

(a) ${}^{187}\text{Re}$ is produced entirely by the r-process. If its initial abundance was $N_0$ and the present abundance is $N(t)$, write the expression for the age $t$ in terms of $N_0$, $N(t)$, and $\lambda$.

(b) The present-day solar ratio is ${}^{187}\text{Re}/{}^{187}\text{Os} = 0.40$. If the initial ratio was $\approx 0.80$, estimate the age.

(c) What complication does the very long half-life of ${}^{187}\text{Re}$ introduce for this chronometer?

Problem 23.23 ⭐⭐⭐ Th/U chronometry of metal-poor stars. The ultra-metal-poor star CS 31082-001 has measured abundances $\log \epsilon(\text{Th}) = -0.98$ and $\log \epsilon(\text{U}) = -1.92$ (where $\log \epsilon(X) = \log_{10}(N_X/N_H) + 12$).

(a) Calculate the present-day number ratio ${}^{232}\text{Th}/{}^{238}\text{U}$ in this star.

(b) Using the r-process production ratio $({}^{232}\text{Th}/{}^{238}\text{U})_0 = 1.65 \pm 0.20$ and the decay constants $\lambda_{232} = 0.0495\,\text{Gyr}^{-1}$, $\lambda_{238} = 0.1551\,\text{Gyr}^{-1}$, calculate the age of the star.

(c) Estimate the uncertainty in the age arising from the $\pm 0.20$ uncertainty in the production ratio.

Synthesis and Comparison

Problem 23.24 ⭐ Create a table comparing the s-process and r-process along the following dimensions: astrophysical site, neutron density, timescale, path on chart of nuclides, mass range of products, and observational evidence.

Problem 23.25 ⭐⭐ Solar r-process residuals. The s-process contribution to the solar abundance of ${}^{130}\text{Te}$ is negligible (it is an "r-only" isotope — shielded from the s-process by ${}^{130}\text{Xe}$). The solar abundance of ${}^{130}\text{Te}$ is $N_\odot = 4.32$ (Si = $10^6$ scale).

(a) The s-process contribution to ${}^{128}\text{Te}$ is estimated as $N_s(128) = 1.85$, while $N_\odot(128) = 5.08$. Calculate the r-process residual for ${}^{128}\text{Te}$.

(b) Compare the r-process and s-process contributions to tellurium ($Z = 52$). Which process dominates?

(c) Explain why isotopes just below a magic neutron number tend to have larger r-process fractions.

Problem 23.26 ⭐⭐⭐ The ${}^{12}\text{C}(\alpha,\gamma){}^{16}\text{O}$ reaction rate affects the r-process — explain how. (Hint: This reaction, discussed in Chapter 22, determines the C/O ratio at the end of helium burning. How does the C/O ratio of a white dwarf affect Type Ia nucleosynthesis? How does the C/O ratio of a massive star's core affect the neutron excess and hence the r-process seed abundances?)

Problem 23.26b ⭐⭐ SN 1987A neutrino detection. The Kamiokande-II detector contained $M_{\text{det}} = 2140$ tonnes of water and detected 12 events from SN 1987A at distance $D = 51.4\,\text{kpc}$.

(a) Calculate the number of free protons (hydrogen nuclei) in the fiducial volume. Water is $\text{H}_2\text{O}$; what fraction of the mass is hydrogen?

(b) The mean anti-electron neutrino energy was $\langle E_{\bar{\nu}_e} \rangle \approx 15\,\text{MeV}$ and the cross section at this energy is $\sigma \approx 1.5 \times 10^{-43}\,\text{cm}^2$. If the total $\bar{\nu}_e$ energy was $E_{\bar{\nu}_e} \approx 5 \times 10^{52}\,\text{erg}$, calculate the neutrino fluence (neutrinos per $\text{cm}^2$) at the detector.

(c) Estimate the expected number of detected events and compare to the 12 observed. What does the agreement tell us about the total neutrino energy budget?

Problem 23.26c ⭐⭐⭐ The s-process neutron exposure. The neutron exposure $\tau$ is defined as $\tau = \int_0^t n_n v_T \, dt'$, measured in $\text{mb}^{-1}$ (with the convention that $1\,\text{mb}^{-1} = 10^{27}\,\text{cm}^2$).

(a) Show that in the local approximation with an exponential exposure distribution $\rho(\tau) = (GN_0/\tau_0)\exp(-\tau/\tau_0)$, the s-process abundance of isotope $A$ is:

$$N_s(A) = \frac{GN_0}{\tau_0 \sigma_A}\prod_{i=56}^{A-1}\frac{1}{1 + 1/(\sigma_i \tau_0)}$$

where $G$ is a normalization constant, $N_0$ is the seed abundance, and $\tau_0$ is the mean exposure.

