Stellar Fusion Lab

mass-coordinate view log-radius view · hover a shell for its numbers

The star in cross-section. Composition regions of the star you selected, at the time you scrubbed to; the area of each ring is proportional to the mass it encloses (mass view). Bright rings mark where fusion is happening now; the thin marker of a burning shell sits at the base of the fuel layer above it and its thickness is not a measurement. Region extents before the presupernova snapshot are illustrative; the snapshot itself uses published model masses. A dashed white ring outlines the region where the reaction shown on the right takes place (for the ejecta after an explosion this is schematic). Use the mass slider and the time slider in the control bar above.

Reaction — pick one below, or let the chamber follow the star

Reaction chamber. Protons are red, neutrons blue. Photons (γ) leave as yellow rays, neutrinos (ν) as faint green streaks that never come back, positrons (e⁺) as cyan sparks that annihilate.

The bridge from hydrogen to iron — a tour of 19 representative reactions. Pick a stage, then a reaction: the chamber plays it and the cross-section jumps to a representative snapshot of that stage, with a dashed ring around the region where the reaction happens. After each core fuel is exhausted, contraction can heat the core enough to ignite another fuel; whether this happens depends on the star.

SECTION 1

Bridge from high school: the twelve ideas everything else is built on

You do not need university physics to follow this page. You need the dozen ideas below, each of which fits on a card. Every later section links back to them. Skim them now, and come back whenever a symbol looks unfamiliar.

1 · Nuclei and the notation ^AX

An atom's nucleus has Z protons (charge +1 each) and N neutrons (no charge). A = Z + N is the mass number. We write ^AX: ¹H is a lone proton, ⁴He has 2 p + 2 n, ¹²C has 6 + 6, ⁵⁶Fe has 26 + 30. Same Z, different N = isotopes (¹²C, ¹³C). Nuclei are tiny: radius

r \approx 1.2 A^{1 / 3} fm, 1 fm = 10^{- 15} m

about 100 000 times smaller than the atom. In a star's core the electrons have been stripped off, so nuclei meet each other bare.

2 · Energy units: eV, MeV, erg, joule

1 electron-volt (eV) is the energy an electron gains crossing 1 volt: $1 eV = 1.602 \times 10^{- 19} J$ . Chemical bonds are a few eV. Nuclear reactions release MeV (10⁶ eV) — a million times more per event, which is why a star can shine for billions of years on a fuel tank that would last a chemical fire only thousands. Astrophysicists still use the cgs unit erg: $1 erg = 10^{- 7} J$ , $1 MeV = 1.602 \times 10^{- 6} erg$ . The Sun radiates $3.83 \times 10^{33} erg s^{- 1}$ .

3 · E = mc² and the mass defect

Mass is a form of energy. Weigh a helium nucleus and it is lighter than two protons plus two neutrons; the missing mass has left as energy when the nucleus was assembled. That missing energy is the binding energy B:

B = [Z m_{p} + N m_{n} - m_{nucleus}] c^{2}

Handy conversion: one atomic mass unit u is $931.494 MeV / c^{2}$ . For ⁴He: 2(938.272) + 2(939.565) − 3727.379 = 28.30 MeV, or 7.07 MeV per nucleon. Section 2 turns this into the most important graph in nuclear astrophysics.

4 · Exponentials, logarithms and log axes

Stars span absurd ranges: densities from 10^-7 to 10¹⁴ g/cm³, times from milliseconds to 10¹⁰ years. On a log axis each tick is ×10, so a straight line means a power law, $y \propto x^{ν}$ , with slope

ν = \frac{d \ln y}{d \ln x}

The other workhorse is the Boltzmann factor $e^{- E / k T}$ : the probability that a particle in a gas at temperature T has energy E falls off exponentially above kT. Reaction rates are exponentials of exponentials — that is why a few per cent more temperature can double a star's power.

5 · Temperature is motion

Temperature measures the average kinetic energy of particles: $⟨ E ⟩ = \frac{3}{2} k T$ , with Boltzmann's constant $k = 8.617 \times 10^{- 5} eV K^{- 1}$ . At the Sun's centre (15.7 million K) that is only kT ≈ 1.35 keV. The energies are spread over the Maxwell–Boltzmann distribution

f (E) d E = \frac{2}{\sqrt{π}} \frac{\sqrt{E}}{(k T)^{3 / 2}} e^{- E / k T} d E

whose high-energy tail is where fusion happens.

6 · Electric repulsion: the Coulomb barrier

Two nuclei with charges Z₁e and Z₂e repel with potential energy

U (r) = \frac{Z_{1} Z_{2} e^{2}}{4 π ε_{0} r} = 1.44 MeV \frac{Z_{1} Z_{2}}{r / fm}

Two protons must approach to about 1 fm before the (short-range, attractive) strong force wins, and that costs roughly 1 MeV — about a thousand times kT in the Sun. Classically the Sun cannot shine. It does, because of card 7.

7 · Particles are waves, so they can tunnel

A proton has a de Broglie wavelength $λ = h / p$ . A wave hitting a barrier it "cannot" climb does not stop dead: it leaks through with a small probability. For a thick barrier the leak is

P \approx \exp (- \frac{2}{ℏ} \int \sqrt{2 m [U (r) - E]} d r)

For the Coulomb barrier this integral gives George Gamow's famous factor $P \approx e^{- \sqrt{E_{G} / E}}$ (section 4). A tiny probability per encounter, times 10⁵⁶ protons trying, is enough.

