====== Stellar nucleosynthesis ======

The stability of a star depends on a continuous **energy balance** between radiative losses at the surface and nuclear energy production at the core. When this balance holds, the star remains in **hydrostatic equilibrium**; when it breaks, the star evolves. The process by which elements are formed and energy is generated within stars is called **stellar nucleosynthesis**.

===== Nuclear reactions and equilibrium feedback =====

The energy radiated from a star’s surface is replenished by **nuclear reactions** occurring deep in the core.  
A stable equilibrium is maintained through a **negative feedback** mechanism:

  * If gravity momentarily exceeds pressure, the star contracts, increasing core temperature and reaction rates. The higher energy output expands the star back toward equilibrium.  
  * If pressure temporarily exceeds gravity, the star expands, cooling the core and reducing nuclear reaction rates, allowing gravity to restore balance.

A star is primarily composed of **hydrogen**, fully ionized into **free protons and electrons**.  
The main mechanism of nuclear burning in such a plasma is the **proton–proton (pp) chain**.  
However, the average kinetic energy of protons is not high enough to overcome the **Coulomb barrier** arising from electrostatic repulsion due to the [[uv:em|electromagnetic force]].  
The required energy is about a **thousand times higher** than what protons typically possess.

{{:courses:ast301:tunneling.webp?nolink&650|}}

The solution lies in **quantum tunneling**.  
In the diagram above, the \(y\)-axis represents proton energy, and the \(x\)-axis represents separation distance between two protons. The square well indicates the **nuclear potential** (strong force), while the rising slope shows the **Coulomb potential** (\( \propto r^{-1} \)).  
Classically, only particles with energy \(E_3\) could enter the well, but at typical core temperatures (\(T \sim 10^7\) K), most protons have energies \(E_1\) or \(E_2\) far below this level.  
Quantum mechanics allows these low-energy protons to **tunnel** through the barrier, enabling fusion even at sub-threshold energies.

At solar core temperatures (≈10 MK), the high-energy tail of the [[uv:mbd|Maxwell–Boltzmann distribution]] contains enough protons capable of tunneling to sustain fusion.

Although the nuclear energy output balances the luminosity, only a **tiny fraction** of nuclear energy replaces what escapes as surface radiation. A star is primarily a **thermal furnace**, with a nuclear "warmer" maintaining stability.  
For the Sun, this nuclear heater operates at roughly **400 yottawatts (YW)**—comparable to its luminosity of **1000 YW**.

===== Proton–proton chain =====

Hydrogen burning via the **proton–proton chain** dominates in stars with core temperatures above about **5 MK**.  
The Sun’s core temperature (\(T_c \approx 16\) MK) is sufficient for this process.  
Initially, the Sun’s composition was about **71% H**, **27% He**, and **2% heavier elements**, but the hydrogen fraction in the core has now dropped to about **36%**.

{{:courses:ast301:pp-chain.png?nolink|}}

The chain of reactions converts hydrogen into helium while releasing neutrinos and gamma photons.  
Each full cycle converts **six protons** into one **helium nucleus** (\(^4\)He), two positrons, and two neutrinos.

The **conservation laws** of physics hold:

  * **Baryon number:** 6 baryons initially → 6 baryons finally (4 protons + 2 neutrons).  
  * **Lepton number:** 6 leptons initially → 6 leptons finally (4 electrons + 2 neutrinos).  
  * **Charge:** Initially neutral; finally neutral (4 positive + 4 negative).  
  * **Energy:** Conserved via conversion of mass deficit into energy.

{{:courses:ast301:pp-chains.png?nolink&600|}}

The diagram above shows the alternative **pp-branches**.  
The **pp I** chain dominates at solar temperatures.  
The **pep** process contributes only ~0.4% but produces a **1.44 MeV neutrino**, compared with **0.42 MeV** from the pp reaction, making it easier to detect.  
The **hep** reaction, though extremely rare, produces the **highest-energy neutrinos (~18 MeV)**.

Solar neutrinos (\(\nu_e\)) are crucial for [[uv:neutrino astronomy]].  
However, detectors observe only about **half** of the expected flux — a discrepancy explained by **neutrino oscillation**, where electron neutrinos transform into other flavors before reaching Earth.

===== The CNO cycle =====

For hotter stars (\(T \gtrsim 20\) MK), the **carbon–nitrogen–oxygen (CNO) cycle** becomes dominant.  
It also fuses four protons into one helium nucleus, but uses pre-existing carbon, nitrogen, and oxygen nuclei as **catalysts**.

