Exercises — Chapter 22

Use the following constants unless otherwise stated: $m_p = 938.272$ MeV/$c^2$, $m_n = 939.565$ MeV/$c^2$, $m_\alpha = 3727.379$ MeV/$c^2$, $m_e = 0.511$ MeV/$c^2$, $k_B = 8.617 \times 10^{-5}$ eV/K, $\hbar c = 197.3$ MeV fm, $e^2/(4\pi\epsilon_0) = 1.440$ MeV fm, $1$ MeV $= 1.602 \times 10^{-13}$ J, $L_\odot = 3.83 \times 10^{33}$ erg/s, $M_\odot = 1.989 \times 10^{33}$ g.

Reference nuclear binding energies (AME2020):

Section A: Hydrogen Burning (Fundamental)

Problem 22.1 ⭐ Calculate the Q-value for each step of the PP-I chain: (a) $p + p \to d + e^+ + \nu_e$ (b) $d + p \to {}^3\text{He} + \gamma$ (c) ${}^3\text{He} + {}^3\text{He} \to {}^4\text{He} + 2p$

Then verify that the total Q-value for the net reaction $4p \to {}^4\text{He} + 2e^+ + 2\nu_e$ is $Q = 26.732$ MeV. (Include the annihilation energy of the two positrons: $2 \times 2m_e c^2 = 2.044$ MeV.)

Problem 22.2 ⭐ The Sun's luminosity is $L_\odot = 3.83 \times 10^{33}$ erg/s, and approximately 98% comes from the pp chain with an average energy release of $\sim 26.1$ MeV per ${}^4$He produced (after subtracting the average neutrino energy).

(a) Calculate the number of ${}^4$He nuclei produced per second.

(b) Calculate the rate of hydrogen consumption (in kg/s and $M_\odot$/yr).

(c) The Sun's total hydrogen mass is approximately $0.7 \times M_\odot$. At the current burning rate, how long could the Sun sustain its luminosity? (In practice, only about 10% of the hydrogen is in the core; adjust your estimate accordingly.)

(d) Calculate the total neutrino flux at the Earth's surface (distance $= 1$ AU $= 1.496 \times 10^{13}$ cm). Compare to the measured value of $\sim 6.5 \times 10^{10}$ cm$^{-2}$ s$^{-1}$.

Problem 22.3 ⭐ The energy generation rate for the pp chain near solar conditions is approximately:

$$\epsilon_{pp} \approx 1.08 \times 10^{-5} \rho X^2 T_6^{3.89} \quad \text{erg g}^{-1}\text{s}^{-1}$$

where $\rho$ is in g/cm$^3$ and $T_6 = T/(10^6 \text{ K})$.

(a) Calculate $\epsilon_{pp}$ at the solar center ($T_6 = 15.7$, $\rho = 150$ g/cm$^3$, $X = 0.34$).

(b) Compare this to the average metabolic rate of the human body ($\sim 1.2$ erg g$^{-1}$ s$^{-1}$). The solar core produces less energy per gram than you do!

(c) If the temperature increases by 10%, by what factor does $\epsilon_{pp}$ change?

Problem 22.4 ⭐ The pp-CNO crossover. Using the approximate energy generation rates:

$$\epsilon_{pp} = C_{pp} \rho X^2 T_6^4, \qquad \epsilon_{\text{CNO}} = C_{\text{CNO}} \rho X X_{\text{CNO}} T_6^{16}$$

with $C_{pp} / C_{\text{CNO}} \approx 9.5 \times 10^{8}$ and $X_{\text{CNO}} = 0.015$ (solar metallicity):

(a) Derive the crossover temperature $T_{\text{cross}}$ where $\epsilon_{pp} = \epsilon_{\text{CNO}}$.

(b) Calculate $T_{\text{cross}}$ numerically. Does it agree with the value $T \approx 17 \times 10^6$ K stated in the text?

(c) What is the crossover temperature for a metal-poor star with $X_{\text{CNO}} = 0.001$? What does this imply about the dominant burning mechanism in the first generation of stars?

Problem 22.5 ⭐⭐ The CNO cycle. (a) Write out all six reactions of the CNO-I cycle and verify that the net reaction is $4p \to {}^4$He $+ 2e^+ + 2\nu_e$.