8 · Pressure, ideal gas and mean molecular weight

A hot gas pushes: $P = n k T$ , with n the number of particles per cm³. In a fully ionised gas the number of particles per unit mass depends on the composition through the mean molecular weight μ:

n = \frac{ρ}{μ m_{u}}, \frac{1}{μ} \approx 2 X + \frac{3}{4} Y + \frac{1}{2} Z

where X, Y, Z are the mass fractions of hydrogen, helium and everything else (μ ≈ 0.62 for the young Sun, 1.34 for pure helium: burning hydrogen halves the number of particles per gram, and the core must get hotter to keep pushing).

9 · Gravity and hydrostatic balance

Every layer of a star is squeezed by the weight of everything above it. Balance means the pressure must rise inward exactly fast enough to carry that weight:

\frac{d P}{d r} = - \frac{G m (r) ρ (r)}{r^{2}}

with m(r) the mass inside radius r and $G = 6.674 \times 10^{- 8} {cm}^{3} g^{- 1} s^{- 2}$ . This one equation (section 3) sets how hot a core of given mass must be — and therefore which fuels it can burn.

10 · Cross-sections and rates

A cross-section σ is the effective target area a nucleus presents for a reaction, measured in barns ( $1 b = 10^{- 24} {cm}^{2}$ , roughly the geometric size of a uranium nucleus). Reactions per cm³ per second between species 1 and 2:

r_{12} = \frac{n_{1} n_{2} ⟨ σ v ⟩}{1 + δ_{12}}

where ⟨σv⟩ averages σ × relative speed over the Maxwell–Boltzmann distribution and δ₁₂ = 1 if the two species are identical (so we don't count each pair twice). Multiply by the energy released per reaction to get the power.

11 · Conservation laws decide what can happen

Every reaction conserves electric charge, the number of nucleons (protons + neutrons), and lepton number. Turning a proton into a neutron ( $p \to n + e^{+} + ν_{e}$ ) needs a positron to carry the charge away and a neutrino to balance lepton number. Neutrinos barely interact with matter, so their share of the energy leaves the star instantly — a leak we must subtract from every energy budget.

12 · Reading a reaction

Nuclear physicists compress "A + b → c + D" to A(b,c)D. So ¹⁴N(p,γ)¹⁵O means a proton is captured by nitrogen-14 and a photon (γ) carries off the excess energy; ¹⁵N(p,α)¹²C means a proton comes in and an alpha particle (⁴He nucleus) goes out. Q is the energy released (negative if the reaction absorbs energy). Decay half-life t_½: time for half the nuclei to decay.

Numbers to keep in your head

Quantity	Symbol	Value
Solar mass	M_☉	1.989 × 10³³ g
Solar radius	R_☉	6.957 × 10¹⁰ cm
Solar luminosity	L_☉	3.828 × 10³³ erg s⁻¹
Sun's central temperature / density today	T_c, ρ_c	1.57 × 10⁷ K, 150 g cm⁻³
Boltzmann constant	k	1.381 × 10⁻¹⁶ erg K⁻¹ = 8.617 × 10⁻⁵ eV K⁻¹
kT at 10⁷ K		0.862 keV
Atomic mass unit	m_u	1.6605 × 10⁻²⁴ g = 931.494 MeV/c²
Proton, neutron, electron rest energies		938.272, 939.565, 0.511 MeV
Avogadro's number	N_A	6.022 × 10²³ mol⁻¹
Energy from burning 1 g of H to He		6.4 × 10¹⁸ erg (0.71 % of mc²)
Year	yr	3.156 × 10⁷ s

SECTION 2

Why fusion releases energy — the binding-energy curve

Nuclei are held together by the strong force, and a bound system weighs less than its parts. The graph below is the entire story of stellar fusion in one line: climbing it releases energy, and it stops climbing at the iron group. Hover any point.

show the semi-empirical mass formula highlight the alpha ladder log A axis · hover a point for Z, N, B/A and the energy released to get there from hydrogen

measured nuclides (AME2020)alpha ladder ⁴He → ⁵⁶Nisemi-empirical formulairon group

Binding energy per nucleon, B/A. Deuterium sits at 1.11 MeV, helium-4 already at 7.07 MeV, carbon-12 at 7.68, oxygen-16 at 7.98, silicon-28 at 8.45, and the curve peaks at nickel-62 (8.795 MeV) with iron-56 (8.790) and iron-58 (8.792) a hair below. Fusing light nuclei moves right and up — energy out. Beyond the peak, fusion would move right and down — energy in. That is why a star's core cannot burn iron.

What the numbers mean

Take the masses of the pieces, subtract the mass of the assembled nucleus, and convert with E = mc²:

Binding energy of a nucleus with Z protons and N neutrons

B (Z, N) = [Z m_{p} + N m_{n} - m (Z, N)] c^{2}, energy released in a reaction: Q = \sum_{products} B - \sum_{reactants} B

Free protons and neutrons have B = 0. Q > 0 means energy comes out (exothermic). For every reaction on this page Q is computed live from the AME2020 binding energies in the data table; the values quoted in textbooks are reproduced to the keV.