(b) For the main s-process component, $\tau_0 \approx 0.30\,\text{mb}^{-1}$. At a magic neutron number where $\sigma_A \approx 4\,\text{mb}$, calculate $\sigma_A \tau_0$. Is this large or small compared to 1, and what does this imply for the abundance at the magic number?

(c) The $\sigma N_s$ product for s-only isotopes near the second s-process peak includes: $\sigma({}^{134}\text{Ba}) \cdot N_s({}^{134}\text{Ba}) \approx 590\,\text{mb} \cdot N_\odot$ and $\sigma({}^{136}\text{Ba}) \cdot N_s({}^{136}\text{Ba}) \approx 455\,\text{mb} \cdot N_\odot$. Are these values approximately equal? Discuss what the deviation from perfect equality tells us about the s-process model.

Problem 23.26d ⭐⭐ Supernova classification. Complete the following table, distinguishing the two main types of supernovae:

Hint for host galaxy preference: core-collapse supernovae require recent star formation (why?), while Type Ia can occur in any galaxy type (why?).

Computational and Research Problems

Problem 23.26e ⭐⭐⭐ r-Process freeze-out and final abundances. When the neutron flux ceases in a neutron star merger, the r-process "freezes out" and the extremely neutron-rich nuclei beta-decay toward stability.

(a) Consider a nucleus at the $N = 82$ waiting point with $Z = 44$ (ruthenium), so $A = 126$. The stable isobar at $A = 126$ is ${}^{126}\text{Xe}$ ($Z = 54$). How many beta decays must occur? If the average beta-decay half-life along this chain is $\sim 0.5\,\text{s}$, estimate the total decay time.

(b) Some of these beta decays may emit one or more neutrons ("beta-delayed neutron emission," with probability $P_n$). If $P_n \approx 0.2$ (averaged over the decay chain), how does this modify the final mass number? Where does the "missing" mass number go?

(c) Explain why beta-delayed neutron emission smooths the r-process abundance peaks and redistributes material to slightly lower $A$ values. Why is this effect important for matching the observed peak shapes?

Problem 23.26f ⭐⭐ The Galactic r-process budget. The total mass of r-process elements in the Milky Way can be estimated from the solar r-process fraction and the total baryonic mass of the Galaxy.

(a) The mass of the Milky Way's interstellar medium plus stars is approximately $M_{\text{baryonic}} \sim 6 \times 10^{10}\,M_\odot$. If the r-process elements constitute approximately $3 \times 10^{-8}$ of the total mass by weight (an estimate from the solar r-process abundances), calculate the total mass of r-process elements in the Galaxy.

(b) If each neutron star merger ejects $M_{\text{ej}} = 0.04\,M_\odot$ of r-process material, how many mergers are needed to produce the total Galactic r-process inventory?

(c) The age of the Galaxy is $\sim 13\,\text{Gyr}$. What average merger rate (per year) is required? Compare to the LIGO/Virgo estimate of $10^{-5}$–$10^{-4}\,\text{yr}^{-1}$ for the Milky Way. Is the budget consistent?

Problem 23.27 ⭐⭐ (Computational) Modify the r_process_path.py script to overlay the solar system abundance pattern on the chart of nuclides, using circle sizes proportional to $\log N_\odot(A)$. Identify the s-process and r-process peaks visually.

Problem 23.28 ⭐⭐⭐ (Computational) Write a simple s-process network code that follows a single neutron exposure on an iron seed. Use the local approximation to calculate $N_s(A)$ for $A = 56$ to $A = 209$ with the following simplifications:

• Path follows the valley of stability (no branching)
• Use experimental cross sections from KADoNiS (Karlsruhe Astrophysical Database of Nucleosynthesis in Stars) for the s-only isotopes
• Assume a single exponential neutron exposure $\rho(\tau) \propto \exp(-\tau/\tau_0)$

Plot the resulting $\sigma N_s$ curve and compare to published solar s-process abundances.

Problem 23.29 ⭐⭐⭐⭐ (Research) Read the original GW170817 discovery paper: Abbott, B.P. et al. (LIGO/Virgo), "GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral," Physical Review Letters 119, 161101 (2017).

(a) How was the chirp mass measured? What is the relationship between the gravitational-wave frequency evolution $\dot{f}$ and the chirp mass?

(b) What constraints did the gravitational-wave signal place on the neutron star equation of state through the tidal deformability parameter $\tilde{\Lambda}$?

(c) How many electromagnetic observatories participated in the follow-up campaign? What wavelengths were covered?

Problem 23.30 ⭐⭐⭐⭐ (Research) The "universality" of the r-process abundance pattern — the observation that r-process-enriched metal-poor stars have remarkably similar abundance patterns for $Z \geq 56$ — is one of the most striking features of r-process nucleosynthesis. Read Sneden et al. (2008, Annual Review of Astronomy and Astrophysics 46, 241) and:

(a) Summarize the evidence for universality in the range $56 \leq Z \leq 76$.