(b) Calculate the total Q-value using the binding energy data above.

(c) In CNO-I equilibrium, nearly all the CNO catalyst nuclei are in the form of ${}^{14}$N. Explain why, using the S-factor values given in the text.

(d) The mass fraction of ${}^{14}$N in the solar photosphere is $\sim 7 \times 10^{-4}$. In the core of a massive star that has completed CNO hydrogen burning, what would you expect the ${}^{14}$N mass fraction to be relative to the original C+N+O mass fraction?

Section B: Helium Burning and the Triple-Alpha Process

Problem 22.6 ⭐ (a) Show that the Q-value for ${}^4$He $+ {}^4$He $\to {}^8$Be is $Q = B({}^8\text{Be}) - 2B({}^4\text{He}) = -0.092$ MeV using the binding energies in the table.

(b) Calculate the lifetime of ${}^8$Be from its width $\Gamma = 5.57$ eV using $\tau = \hbar / \Gamma$. Express your answer in seconds.

(c) At $T = 10^8$ K and $\rho = 10^5$ g/cm$^3$ (typical helium-burning conditions), the equilibrium ratio $n({}^8\text{Be})/n(\alpha) \sim 10^{-9}$. If the helium core has a mass of $\sim 0.5 \, M_\odot$, estimate the total mass of ${}^8$Be present at any instant (in grams).

Problem 22.7 ⭐⭐ The Q-value for the complete triple-alpha process is:

$$3 \, {}^4\text{He} \to {}^{12}\text{C} + \gamma \qquad Q = B({}^{12}\text{C}) - 3B({}^{4}\text{He})$$

(a) Calculate $Q$ numerically.

(b) The Hoyle state is at $E_x = 7.654$ MeV in ${}^{12}$C. Calculate the energy of the Hoyle state above the $3\alpha$ threshold. (The threshold is at $Q_{3\alpha} = -Q$ above the ${}^{12}$C ground state.)

(c) The resonance energy is $E_r = 379.47$ keV above the ${}^8$Be$+\alpha$ threshold. At what temperature does $k_BT = E_r$? At what temperature does $E_r/(k_BT) = 10$ (a rough criterion for significant but not dominant resonance contribution)?

Problem 22.8 ⭐⭐ The triple-alpha energy generation rate. Near $T_8 = 1$, the energy generation rate is often parameterized as:

$$\epsilon_{3\alpha} = 5.1 \times 10^8 \rho^2 Y^3 T_8^{-3} \exp(-44.0/T_8) \quad \text{erg g}^{-1}\text{s}^{-1}$$

(a) Calculate $\epsilon_{3\alpha}$ at $T_8 = 1$ and $T_8 = 2$ for $\rho = 10^5$ g/cm$^3$, $Y = 1.0$.

(b) Compute the ratio $\epsilon_{3\alpha}(T_8 = 2) / \epsilon_{3\alpha}(T_8 = 1)$. What effective power-law exponent $n$ in $\epsilon \propto T^n$ does this correspond to?

(c) Show that the local power-law exponent at temperature $T_8$ is $n = -3 + 44.0/T_8$.

(d) Calculate $n$ at $T_8 = 1$ and $T_8 = 2$. Compare to the exponents for pp ($\sim 4$) and CNO ($\sim 16$).

Problem 22.9 ⭐⭐⭐ The anthropic argument for the Hoyle state. Suppose the Hoyle state were located at $E_x' = E_x + \Delta E$ instead of its actual $E_x = 7654.2$ keV.

(a) Write an expression for the triple-alpha rate as a function of $\Delta E$, assuming the resonance energy above the $3\alpha$ threshold shifts by $\Delta E$.

(b) By what factor does the triple-alpha rate change if $\Delta E = +300$ keV (Hoyle state is higher)?

(c) By what factor does it change if $\Delta E = -300$ keV? (If the resonance moves below threshold, it becomes a subthreshold resonance. Qualitatively, what happens to the resonance contribution?)

(d) Discuss: does the sensitivity of the triple-alpha rate to the Hoyle state energy constitute evidence for fine-tuning? What are the scientific limits of this argument?

Problem 22.10 ⭐⭐ ${}^{12}$C$(\alpha,\gamma){}^{16}$O. (a) Calculate the Q-value for ${}^{12}$C$(\alpha,\gamma){}^{16}$O.