Example, the very first step of hydrogen burning in the Sun: p + p → d + e⁺ + ν_e. The deuteron has B = 2.224 MeV, but making a neutron out of a proton costs 1.804 MeV of rest mass (neutron − proton − electron = 0.782 MeV, plus the positron's 0.511 MeV, plus another 0.511 MeV because in the atomic-mass convention the electron bookkeeping shifts by one), so Q = 2.224 − 1.804 = MeV. When the positron meets an electron and annihilates, 1.022 MeV more appears as photons, so the star gains 1.442 MeV per reaction — minus the neutrino's share, which escapes.

Where the 1.804 MeV comes from (β⁺ decay bookkeeping)

Start from: Q-values from masses — Q is the rest energy that disappears — and B = ZΔ_H + NΔ_n − Δ from binding energy. Assumptions: atomic mass excesses (each entry carries its Z electrons), the difference in atomic electron binding between parent and daughter neglected (eV to keV), massless neutrino, parent and daughter in their ground states, recoil energy negligible. Symbols: M(Z,N) = atomic mass, m(Z,N) = bare nuclear mass; m_ec² = 0.51100 MeV, so 2m_ec² = 1.0220 MeV; Δ = mass excess; ΔB = B(daughter) − B(parent); Q_β⁺, Q_β⁻, Q_EC = energy released to the lepton pair and the recoiling daughter; Q_dep = what is left in the gas after the neutrino escapes and any positron annihilates.

1. β⁺ decay at the nuclear level: a proton inside the nucleus becomes a neutron, and a positron and a neutrino are created. The positron's rest mass has to be paid for out of the mass difference. definition

(Z, N) \to (Z - 1, N + 1) + e^{+} + ν, Q_{β^{+}} = [m (Z, N) - m (Z - 1, N + 1) - m_{e}] c^{2}

2. Relate nuclear to atomic masses. A neutral atom is its nucleus plus Z electrons, up to the electron binding energy, which reaches about 0.8 MeV summed over all the electrons of the heaviest atoms but differs between parent and daughter by only tens of keV, since they differ by one electron. approximation

M (Z, N) = m (Z, N) + Z m_{e} - \frac{B_{e} (Z)}{c^{2}} \approx m (Z, N) + Z m_{e}

3. Substitute for both nuclei and count the electrons: −Z from the parent, +(Z − 1) from the daughter, −1 for the positron. Two electron masses survive. identity

\begin{matrix} Q_{β^{+}} & = [M (Z, N) - Z m_{e} - M (Z - 1, N + 1) + (Z - 1) m_{e} - m_{e}] c^{2} \\ = [M (Z, N) - M (Z - 1, N + 1)] c^{2} - 2 m_{e} c^{2} \end{matrix}

4. So β⁺ decay has a threshold: the atomic mass difference must exceed 1.0220 MeV, or the decay cannot happen at all however favourable the nuclear physics. numerical

2 m_{e} c^{2} = 2 \times 0.51100 = 1.0220 MeV

5. Electron capture competes for the same transition: instead of creating a positron the nucleus swallows one of its own atomic electrons. Now count again — −Z from the parent, +1 for the captured electron, +(Z − 1) from the daughter — and the electron masses cancel completely. identity

\begin{matrix} (Z, N) + e^{-} & \to (Z - 1, N + 1) + ν \\ Q_{EC} & = [m (Z, N) + m_{e} - m (Z - 1, N + 1)] c^{2} \\ = [M (Z, N) - M (Z - 1, N + 1)] c^{2} = Q_{β^{+}} + 2 m_{e} c^{2} \end{matrix}

6. Hence capture is open whenever the atomic mass difference is positive at all, while β⁺ needs 1.022 MeV more. ⁷Be is the case that matters for the Sun: its atomic mass difference from ⁷Li is 0.862 MeV, below the threshold, so it can only capture — and because the capture rate depends on the electron density, its lifetime depends on where in the star it sits (⁷Be capture). numerical

Δ (^{7} Be) - Δ (^{7} Li) = 15769.0 - 14907.1 = 861.9 keV < 1022.0 keV

7. β⁻ decay: a neutron becomes a proton, and the emitted electron is exactly the one the daughter atom now needs to stay neutral. Counting: −Z from the parent, +(Z + 1) from the daughter, −1 for the emitted electron — everything cancels, with no 2m_ec² penalty. identity

\begin{matrix} (Z, N) & \to (Z + 1, N - 1) + e^{-} + \bar{ν} \\ Q_{β^{-}} & = [m (Z, N) - m (Z + 1, N - 1) - m_{e}] c^{2} \\ = [M (Z, N) - M (Z + 1, N - 1)] c^{2} \end{matrix}

8. Now put these in terms of binding energies, which is how one sees at a glance which way a nuclide will go. Write Δ = ZΔ_H + NΔ_n − B for parent and daughter and subtract: for β⁺ the daughter has one proton fewer and one neutron more. identity

\begin{matrix} Δ (Z, N) - Δ (Z - 1, N + 1) & = [Z Δ_{H} + N Δ_{n} - B] - [(Z - 1) Δ_{H} + (N + 1) Δ_{n} - B^{'}] \\ = Δ_{H} - Δ_{n} + (B^{'} - B) = Δ B + (Δ_{H} - Δ_{n}) \end{matrix}

