Nuclear Fusion and the Four Forces

Show explicit units and run a sanity check on every result. A scaling answer is judged by its exponents, not its coefficient. Worked solutions are released after the homework due date.

Useful constants (CGS):

Constant	Value
$k_B$	$1.38 \times 10^{-16}~\mathrm{erg\,K^{-1}}$ ( $= 8.617 \times 10^{-5}~\mathrm{eV\,K^{-1}}$ )
$1~\mathrm{eV}$	$1.602 \times 10^{-12}~\mathrm{erg}$ ; $1~\mathrm{MeV} = 10^6~\mathrm{eV}$
$1~\mathrm{amu}$	$1.661 \times 10^{-24}~\mathrm{g} = 931.5~\mathrm{MeV}/c^2$
$c$	$3.0 \times 10^{10}~\mathrm{cm\,s^{-1}}$
$h$	$6.626 \times 10^{-27}~\mathrm{erg\,s}$
$m_p$	$1.67 \times 10^{-24}~\mathrm{g}$
$e$	$4.803 \times 10^{-10}~\mathrm{esu}$ (so $e^2$ is in $\mathrm{erg\,cm}$ )
$r_{\rm nuc}$	$\approx 1 \times 10^{-13}~\mathrm{cm}$
$T_c$ (solar core)	$\approx 1.5 \times 10^{7}~\mathrm{K}$
$E_{\rm barrier}$ (p–p, head-on)	$\approx 1.44~\mathrm{MeV}$
$L_\odot$	$3.8 \times 10^{33}~\mathrm{erg\,s^{-1}}$
$M_\odot$	$2.0 \times 10^{33}~\mathrm{g}$

Conceptual

Problem

⭐⭐ Right math, wrong conclusion. A student argues: “Fusion happens only in the narrow Gamow window, and only a vanishing fraction of protons ever reach it — so the Sun should barely glow.” The fraction really is minuscule. Yet the conclusion is false.

(a) What did the estimate leave out — why does a tiny fraction still produce an enormous rate?
(b) The energy a reaction releases takes $\sim 10^5\,\mathrm{yr}$ to diffuse to the surface (Reading 4). How does that change what “the Sun’s brightness right now” is actually measuring?
(c) State the general lesson and connect it to Kelvin’s age-of-the-Sun argument from Reading 1.

Problem

⭐⭐ Where the window sits. Carbon burning fuses two ${}^{12}\mathrm{C}$ nuclei (each of charge $6e$ ), so its Coulomb barrier is far taller than the proton–proton barrier. Reason with the same two-exponential picture, $\text{rate}(E) \propto e^{-E/k_B T}\,e^{-\sqrt{E_G/E}}$ .

(a) Does carbon’s Gamow window sit at higher or lower energy than hydrogen’s? Tie your answer to the Gamow energy $E_G$ .
(b) What does that imply about the core temperature a star needs to burn carbon versus hydrogen?
(c) A massive star burns H, then He, then C, then heavier fuels in sequence. Explain why each stage demands a hotter core than the one before.

Problem

⭐⭐⭐ Turn up the weak force. Imagine the weak interaction were $100\times$ stronger, with everything else unchanged.

(a) At a fixed core temperature and density, what happens to the rate of the first step $p + p \rightarrow d$ ?
(b) A student concludes, “so the Sun would be $100\times$ brighter and burn out $100\times$ faster.” Identify the flaw in that reasoning for a main-sequence star.
(c) Using the stellar thermostat (Reading 5), state what actually adjusts — and whether the Sun’s luminosity and lifetime change much.

Calculation

Problem

⭐⭐ A higher wall for helium. The Coulomb barrier between charges $Z_1 e$ and $Z_2 e$ at separation $r$ is $E_{\rm C} = Z_1 Z_2 e^2/r$ .

(a) Compute the barrier for a proton and a ${}^4\mathrm{He}$ nucleus ( $Z=2$ ) at $r_{\rm nuc} = 10^{-13}\,\mathrm{cm}$ , in erg and in MeV.
(b) Compare it to the proton–proton barrier ( $\approx 1.44\,\mathrm{MeV}$ ).
(c) Explain why this means helium burning requires a hotter core than hydrogen burning.