(b) The Gamow energy for this reaction at $T = 2 \times 10^8$ K is approximately $E_0 \approx 300$ keV. Verify this using $E_0 = (bk_BT/2)^{2/3}$ with $b = 31.29 Z_1 Z_2 (\mu/\text{MeV})^{1/2}$ keV$^{1/2}$, where $\mu = m_1 m_2/(m_1 + m_2)$ is the reduced mass.

(c) The $1^-$ subthreshold state in ${}^{16}$O is at $E_x = 7.117$ MeV, which is $45$ keV below the ${}^{12}$C$+\alpha$ threshold ($S_\alpha = 7.162$ MeV). Explain qualitatively why this state significantly enhances the E1 component of the S-factor even though it is below threshold.

Section C: Advanced Burning Stages

Problem 22.11 ⭐ For carbon burning, calculate the Q-values for the three main channels:

(a) ${}^{12}\text{C} + {}^{12}\text{C} \to {}^{20}\text{Ne} + \alpha$

(b) ${}^{12}\text{C} + {}^{12}\text{C} \to {}^{23}\text{Na} + p$

(c) ${}^{12}\text{C} + {}^{12}\text{C} \to {}^{23}\text{Mg} + n$ (Use $B({}^{23}\text{Mg}) = 181.726$ MeV.)

Verify that the dominant channels (a) and (b) are exothermic.

Problem 22.12 ⭐⭐ Neon burning as photodisintegration.

(a) Calculate the Q-value for $\gamma + {}^{20}\text{Ne} \to {}^{16}\text{O} + \alpha$.

(b) Calculate the Q-value for ${}^{20}\text{Ne} + \alpha \to {}^{24}\text{Mg} + \gamma$.

(c) Show that the net reaction $2 \, {}^{20}\text{Ne} \to {}^{16}\text{O} + {}^{24}\text{Mg}$ is exothermic, and calculate $Q_{\text{net}}$.

(d) At $T = 1.5 \times 10^9$ K, calculate the ratio $Q_{\text{photodisintegration}} / (k_BT)$. What does this tell you about the photodisintegration rate?

Problem 22.13 ⭐⭐ Coulomb barriers and burning temperatures. For each of the following reactions, calculate the Coulomb barrier height $V_C = Z_1 Z_2 e^2 / [4\pi\epsilon_0 (R_1 + R_2)]$ using $R_i = 1.2 A_i^{1/3}$ fm:

(a) $p + p$ (b) ${}^4$He $+ {}^4$He (c) ${}^{12}$C $+ {}^{12}$C (d) ${}^{16}$O $+ {}^{16}$O (e) ${}^{28}$Si $+ {}^{28}$Si

Then calculate the Gamow peak energy $E_0$ at the characteristic temperature for each burning stage and the ratio $E_0/V_C$. Verify that all reactions proceed by quantum tunneling ($E_0 \ll V_C$).

Problem 22.14 ⭐⭐ Silicon burning and NSE. (a) Explain why ${}^{28}$Si $+ {}^{28}$Si $\to {}^{56}$Ni does not occur, despite the large positive Q-value.

(b) Calculate $Q$ for $\gamma + {}^{28}\text{Si} \to {}^{24}\text{Mg} + \alpha$.

(c) At $T = 3.5 \times 10^9$ K, estimate the photodisintegration rate parameter $\exp(-Q/k_BT)$ for this reaction.

(d) In NSE at $T = 4 \times 10^9$ K with $Y_e = 0.50$, the composition is dominated by ${}^{56}$Ni. Using the binding energies in the table, show that ${}^{56}$Ni has a higher binding energy than ${}^{56}$Fe but a lower $B/A$.

Problem 22.15 ⭐⭐ The ${}^{56}$Ni decay chain.

(a) Write the full decay chain ${}^{56}$Ni $\to {}^{56}$Co $\to {}^{56}$Fe, identifying each decay mode (electron capture).

(b) If a supernova produces $0.6 \, M_\odot$ of ${}^{56}$Ni, calculate the initial radioactive power (in $L_\odot$) from ${}^{56}$Ni decay. The average energy deposited per decay is $\sim 1.7$ MeV.