9. Evaluate the constant Δ_H − Δ_n once; it is the mass difference that makes a free neutron unstable and a free proton stable. numerical

Δ_{H} - Δ_{n} = 7288.971 - 8071.318 = - 782.35 keV

10. Assemble the three rules. Each is the change in binding energy, offset by a fixed constant of the weak interaction's bookkeeping. identity

\begin{matrix} Q_{β^{+}} & = Δ B - 0.782 - 1.022 = Δ B - 1.804 MeV \\ Q_{EC} & = Δ B - 0.782 MeV \\ Q_{β^{-}} & = Δ B + 0.782 MeV \end{matrix}

11. Check the β⁻ rule on the simplest possible case, the free neutron, where parent and daughter are both unbound so ΔB = 0. The measured value is 0.7823 MeV — the rule reproduces it because that is the same number read the other way. numerical

Q_{β^{-}} (n \to p + e^{-} + \bar{ν}) = 0 + 0.782 MeV

12. Work the CNO example: ¹³N → ¹³C + e⁺ + ν, with Δ(¹³N) = 5345.48 keV and Δ(¹³C) = 3125.01 keV. numerical

Q_{β^{+}} = 5345.48 - 3125.01 - 1022.0 = 1198.5 keV = 1.1985 MeV

13. Add what the star gets back. The positron annihilates with an ambient electron and returns 2m_ec², so the budget before the neutrino is subtracted is the plain atomic mass difference — the same for a β⁺ decay and for the electron capture that competes with it. The two routes differ only in how much the neutrino takes. identity

\begin{matrix} Q_{β^{+}} + 2 m_{e} c^{2} & = 1.1985 + 1.022 = 2.2205 \approx 2.221 MeV = Δ B - 0.782 MeV = Q_{EC} \\ Q_{dep} & = Q_{β^{+}} + 2 m_{e} c^{2} - ⟨ E_{ν} ⟩ \end{matrix}

Result: Q_β⁺ = [M(Z,N) − M(Z−1,N+1)]c² − 2m_ec², Q_EC = [M(Z,N) − M(Z−1,N+1)]c², Q_β⁻ = [M(Z,N) − M(Z+1,N−1)]c²; in binding energies, ΔB − 1.804, ΔB − 0.782 and ΔB + 0.782 MeV. Derived: the electron counting in all three cases and the equality of the total budget for β⁺ and capture. Measured: the mass excesses (AME2020), m_ec², and the mean neutrino energies, which depend on the shape of each β spectrum and are not fixed by the masses. Calculated: 2m_ec² = 1.0220 MeV, the ⁷Be difference of 0.862 MeV that closes the β⁺ channel, Q_β⁺(¹³N) = 1.1985 MeV and 2.221 MeV including annihilation. How fast a decay goes is a different question entirely — phase space (∝ Q⁵ for allowed decays) and nuclear matrix elements — and the page imports those half-lives from NUBASE2020.

Why the curve has this shape: the semi-empirical mass formula

In 1935 Carl Friedrich von Weizsäcker noticed that a nucleus behaves like a charged liquid drop. Four effects compete:

Semi-empirical mass formula (Weizsäcker), coefficients in MeV

B (A, Z) = a_{V} A - a_{S} A^{2 / 3} - a_{C} \frac{Z (Z - 1)}{A^{1 / 3}} - a_{A} \frac{(A - 2 Z)^{2}}{A} + δ (A, Z)

a_V = 15.75 (volume: every nucleon bonds to its neighbours), a_S = 17.8 (surface: nucleons on the skin have fewer neighbours), a_C = 0.711 (Coulomb: protons repel), a_A = 23.7 (asymmetry: the Pauli principle punishes N ≠ Z), δ = ±11.18/√A pairing term (even–even nuclei are bonus-bound, odd–odd penalised). Toggle the overlay in the figure: the smooth curve is this formula; the wiggles of the real data are shell effects the drop model ignores — notably the extra-tight ⁴He, ¹²C, ¹⁶O "alpha-cluster" nuclei.

Why the maximum lands near iron

Divide by A to get binding per nucleon: $B / A = a_{V} - a_{S} A^{- 1 / 3} - a_{C} Z (Z - 1) A^{- 4 / 3} - a_{A} (1 - 2 Z / A)^{2}$ . The surface term fades as A grows (a big drop has proportionally less skin), which is why the curve rises for light nuclei. The Coulomb term grows as Z²/A^4/3 ∝ A^2/3 for Z ≈ A/2, which is why it eventually falls. Setting the derivative to zero (with the asymmetry term choosing the best Z for each A) puts the maximum around A ≈ 56–62. Measured: ⁶²Ni 8.795 MeV, ⁵⁸Fe 8.792, ⁵⁶Fe 8.790. Stars end at ⁵⁶Fe rather than ⁶²Ni because of how they get there (silicon burning builds ⁵⁶Ni, which decays to ⁵⁶Fe; see section 9), not because ⁵⁶Fe is the absolute maximum.