Problem

⭐⭐ de Broglie wavelength of a Gamow-window proton. Take a proton at energy $E \approx 5\,\mathrm{keV}$ — a representative Gamow-window energy, above the $\sim 2\,\mathrm{keV}$ mean.

(a) Find its speed from the non-relativistic relation $E = \tfrac{1}{2} m_p v^2$ .
(b) Find its de Broglie wavelength $\lambda = h/(m_p v)$ .
(c) Compare $\lambda$ to the nuclear scale $r_{\rm nuc} = 10^{-13}\,\mathrm{cm}$ , and say what the ratio means for the classical picture.

Problem

⭐⭐ Fusion versus fire. The Sun converts about $0.71\%$ of fused hydrogen mass into energy.

(a) Using $E = 0.0071\,m c^2$ , find the energy released by fusing $1\,\mathrm{g}$ of hydrogen into helium.
(b) Burning $1\,\mathrm{g}$ of coal releases $\approx 3\times10^{11}\,\mathrm{erg}$ . By what factor does fusion beat chemical burning?
(c) Comment on what this factor means for how long fusion can power a star compared with any chemical source.

Problem

⭐⭐ The solar neutrino flood. Each net pp reaction releases $26.7\,\mathrm{MeV}$ (about $26.2\,\mathrm{MeV}$ retained after neutrinos escape) and emits $2$ neutrinos.

(a) What fraction of the released energy is carried off by neutrinos ( $\approx 0.5\,\mathrm{MeV}$ )?
(b) Using $L_\odot = 3.8\times10^{33}\,\mathrm{erg\,s^{-1}}$ and the retained $26.2\,\mathrm{MeV}$ , estimate the net reaction rate (reactions per second).
(c) Estimate the total solar neutrino production rate (neutrinos per second).

Synthesis

Problem

⭐⭐ How long can fusion run the Sun? Use the per-gram fusion energy from Problem 6 ( $\approx 6.4\times10^{18}\,\mathrm{erg\,g^{-1}}$ ).

(a) With hydrogen mass fraction $X = 0.70$ and only the inner $\sim 10\%$ of the Sun’s mass hot enough to burn, find the total fusible energy ( $M_\odot = 2\times10^{33}\,\mathrm{g}$ ).
(b) Divide by $L_\odot = 3.8\times10^{33}\,\mathrm{erg\,s^{-1}}$ to get the nuclear timescale, in seconds and years.
(c) Compare to Kelvin’s gravitational estimate of $\approx 30\,\mathrm{Myr}$ — by what factor does fusion extend the Sun’s life?

Problem

⭐⭐⭐ The iron wall. Binding energy per nucleon rises from $0$ ( ${}^1\mathrm{H}$ ) to $\approx 7.1\,\mathrm{MeV}$ ( ${}^4\mathrm{He}$ ) to a broad maximum $\approx 8.8\,\mathrm{MeV}$ near iron.

(a) Explain, in terms of binding energy per nucleon, why $\mathrm{H}\rightarrow\mathrm{He}$ fusion releases energy.
(b) A massive star’s core fuses silicon to iron. Predict what happens to nuclear energy generation once the core is iron.
(c) Connect (b) to why the most massive stars end in catastrophic core collapse rather than a gentle fade.

Problem

⭐⭐⭐ Why massive stars switch to CNO. The pp rate scales as $\epsilon_{\rm pp} \propto T^4$ ; the CNO rate as $\epsilon_{\rm CNO} \propto T^{18}$ . Core temperature rises only weakly with mass, $T_c \propto M^{0.2}$ (Reading 2).

(a) A $10\,M_\odot$ star’s core is only $\approx 1.6\times$ hotter than the Sun’s. By what factor does each rate change between the Sun and this star?
(b) Explain why such a modest temperature increase flips the dominant pathway to CNO.
(c) The CNO energy release is sharply concentrated where $T$ is highest. Argue why this drives a convective core — and why that matters for how the energy escapes (Reading 4).