(b) Does universality extend to the actinides ($Z \geq 89$)? What is the "actinide boost" phenomenon?

(c) What does universality (or its breakdown) tell us about the r-process site and conditions?

Worked Solutions to Selected Problems

Solution 23.1(a). $M_{\text{Ch}} = 1.44(0.42/0.5)^2 = 1.44 \times 0.706 = 1.02\,M_\odot$.

Solution 23.1(b). In nuclear statistical equilibrium at $T \sim 4 \times 10^9\,\text{K}$, the dominant iron-peak nuclei include ${}^{56}\text{Fe}$ ($Y_e = 26/56 = 0.464$), ${}^{54}\text{Fe}$ ($Y_e = 0.481$), ${}^{58}\text{Ni}$ ($Y_e = 0.483$), and ${}^{52}\text{Cr}$ ($Y_e = 0.462$). These nuclei all have $N > Z$ (more neutrons than protons), so the electron fraction $Y_e$ is less than 0.5. NSE favors these neutron-rich species because the asymmetry energy is minimized when $N \approx Z + (A/200)^2$ (from the SEMF), and for iron-peak nuclei the Coulomb energy penalty for higher $Z$ pushes the equilibrium toward lower $Y_e$.

Solution 23.1(c). At $Y_e = 0.36$: $M_{\text{Ch}} = 1.44(0.36/0.5)^2 = 1.44 \times 0.518 = 0.75\,M_\odot$. The ratio is $0.75/1.02 = 0.73$ — a 27% reduction. This is why electron capture accelerates the collapse: it reduces both the Chandrasekhar mass and the degeneracy pressure simultaneously.

Solution 23.8(a). $Q = [M({}^{13}\text{C}) + M({}^{4}\text{He}) - M({}^{16}\text{O}) - m_n] \times 931.494\,\text{MeV/u}$ $= [13.00335 + 4.00260 - 15.99491 - 1.00866]\,\text{u} \times 931.494$ $= [0.00238]\,\text{u} \times 931.494 = +2.22\,\text{MeV}$. The reaction is exothermic with $Q = +2.22\,\text{MeV}$. However, the Coulomb barrier between ${}^{13}\text{C}$ and ${}^{4}\text{He}$ requires temperatures of $T \sim 10^8\,\text{K}$ ($kT \sim 10\,\text{keV}$) for significant Gamow-peak penetration.

Solution 23.13(a). $\tau_n = 1/(n_n \sigma_n v_T) = 1/(10^{22} \times 100 \times 10^{-27} \times 10^8) = 1/(10^{22} \times 10^{-25} \times 10^8) = 1/(10^5) = 10^{-5}\,\text{s} = 10\,\mu\text{s}$.

Solution 23.13(b). For $t_{1/2} = 100\,\text{ms}$: $\lambda_\beta = \ln 2 / 0.1\,\text{s} = 6.93\,\text{s}^{-1}$. Meanwhile, $\lambda_n = 1/\tau_n = 10^5\,\text{s}^{-1}$. Thus $\lambda_n / \lambda_\beta \approx 1.4 \times 10^4$, confirming $\lambda_n \gg \lambda_\beta$.

Solution 23.13(c). At $\tau_n = 10\,\mu\text{s}$ per capture, 140 captures take $140 \times 10\,\mu\text{s} = 1.4\,\text{ms}$. The entire r-process, from seed to actinide, runs to completion in about a millisecond. This is why the r-process requires only seconds of neutron-rich conditions — the nuclear reactions themselves are fast; it is the supply of free neutrons that must be maintained.

Solution 23.17(a). For equal masses: $\mathcal{M} = m \cdot 2^{-1/5}$, so $m = \mathcal{M} \cdot 2^{1/5} = 1.188 \times 1.149 = 1.365\,M_\odot$. Each neutron star has mass $\approx 1.37\,M_\odot$.

Solution 23.22(b). $N(t) = N_0 e^{-\lambda t}$, so $t = (1/\lambda)\ln(N_0/N(t))$. If ${}^{187}\text{Re}/{}^{187}\text{Os} = 0.40$ now and was $0.80$ initially, then $N_{\text{Re}}(t)/N_{\text{Re}}(0) = 0.40/0.80 = 0.50$ (assuming the osmium increase matches the rhenium decrease). Then $t = (t_{1/2}/\ln 2)\ln(1/0.50) = 41.6\,\text{Gyr} \times 1.0 = 41.6\,\text{Gyr}$. This is unreasonably large — it exceeds the age of the universe! The resolution is that ${}^{187}\text{Os}$ has other sources besides ${}^{187}\text{Re}$ decay (it is produced by the s-process), making the Re-Os system more complex than a simple parent-daughter pair. This is the complication asked for in part (c).