Energy released along the road from hydrogen to iron

Read off the curve: from H (B/A = 0) to ⁴He releases 7.07 MeV per nucleon; from ⁴He all the way to ⁵⁶Fe only 1.72 MeV more. Hydrogen burning yields about 80 % of all the nuclear energy a star will ever release — which is why stars spend ~90 % of their lives on it, and why the later stages are so brief.

Step	Net reaction	Q per reaction	Energy per gram of fuel	Fraction of the H→Fe total

Computed live from the binding energies; per-gram values are before neutrino losses (hydrogen burning loses 2–7 % to neutrinos depending on the chain; the deposited energy is what actually heats the star). The helium row uses the C/O mixture produced by the helium-burning network for the selected star.

Background: mass fractions, number densities and molar abundances

Three ways to count nuclei, all used on this page. The mass fraction X_i is the fraction of the gas's mass in species i (ΣX_i = 1). The number density is nuclei per cm³: $n_{i} = ρ X_{i} / (A_{i} m_{u})$ . The molar abundance $Y_{i} = X_{i} / A_{i}$ (mol per gram) is what reaction networks integrate, because reactions conserve nuclei, not mass fractions: when a proton is captured, X_i changes but the count of catalytic CNO nuclei does not. Identical particles are counted once: the number of distinct p–p pairs is n²/2, which is the 1/(1+δ₁₂) factor in the rate.

SECTION 3

Stars: a fight between gravity and pressure that gravity slowly wins

A star is a ball of gas that would collapse under its own weight in about an hour if nothing pushed back. What pushes back is pressure, and pressure needs heat. Fusion supplies the heat — but only because gravity first squeezed the core hot enough to start it. This section explains why the core temperature is set by the star's mass, why a star that loses energy gets hotter, and how that turns into a ladder of ever-heavier fuels.

Hydrostatic equilibrium and mass continuity

\frac{d P}{d r} = - \frac{G m (r) ρ}{r^{2}}, \frac{d m}{d r} = 4 π r^{2} ρ

P pressure, ρ density, m(r) the mass inside radius r. A quick estimate: replace dP/dr by −P_c/R and m by M, ρ by M/R³ to get the central pressure P_c ~ GM²/R⁴ (the exact bound is P_c > GM²/8πR⁴). For the Sun that is ~10¹⁵–10¹⁷ dyn cm⁻², about 10¹⁰ atmospheres.

Central temperature from the ideal-gas law

T_{c} \sim \frac{μ m_{u}}{k} \frac{G M}{R} \times (number of order 1)

With μ ≈ 0.6 and the Sun's M and R: T ≈ 10⁷ K. The ratio M/R is what matters — and for main-sequence stars R grows more slowly than M, so heavier stars have hotter cores. For your star: T_c = MK, ρ_c = g cm⁻³ ().

Schematic run of density, temperature and energy generation through your star (polytrope shape n = , scaled to the star's mass and radius; the central values are taken from detailed models, the shape is illustrative). The shaded region produces 90 % of the luminosity.

The virial theorem: why losing energy makes a star hotter

Multiply hydrostatic equilibrium by the volume and integrate over the star and you get a relation between the total thermal energy K and the gravitational energy Ω:

Virial theorem for an ideal monatomic gas

2 K + Ω = 0, E_{tot} = K + Ω = - K = \frac{1}{2} Ω

Ω ≈ −(3/5)GM²/R is negative. If the star radiates energy away (E_tot decreases), Ω must become more negative — the star shrinks — and K must increase: it gets hotter. A star has negative heat capacity. This is the engine behind everything in sections 8–11: exhaust a fuel, lose the pressure support, contract, heat up, ignite the next fuel.

Derivation of the virial theorem from hydrostatic equilibrium

Start from: hydrostatic equilibrium; the gravitational energy of a shell brought in from infinity (Newtonian gravity); the internal energy of a monatomic ideal gas; integration by parts (integrals). Symbols: m = m(r) the mass inside radius r; Ω the gravitational potential energy (negative); K the total thermal energy of the gas; u the internal energy per volume; dV = 4πr²dr; L the total rate of energy loss (photons and, late in life, neutrinos).

1. Hydrostatic equilibrium. imported

\frac{d P}{d r} = - \frac{G m ρ}{r^{2}}

2. Multiply both sides by the volume 4πr³ and integrate from the centre to the surface; on the right, 4πr²ρ dr is the shell mass dm. identity

\int_{0}^{R} 4 π r^{3} \frac{d P}{d r} d r = - \int_{0}^{R} \frac{G m}{r} 4 π r^{2} ρ d r = - \int_{0}^{M} \frac{G m}{r} d m

3. The right-hand side is the gravitational potential energy: the work needed to bring each shell dm in from infinity to radius r against the pull of the mass m inside is −Gm dm/r. definition

Ω \equiv - \int_{0}^{M} \frac{G m}{r} d m

4. Integrate the left side by parts; the boundary term vanishes because r = 0 at the centre and P = 0 at the surface. identity

\begin{matrix} \int_{0}^{R} 4 π r^{3} \frac{d P}{d r} d r & = [4 π r^{3} P]_{0}^{R} - \int_{0}^{R} 12 π r^{2} P d r \\ = 0 - 3 \int_{0}^{R} P 4 π r^{2} d r = - 3 \int P d V \end{matrix}