(c) Show that after $\sim 60$ days, the dominant power source transitions from ${}^{56}$Ni decay ($\tau_{1/2} = 6.08$ d) to ${}^{56}$Co decay ($\tau_{1/2} = 77.24$ d). Sketch the expected luminosity decline on a semi-logarithmic plot.

(d) The observed late-time decline rate of SN 1987A was $0.98 \pm 0.01$ magnitudes per 100 days. Calculate the expected decline rate from ${}^{56}$Co decay. (Recall: 1 magnitude corresponds to a factor of $10^{0.4} = 2.512$ in luminosity.)

Section D: Timescales and Stellar Structure

Problem 22.16 ⭐ The timescale cascade. For a $25 \, M_\odot$ star, the burning timescales are approximately: H: $7 \times 10^6$ yr, He: $5 \times 10^5$ yr, C: 600 yr, Ne: 1 yr, O: 6 months, Si: 1 day.

(a) Express all timescales in seconds.

(b) Calculate the ratio of each successive timescale to the previous one. Is the acceleration approximately geometric (constant ratio)?

(c) If you plotted the burning timescale as a function of the core temperature (or the burning stage index) on a semi-logarithmic scale, what would the plot look like?

Problem 22.17 ⭐⭐ Energy budget of a massive star. A $25 \, M_\odot$ star has a hydrogen mass fraction $X = 0.70$ and a core mass of $\sim 10 \, M_\odot$ that undergoes nuclear processing.

(a) Calculate the total nuclear energy available from hydrogen burning: $E_H = 0.007 \times M_{\text{core}} \times X \times c^2$. (The factor 0.007 is the fraction of rest mass converted to energy in H $\to$ He.)

(b) Calculate the equivalent for helium burning (He $\to$ C/O), using $\Delta(B/A) \approx 0.6$ MeV/nucleon.

(c) Calculate the equivalent for all burning from C to Fe, using $\Delta(B/A) \approx 0.3$ MeV/nucleon for the $\sim 2 \, M_\odot$ that undergoes these stages.

(d) Show that hydrogen burning provides $\sim 90\%$ of the total nuclear energy, yet the post-hydrogen stages collectively last only $\sim 7\%$ of the stellar lifetime. Explain this paradox in terms of neutrino losses.

Problem 22.18 ⭐⭐ Neutrino luminosity. At temperatures above $\sim 10^9$ K, neutrino losses from thermal processes (pair annihilation, plasmon decay, etc.) dominate the energy budget.

(a) The pair-annihilation neutrino energy loss rate is approximately $\epsilon_\nu^{\text{pair}} \sim 4.9 \times 10^{18} T_9^9$ erg g$^{-1}$ s$^{-1}$ at low density. Calculate $\epsilon_\nu^{\text{pair}}$ at $T_9 = 2$ (oxygen burning) and $T_9 = 4$ (silicon burning).

(b) Compare these neutrino loss rates to the nuclear energy generation rates at the same temperatures (estimate from the burning timescales and available fuel).

(c) During silicon burning, approximately what fraction of the nuclear energy generated is lost to neutrinos rather than radiated as photons?

Problem 22.19 ⭐⭐⭐ The helium flash. In a low-mass star ($M \approx 1 \, M_\odot$), helium ignites in a degenerate core.

(a) Explain why degenerate matter does not expand when heated. (Hint: the pressure of a degenerate electron gas is independent of temperature.)

(b) In non-degenerate matter, heating increases pressure, which causes expansion, which causes cooling — a thermostat. Why does this mechanism fail in degenerate matter?

(c) The triple-alpha rate scales as $\sim T^{41}$ near $T_8 = 1$. If the temperature increases by 1% in degenerate matter, by what factor does the energy generation rate increase?

(d) Explain qualitatively how this leads to a thermonuclear runaway (the helium flash). Why does the flash eventually terminate? (Hint: at sufficiently high temperature, the degeneracy is lifted.)

Section E: Synthesis and Analysis

Problem 22.20 ⭐⭐⭐ The C/O ratio and white dwarf composition. A $3 \, M_\odot$ star ends its life with a C/O white dwarf of mass $\sim 0.6 \, M_\odot$.

(a) If the ${}^{12}$C$(\alpha,\gamma){}^{16}$O rate were 50% higher, qualitatively how would the C/O ratio change?

