43-5 Thermonuclear Fusion in the Sun and Other Stars#

Prompts

How does gravitational confinement create the conditions for fusion in a star? What provides the pressure and temperature?
The Sun’s core is at \(T \sim 10^7\) K. Why is the proton-proton chain (not D–T) the main fusion path? What makes the first step \(p + p \to d + e^+ + \nu\) especially slow?
Outline the proton-proton chain from hydrogen to helium. What is the net reaction? Where does the energy go?
What is the CNO cycle? Why does it dominate in hotter, more massive stars? What role do C, N, O play?
In massive stars, fusion continues beyond helium to carbon, oxygen, and eventually iron. Why does fusion stop at iron?

Lecture Notes#

Overview#

Stars sustain fusion by gravitational confinement: gravity compresses the core, raising density and temperature (\(T \sim 10^7\) K in the Sun) to the values needed for thermonuclear reactions (section 43-4).
The proton-proton chain fuses hydrogen to helium in the Sun and similar low-mass stars. The CNO cycle does the same but uses C, N, O as catalysts; it dominates in hotter, more massive stars.
Massive stars fuse heavier elements (C, O, …) up to iron, where fusion no longer releases energy.

Gravitational Confinement#

A star is a self-gravitating ball of gas. Gravity compresses the core; the pressure and temperature rise until fusion begins. The energy released supports the star against further collapse — hydrostatic equilibrium.

Gravitational confinement means the plasma is held and heated by gravity alone — no external magnetic fields or lasers. The core reaches \(T \sim 10^7\) K (\(k_B T \sim 1\) keV) and high density, enabling fusion despite the Coulomb barrier (section 43-4).

The Proton-Proton Chain#

In the Sun, the main path from hydrogen to helium is the proton-proton (p–p) chain:

\(p + p \to d + e^+ + \nu_e\) (slow: weak interaction)
\(d + p \to {}^3\text{He} + \gamma\)
\({}^3\text{He} + {}^3\text{He} \to {}^4\text{He} + 2p\)

Net: \(4p \to {}^4\text{He} + 2e^+ + 2\nu_e + \gamma\)’s. The positrons annihilate with electrons; the neutrinos escape; the photons thermalize and eventually leave as sunlight.

Why p–p is slow

The first step converts two protons to deuterium via the weak interaction (one proton becomes a neutron). The weak cross section is tiny, so this step limits the rate. The Sun’s power output is set by this bottleneck. D–T would be faster, but the Sun has essentially no tritium.

The CNO Cycle#

In hotter stars (\(T \gtrsim 1.7\times 10^7\) K), the CNO cycle dominates. Carbon, nitrogen, and oxygen act as catalysts — they are consumed and regenerated:

\[ ^{12}\text{C} + p \to {}^{13}\text{N} + \gamma \to {}^{13}\text{C} + e^+ + \nu \]

\[ ^{13}\text{C} + p \to {}^{14}\text{N} + \gamma \]

\[ ^{14}\text{N} + p \to {}^{15}\text{O} + \gamma \to {}^{15}\text{N} + e^+ + \nu \]

\[ ^{15}\text{N} + p \to {}^{12}\text{C} + {}^4\text{He} \]

Net: \(4p \to {}^4\text{He} + 2e^+ + 2\nu + \gamma\)’s — same as the p–p chain, but the rate is more sensitive to temperature. CNO dominates in massive stars because their cores are hotter.

Path	Dominates when	Note
p–p chain	Sun, low-mass stars	Weak first step; slow
CNO cycle	Hotter, massive stars	\(T\)-sensitive; C,N,O catalysts

Fusion Beyond Helium#

In massive stars, after hydrogen is exhausted, helium fuses to carbon (triple-alpha), then carbon and oxygen fuse to heavier elements. Fusion continues up to iron — the peak of the binding-energy-per-nucleon curve (section 42-2). Beyond iron, fusion would absorb energy, so it does not occur. The iron core eventually collapses, leading to supernova and neutron star or black hole formation.

Summary#

Gravitational confinement: Gravity compresses and heats the core; \(T \sim 10^7\) K in the Sun.
p–p chain: \(4p \to {}^4\text{He}\); first step (weak) is the bottleneck.
CNO cycle: Same net result; C,N,O as catalysts; dominates in hotter stars.
Fusion stops at iron; heavier elements form in supernovae.