5. For a monatomic ideal gas the thermal energy per volume is (3/2)nkT = (3/2)P, so the pressure integral is two thirds of the total thermal energy. imported

u = \frac{3}{2} P \Rightarrow K = \int u d V = \frac{3}{2} \int P d V \Rightarrow \int P d V = \frac{2}{3} K

6. Combine steps 2–5: the virial theorem, and the total energy that follows from it. identity

- 3 \times \frac{2}{3} K = Ω \Rightarrow 2 K + Ω = 0, E_{tot} = K + Ω = - K = \frac{1}{2} Ω

7. No nuclear energy is being released, and L stands for every loss the star suffers (photons and neutrinos). assumption

8. Negative heat capacity: the total energy falls at the rate of the losses; since E_tot = −K the thermal energy grows at exactly that rate, and Ω falls twice as fast (the star contracts). Heat it and it expands and cools; cool it and it contracts and warms. identity

\frac{d E_{tot}}{d t} = - L \Rightarrow \frac{d K}{d t} = + L, \frac{d Ω}{d t} = - 2 L

9. Ω for a uniform sphere: m = M(r/R)³ and dm = 3Mr²dr/R³; the integrand becomes a power of r. Real stars are centrally concentrated: for a polytrope of index n, Ω = −3GM²/((5 − n)R), i.e. −(3/2)GM²/R for n = 3 (imported). identity

Ω = - \int_{0}^{R} \frac{G M (r / R)^{3}}{r} \frac{3 M r^{2}}{R^{3}} d r = - \frac{3 G M^{2}}{R^{6}} \int_{0}^{R} r^{4} d r = - \frac{3 G M^{2}}{R^{6}} \frac{R^{5}}{5} = - \frac{3}{5} \frac{G M^{2}}{R}

10. The Kelvin–Helmholtz time: the thermal reservoir is K = |Ω|/2, so a star can shine on gravity alone for |Ω|/(2L). For the Sun: GM²/R = 6.674×10⁻⁸ × (1.989×10³³)²/6.957×10¹⁰ = 3.80×10⁴⁸ erg, |Ω| = 2.3×10⁴⁸ erg (uniform) and 2.3×10⁴⁸/(2 × 3.83×10³³) = 3.0×10¹⁴ s = 9.5 Myr, or 24 Myr for n = 3 — the time the core takes to contract and heat between burning stages, and the reason Kelvin's 1862 age for the Sun was so short. numerical

t_{K H} = \frac{| Ω |}{2 L} = \frac{3 G M^{2}}{10 R L} \approx 9 Myr (uniform), \frac{3 G M^{2}}{4 R L} \approx 24 Myr (n = 3)

The thermostat: fusion rates rise as T^ν with ν between 4 and 40. If the core over-produced energy, K would exceed its virial value; the excess pressure expands the star, which by step 8 cools the gas and throttles the reactions. The feedback works only while the pressure is thermal — degenerate cores have no thermostat, which is what makes the helium flash and Type Ia supernovae. Every burning stage on this page is set up this way: the fuel runs out, the core contracts on the Kelvin–Helmholtz time and heats until the next fuel ignites.

Result: 2K + Ω = 0, E_tot = −K = Ω/2, dK/dt = +L when nothing burns; Ω = −(3/5)GM²/R for a uniform sphere. Derived: all of it from hydrostatic equilibrium and u = (3/2)P; Imported: the polytropic factor 3/(5 − n); Calculated: 2.3×10⁴⁸ erg and 9 Myr for the Sun.

The thermostat: why main-sequence stars burn steadily

Fusion rates rise steeply with temperature (ε ∝ T⁴ for the pp chain, T¹⁸ for CNO, T⁴⁰ for helium). Such a fuel should be explosive — yet the Sun's output is stable to 0.1 %. The reason is the negative heat capacity again: if the core overproduced energy, it would expand, and by the virial theorem expansion cools it, throttling the reactions. This feedback holds as long as the pressure is thermal. It fails when the pressure comes from electron degeneracy, which does not depend on temperature — then heating does not cause expansion, the reaction runs away, and you get a helium flash (section 8) or a Type Ia supernova. It also fails in the last second of a massive star's life, when the pressure support is removed faster than the core can respond (section 11).

Background: three kinds of pressure

Thermal (ideal gas)

P = \frac{ρ k T}{μ m_{u}}

Depends on T. Has a thermostat. Dominates in the Sun and in all main-sequence cores.

Electron degeneracy

P_{e} = K_{nr} {(\frac{ρ}{μ_{e}})}^{5 / 3} (non-rel.), P_{e} = K_{rel} {(\frac{ρ}{μ_{e}})}^{4 / 3} (rel.)

Quantum mechanics forbids two electrons in the same state; squeeze them and they push back regardless of T (K_nr = 1.004×10¹³, K_rel = 1.243×10¹⁵ cgs). No thermostat. Holds up white dwarfs; sets the Chandrasekhar limit. Electrons are degenerate when $ρ / μ_{e} ≳ 2.4 \times 10^{- 8} T^{3 / 2}$ g cm⁻³.