(b) How would this affect: (i) the cooling rate of the white dwarf, (ii) the ignition conditions for a Type Ia supernova if the white dwarf accretes to the Chandrasekhar mass, and (iii) the nucleosynthesis yields of the supernova?

(c) Why has the ${}^{12}$C$(\alpha,\gamma){}^{16}$O rate been called "the most important reaction in nuclear astrophysics"? Summarize the downstream consequences in 3–4 sentences.

Problem 22.21 ⭐⭐⭐ Why not fuse iron? Consider the hypothetical reaction ${}^{56}$Fe $+ {}^{56}$Fe $\to {}^{112}$something.

(a) Using the SEMF (with $a_V = 15.75$, $a_S = 17.80$, $a_C = 0.711$, $a_{\text{sym}} = 23.7$ MeV), estimate $B(Z=52, A=112)$. (Choose $Z = 52$ as a compromise between the SEMF's predicted most stable isobar and charge conservation.)

(b) Calculate the Q-value for ${}^{56}$Fe $+ {}^{56}$Fe $\to {}^{112}$Te (using your SEMF estimate for ${}^{112}$Te).

(c) Is this reaction exothermic or endothermic? Explain the result in terms of the $B/A$ curve.

(d) Calculate the Coulomb barrier for ${}^{56}$Fe $+ {}^{56}$Fe. At what temperature would the Gamow peak energy reach 1% of the barrier? Is this temperature physically achievable in a star?

Problem 22.22 ⭐⭐ Reading the onion shells. A pre-supernova stellar model gives the following composition at different enclosed masses:

(a) Identify the burning stage that produced each shell.

(b) Why does the O shell contain ${}^{24}$Mg but not ${}^{28}$Si? What is the boundary temperature between neon/carbon burning products and oxygen burning products?

(c) Why is ${}^{14}$N prominent in the helium shell? Where did it come from?

(d) Estimate the iron core mass. Is it near the Chandrasekhar mass ($\sim 1.4 \, M_\odot$)? What is the significance of this?

Problem 22.23 ⭐⭐⭐ Nuclear statistical equilibrium. In NSE, the abundance of nucleus $(Z, A)$ relative to free nucleons is set by the binding energy.

(a) At $T = 5 \times 10^9$ K ($k_BT = 431$ keV) and $Y_e = 0.50$, which nucleus is most favored by the factor $\exp(B/k_BT)$: ${}^{56}$Ni or ${}^{56}$Fe? Calculate the ratio.

(b) At $T = 10 \times 10^9$ K, is the NSE composition dominated by heavy nuclei or by free nucleons? Estimate $B/(k_BT)$ for ${}^{56}$Ni and consider the entropy/partition function effects.

(c) At $T = 3 \times 10^9$ K, the NSE composition is strongly peaked around $A \sim 56$. Why is the distribution so narrow? (Hint: consider how steeply $B/A$ varies near the peak.)

Problem 22.24 ⭐⭐⭐ Minimum stellar mass for each burning stage. Burning ignites when the core temperature reaches the threshold. Stars below a critical mass never reach that temperature.

(a) The minimum main-sequence mass for carbon ignition is $\sim 8 \, M_\odot$. Stars below this mass become C/O white dwarfs. Calculate the Chandrasekhar mass $M_{\text{Ch}} = 1.44 (2Y_e)^2 \, M_\odot$ for $Y_e = 0.50$ (equal C and O) and argue that the C/O core mass at the end of helium burning in a $\sim 8 \, M_\odot$ star is near $M_{\text{Ch}}$.

(b) The minimum mass for neon ignition is about the same as for carbon ignition. Why? (Consider: does the core structure change qualitatively between carbon and neon burning?)

(c) Stars of $8$–$10 \, M_\odot$ may ignite carbon in a degenerate core (similar to the helium flash in lower-mass stars). What are the possible outcomes of degenerate carbon ignition?

Section F: Computational and Estimation Problems

Problem 22.25 ⭐ Quick estimates. For each of the following, give an order-of-magnitude estimate and briefly explain your reasoning:

(a) The number of pp reactions per second in the solar core.

(b) The mass of ${}^{12}$C produced per second in a red giant star with helium-burning luminosity $L_{\text{He}} = 100 \, L_\odot$.