{{https://upload.wikimedia.org/wikipedia/commons/thumb/2/21/CNO_Cycle.svg/768px-CNO_Cycle.svg.png?nolink&500}}

The sequence of reactions begins when \(^{12}\text{C}\) captures a proton to form \(^{13}\text{N}\) and emits a gamma photon.  
\(^{13}\text{N}\) undergoes **beta-plus decay**, producing a positron and an electron neutrino:

$$
^{13}\text{N} \rightarrow\ ^{13}\text{C} + e^+ + \nu_e
$$

The cycle proceeds through a chain of proton captures and decays, returning to \(^{12}\text{C}\), which acts purely as a **catalyst**.

===== Energy production and efficiency =====

The atomic masses of hydrogen and helium are \(M_H = 1.00783\,\text{amu}\) and \(M_{He} = 4.00260\,\text{amu}\), respectively, where \(1\,\text{amu} = 1.66053\times10^{-27}\,\text{kg}\).  
The energy released in one full pp or CNO fusion cycle is

$$
E_r = (4M_H - M_{He})c^2 = 4.29\times10^{-12}\ \text{J} = 26.75\ \text{MeV}.
$$

This corresponds to **0.71%** of the rest-mass energy of the original hydrogen atoms.

The **specific energy yield** is

$$
\frac{E_r}{4M_H} = 0.0071\, c^2 = 6.4\times10^{14}\ \text{J/kg},
$$

equivalent to **640 terajoules per kilogram** — roughly **10 million times more efficient** than chemical energy.

Only about **10%** of hydrogen resides in the Sun’s core where conditions allow fusion, and about **2%** of energy is lost via neutrinos.  
Thus, the total hydrogen-burning energy budget is

$$
E_H \approx (0.98 \times 6.4\times10^{14}\ \text{J/kg}) (0.1\times 2\times10^{30}\ \text{kg}) = 1.3\times10^{44}\ \text{J}.
$$

The corresponding **stellar lifetime** is

$$
\tau_\odot = \frac{E_H}{L_\odot} \approx 10^{10}\ \text{yr} = 10\ \text{Gyr},
$$

which is much longer than the **thermal (Kelvin–Helmholtz)** timescale, confirming that nuclear fusion sustains stars over billions of years.

===== Energy generation rate =====

The rate of energy production per unit mass (\(\epsilon\)) depends on composition, density, and temperature:

$$
\epsilon = \epsilon_0 X^2 \left( \frac{\rho}{10^5\ \mathrm{kg\,m^{-3}}} \right)
\left( \frac{T}{10^7\ \mathrm{K}} \right)^\beta,
$$

where  
\(X\) = hydrogen mass fraction,  
\(\rho\) = density,  
\(T\) = temperature, and  
\(\beta\) = temperature exponent that varies by reaction type.

^ Chain/Cycle ^ Dominant Temperature (MK) ^ $\beta$ ^ Dominant in Stars ^
| pp | 5–15 | 4 | Sun and less massive stars |
| CNO | ≥20 | 15 | Type A and more massive stars |

For the Sun (\(X=0.71\), \(\rho = 150\,\mathrm{Mg\,m^{-3}}\), \(T = 16\,\mathrm{MK}\), \(\beta = 4\)):

$$
\epsilon_{pp} \approx 2.4\times10^{-3}\ \mathrm{W\,kg^{-1}} = 2.4\ \mathrm{mW/kg}.
$$

At higher core temperatures, the CNO cycle dominates due to its **extreme temperature sensitivity** (\(\epsilon \propto T^{15}\)).

===== Insights =====
  * The Sun’s stability arises from nuclear reactions finely balanced with radiative losses.  
  * Quantum tunneling allows fusion at energies far below the Coulomb barrier.  
  * Both pp and CNO processes fuse four protons into one helium nucleus, releasing ~27 MeV per reaction.  
  * Only 10% of hydrogen is available for fusion, leading to a main-sequence lifetime of ~10 Gyr.  
  * The transition from pp to CNO dominance marks the shift from solar-type to massive stellar cores.  

===== Inquiries =====
  - Why is quantum tunneling essential for fusion inside stars?  
  - Compare the efficiency of the pp chain and the CNO cycle at different temperatures.  
  - How does the Sun’s observed neutrino flux support or challenge stellar models?  
  - Derive the expression for stellar lifetime from total fusion energy and luminosity.  
  - Explain why the CNO cycle requires pre-existing heavy elements and how this relates to stellar generations.