Radiation

P_{rad} = \frac{1}{3} a T^{4}

Photons carry momentum. Negligible in the Sun (0.05 %), dominant above ~100 M☉, where it makes the star barely bound (the Eddington limit, and pair-instability at >140 M☉).

Mass sets everything: luminosity and lifetime

Combining hydrostatic equilibrium with radiative energy transport (photons diffusing outward against the opacity κ) gives the famous mass–luminosity relation: for a star supported by ideal-gas pressure with electron-scattering opacity, L ∝ μ⁴M³ , independent of how the energy is generated. Real stars: L ∝ M^3.5 between about 1 and 20 M☉, steeper below, flatter above as radiation pressure takes over.

Main-sequence lifetime

t_{MS} = \frac{X_{0} q_{c} M q_{H}}{⟨ L ⟩} \approx 10^{10} yr {(\frac{M}{M_{⊙}})}^{- 2.5} q_{H} \approx 6.0 – 6.4 \times 10^{18} erg g^{- 1}

q_H is the energy per gram of hydrogen burned (after neutrino losses) and q_c the fraction of the star's hydrogen that actually gets burned (~10 % for the Sun, whose core is not mixed; up to ~80 % for the most massive stars, whose convective cores keep refuelling the centre). Your star: L = L☉, R = R☉, T_eff = K, main-sequence lifetime .

Log-log chart · the dot is your star · hover for values · show the published model points

Luminosity, core temperature, core density and lifetime versus mass. Solid lines: the interpolation used on this page through published models (Woosley, Heger & Weaver 2002 for 13–75 M☉; Hansen & Kawaler ZAMS models for 1–10 M☉; the present Sun; Chabrier & Baraffe scalings below 0.5 M☉ — those low-mass points are approximate). Dashed: homology power laws. Note the kink near 1.3 M☉ where the CNO cycle takes over: hotter cores are then needed only slightly to make much more power, so T_c flattens.

The contraction ladder in one picture

Plot the centre of a star in the density–temperature plane. As a core of fixed mass contracts, the virial theorem gives T_c ∝ ρ_c^1/3 M^2/3 — a straight line of slope 1/3 whose height depends on the mass. Each fuel ignites when the line crosses its ignition temperature. But there is a wall: where electrons become degenerate, contraction no longer heats the gas, and the track bends over and cools. A low-mass core hits the wall before the next ignition line; a massive core clears every line up to silicon.

H ignitionHeCNeOSiyour star's contraction track and burning stagesdegenerate electrons to the right

Central temperature versus central density. Ignition temperatures are the burning-stage temperatures of detailed models; the degeneracy boundary is where electron degeneracy pressure equals the ideal-gas pressure. The white track is the T ∝ ρ^1/3 contraction line for your star's core mass through its tabulated burning stages.

Why the centre alone cannot tell you the luminosity, the lifetime or the shells

Everything a reaction network computes at the centre — rates, branching ratios, energy per gram — is local. The star's luminosity is an integral over all the burning gas, $L = \int ε d m$ ; its lifetime depends on how convection refuels the centre; its shell structure depends on how burning migrates outward as each fuel is exhausted. None of that follows from one parcel of gas. So this page does two separate things and says which is which: the global properties (L, R, lifetimes, stage temperatures and densities, shell masses) are taken from published full-structure models and interpolated; the nuclear physics (rates, branching, energy release, abundance evolution) is computed live from the formulas at those conditions. Where the two are joined by an assumption — for example converting "fraction of central hydrogen consumed" into "age" — the assumption is printed beside the number.

Graduate note: homology relations and what sets the T_c(M) slopes

For chemically homogeneous stars in radiative equilibrium, scaling the structure equations (hydrostatic equilibrium, radiative transport with κ ∝ ρ^aT^−b, and ε ∝ ρT^ν) gives power laws. With electron-scattering opacity (a = b = 0): L ∝ μ⁴M³; R ∝ M^{(ν−1)/(ν+3)}; T_c ∝ μ M/R ∝ M^4/(ν+3); ρ_c ∝ M/R³. For the pp chain (ν ≈ 4): T_c ∝ M^0.57, ρ_c ∝ M^−0.3; for the CNO cycle (ν ≈ 18): T_c ∝ M^0.21, ρ_c ∝ M^−1.4 (Pols, Stellar Structure and Evolution, §9.2). The higher the temperature sensitivity, the less T_c needs to rise to feed a much brighter star — the flattening you see in the chart. Below ~0.35 M☉ stars are fully convective and closer to n = 1.5 polytropes; there ρ_c rises again toward the hydrogen-burning limit (ρ_c ~ 10³ g cm⁻³ at 0.075 M☉).

SECTION 4

How nuclei tunnel: the Gamow peak, the S-factor and the reaction rate

Two protons in the Sun's core have about 1 keV of thermal energy and need roughly 1000 keV to touch. In classical physics they never meet. Quantum mechanics lets them cheat — rarely — and the mathematics of that cheating explains why fusion rates are so violently sensitive to temperature. We build the idea up in five steps; open the boxes for the algebra.