(c) The total mass of iron-peak elements produced by a single core-collapse supernova.

(d) The number of neutrinos passing through your body per second from the Sun (assume cross-sectional area $\sim 1$ m$^2$).

Problem 22.26 ⭐⭐ (Programming) Use or extend the stellar_burning.py code to:

(a) Plot $\epsilon_{pp}$ and $\epsilon_{\text{CNO}}$ as functions of temperature from $T_6 = 5$ to $T_6 = 40$. Identify the crossover temperature.

(b) Plot the triple-alpha rate $\epsilon_{3\alpha}$ as a function of $T_8$ from 0.5 to 3.0 on a logarithmic scale. Verify the extreme temperature sensitivity.

(c) Compare the temperature dependence (power-law exponent $n$) of all three rates on a single plot of $n$ vs. $T$.

Problem 22.27 ⭐⭐ (Programming) Implement a simple calculation of the ${}^8$Be equilibrium abundance:

(a) Plot $n({}^8\text{Be})/n(\alpha)$ as a function of temperature from $T_8 = 0.5$ to $T_8 = 3.0$ at $\rho = 10^5$ g/cm$^3$.

(b) At what temperature does $n({}^8\text{Be})/n(\alpha) = 10^{-8}$? At what temperature does it reach $10^{-6}$?

(c) Overlay the triple-alpha rate on the same plot (with a secondary y-axis) and discuss the correlation.

Section G: Integrated Quantitative Problems

Problem 22.25a ⭐⭐⭐ Energy yield per burning stage. For each stellar burning stage, calculate the energy released per nucleon processed:

(a) Hydrogen burning: the initial fuel is ${}^1$H ($B/A = 0$) and the product is ${}^4$He ($B/A = 7.074$ MeV). Calculate $\Delta(B/A)$ and hence the fractional rest-mass energy release $\Delta m/m$.

(b) Helium burning: the fuel is ${}^4$He ($B/A = 7.074$ MeV) and the products are roughly equal parts ${}^{12}$C ($B/A = 7.680$ MeV) and ${}^{16}$O ($B/A = 7.976$ MeV). Calculate the average $\Delta(B/A)$.

(c) Carbon burning: fuel ${}^{12}$C ($B/A = 7.680$), primary product ${}^{20}$Ne ($B/A = 8.032$). Calculate $\Delta(B/A)$.

(d) Silicon burning (to Fe-peak): fuel ${}^{28}$Si ($B/A = 8.448$), product ${}^{56}$Ni ($B/A = 8.643$). Calculate $\Delta(B/A)$.

(e) Plot $\Delta(B/A)$ vs. burning stage. Show that hydrogen burning provides the overwhelming majority of the total nuclear energy and that each successive stage contributes less. Explain why this, combined with neutrino losses, drives the timescale cascade.

Problem 22.25b ⭐⭐ Solar neutrino energetics. The Sun produces neutrinos with different energy spectra from each branch of the pp chain.

(a) For PP-I, each completed cycle produces two neutrinos from $p + p \to d + e^+ + \nu_e$, each with a continuous spectrum up to $E_{\nu,\text{max}} = 0.420$ MeV. The average neutrino energy is $\langle E_\nu \rangle \approx 0.265$ MeV. Calculate the total energy carried away by neutrinos per PP-I cycle as a fraction of the total Q-value (26.732 MeV).

(b) For PP-II, the ${}^7$Be electron capture produces a monoenergetic neutrino at 0.862 MeV (90%) or 0.384 MeV (10%). Calculate the average neutrino energy loss per PP-II cycle (include the pp neutrino from the initial step).

(c) For PP-III, the ${}^8$B beta decay produces neutrinos with average energy $\langle E_\nu \rangle \approx 6.7$ MeV. What fraction of the total energy is carried away by neutrinos in this branch?

(d) Using the solar branching ratios (PP-I: 83.3%, PP-II: 16.7%, PP-III: 0.015%), calculate the luminosity-weighted average neutrino energy loss fraction for the Sun.

Problem 22.25c ⭐⭐⭐ Sensitivity of the C/O ratio. The C/O ratio at the end of helium burning depends on the competition between the triple-alpha process (which creates ${}^{12}$C) and the ${}^{12}$C$(\alpha,\gamma){}^{16}$O reaction (which destroys ${}^{12}$C and creates ${}^{16}$O).

(a) Write a simplified differential equation system for the mass fractions $X_{12}$ (carbon-12) and $X_{16}$ (oxygen-16) during helium burning:

$$\frac{dX_{12}}{dt} = f_{3\alpha}(\rho, T, Y) - f_{12\text{C}(\alpha,\gamma)}(\rho, T, X_{12}, Y)$$

Identify the terms that produce and destroy carbon.

(b) Qualitatively, the C/O ratio at helium exhaustion depends on the ratio $r = \langle \sigma v \rangle_{12\text{C}(\alpha,\gamma)} / \langle \sigma v \rangle_{3\alpha}$. If $r$ is large, is the result carbon-rich or oxygen-rich? Explain.

(c) The recommended ${}^{12}$C$(\alpha,\gamma){}^{16}$O S-factor has an uncertainty of $\sim 24\%$. If $S(300)$ increases from 162 to 200 keV b (a $\sim 23\%$ increase), estimate the fractional change in the C/O ratio, given the approximate relation C/O $\propto S(300)^{-1.5}$ found in stellar models.

(d) Why would a $\sim 24\%$ uncertainty in a single nuclear reaction rate be considered "the most important unsolved problem in nuclear astrophysics"? List three astrophysical observables that depend on the C/O ratio.

Problem 22.25d ⭐⭐ The odd-even effect in nucleosynthesis products. The cosmic abundance pattern shows a pronounced "sawtooth" in which elements with even $Z$ are more abundant than their odd-$Z$ neighbors.

(a) Using the binding energy table, compare $B/A$ for neighboring even-even and odd-$A$ nuclei: ${}^{16}$O vs. ${}^{14}$N, ${}^{20}$Ne vs. ${}^{23}$Na, ${}^{24}$Mg vs. ${}^{23}$Na, ${}^{28}$Si vs. ${}^{31}$P, ${}^{32}$S vs. ${}^{31}$P. Which has higher $B/A$ in each pair?

(b) In nuclear statistical equilibrium (NSE), the abundance depends exponentially on $B$: $Y \propto \exp(B/k_BT)$. For a typical difference $\Delta B \sim 5$ MeV between an even-even and an odd-$A$ neighbor at $A \sim 28$, calculate the abundance ratio at $T = 4 \times 10^9$ K.

(c) Explain why the odd-even effect is a natural prediction of stellar nucleosynthesis and NSE, and why it constitutes evidence that the cosmic abundances were set by nuclear burning processes.

Section H: Research and Open-Ended Problems

Problem 22.28 ⭐⭐⭐ (Research) The Hoyle state has been the subject of intense theoretical study using ab initio nuclear structure methods. Read one of the following papers and write a one-page summary:

• Epelbaum, Krebs, Lee, and Meissner, Physical Review Letters 106, 192501 (2011) — lattice effective field theory calculation of the Hoyle state.
• Chernykh et al., Physical Review Letters 98, 032501 (2007) — no-core shell model/resonating group method.

What is the current theoretical precision for the Hoyle state energy? How sensitive is it to the nuclear force parameters?

Problem 22.29 ⭐⭐⭐ (Research) The ${}^{12}$C$+{}^{12}$C cross section at astrophysical energies. Review the current experimental status:

(a) What is the lowest center-of-mass energy at which the ${}^{12}$C$+{}^{12}$C cross section has been measured?

(b) What experimental techniques are used (particle detection, gamma detection, particle-gamma coincidences)?

(c) What is the role of underground accelerator facilities in improving these measurements?

(d) How does the current uncertainty in the ${}^{12}$C$+{}^{12}$C rate affect stellar evolution models?

Problem 22.30 ⭐⭐⭐ (Research) Solar neutrino spectroscopy. The Borexino experiment at Gran Sasso has measured solar neutrinos from the pp chain, ${}^7$Be, pep, ${}^8$B, and (for the first time) the CNO cycle.

(a) Summarize the Borexino CNO result (Nature 587, 577, 2020). What was measured and what was the significance?

(b) How does the measured CNO neutrino flux constrain the solar metallicity problem (the disagreement between helioseismology and solar photospheric abundances)?

(c) What nuclear physics inputs are needed to interpret the Borexino data?