Step 1 — the classical dead end

The Maxwell–Boltzmann distribution (section 1, card 5) puts a fraction ~e^−E/kT of particles above energy E. Reaching the top of the Coulomb barrier at E ≈ 1 MeV with kT = 1.35 keV means e⁻⁷⁴⁰ ≈ 10⁻³²⁰. The Sun contains 10⁵⁷ protons. Zero of them are that fast.

Step 2 — tunnelling through the Coulomb barrier

A proton is also a wave. Solving the Schrödinger equation for a wave that meets the 1/r Coulomb hill gives a transmission probability that George Gamow wrote down in 1928:

Gamow factor

P (E) = \exp (- 2 π η) = \exp (- \sqrt{\frac{E_{G}}{E}}), E_{G} = 2 μ c^{2} (π α Z_{1} Z_{2})^{2} = 0.979 MeV \times μ Z_{1}^{2} Z_{2}^{2}

η = Z₁Z₂e²/ħv is the Sommerfeld parameter, α = 1/137.036 the fine-structure constant, μ the reduced mass in atomic mass units (μ = m₁m₂/(m₁+m₂)). E_G is the Gamow energy: 490 keV for p + p, 32 MeV for ⁴He + ¹²C, 2.0 GeV for ¹²C + ¹²C. At E = 10 keV two protons tunnel with probability e⁻⁷ ≈ 10⁻³: enormous compared with 10⁻³²⁰.

Derivation of the Gamow factor (WKB)

Start from: the WKB transmission probability through a barrier V(r) higher than the energy E; the Coulomb potential V = Z₁Z₂e²/r between the nuclear radius R and the classical turning point; and the fact that the relative motion of two bodies is that of one body of reduced mass (Newton's laws). Symbols: μ_red = m₁m₂/(m₁ + m₂) the reduced mass (written μ below; it is never the mean molecular weight here); e² in Gaussian units (e²/4πε₀ in SI); α = e²/ħc = 1/137.036; v the relative speed; η the Sommerfeld parameter; E_G the Gamow energy; μ_u = μ/m_u the reduced mass in atomic mass units.

1. The WKB result gives the tunnelling probability as the exponential of an integral of the imaginary momentum across the forbidden region. imported

P \approx \exp [- \frac{2}{ℏ} \int_{R}^{r_{c}} \sqrt{2 μ (\frac{Z_{1} Z_{2} e^{2}}{r} - E)} d r]

2. The classical turning point r_c is where the Coulomb energy equals E; with it the bracket becomes E(r_c/r − 1). definition

\frac{Z_{1} Z_{2} e^{2}}{r_{c}} = E \Rightarrow r_{c} = \frac{Z_{1} Z_{2} e^{2}}{E}, \frac{Z_{1} Z_{2} e^{2}}{r} - E = E (\frac{r_{c}}{r} - 1)

3. Substitute r = r_c sin²θ, so that the awkward square root becomes a cotangent. identity

\begin{matrix} d r & = 2 r_{c} \sin θ \cos θ d θ \\ \frac{r_{c}}{r} - 1 & = \frac{1 - \sin^{2} θ}{\sin^{2} θ} = \cot^{2} θ \\ \sqrt{2 μ E (\frac{r_{c}}{r} - 1)} & = \sqrt{2 μ E} \cot θ \end{matrix}

4. Multiply the two factors of the integrand: cotθ × sinθ cosθ = cos²θ. identity

\sqrt{2 μ E} \cot θ \times 2 r_{c} \sin θ \cos θ d θ = 2 r_{c} \sqrt{2 μ E} \cos^{2} θ d θ

5. Change the limits (r = R gives θ_R = arcsin√(R/r_c); r = r_c gives θ = π/2) and integrate cos²θ = (1 + cos 2θ)/2 (integrals). identity

\int_{R}^{r_{c}} \sqrt{2 μ (\frac{Z_{1} Z_{2} e^{2}}{r} - E)} d r = 2 r_{c} \sqrt{2 μ E} {[\frac{θ}{2} + \frac{\sin 2 θ}{4}]}_{θ_{R}}^{π / 2} = r_{c} \sqrt{2 μ E} [\frac{π}{2} - θ_{R} - \frac{1}{2} \sin 2 θ_{R}]

6. For R ≪ r_c the angle θ_R ≈ √(R/r_c) is small and sin 2θ_R ≈ 2θ_R, so the bracket is π/2 minus a small correction (for p + p at 6 keV, r_c = 240 fm against R ≈ 2 fm: the correction is (4/π)√(2/240) ≈ 0.12 of the leading term, and it is carried by the measured S-factor). Keep the leading term. approximation

\frac{π}{2} - θ_{R} - \frac{1}{2} \sin 2 θ_{R} \approx \frac{π}{2} - 2 \sqrt{\frac{R}{r_{c}}} = \frac{π}{2} [1 - \frac{4}{π} \sqrt{\frac{R}{r_{c}}}] \approx \frac{π}{2}

7. Insert r_c = Z₁Z₂e²/E into the exponent and simplify √E in numerator and denominator. identity

\frac{2}{ℏ} \times \frac{π}{2} \frac{Z_{1} Z_{2} e^{2}}{E} \sqrt{2 μ E} = \frac{π Z_{1} Z_{2} e^{2}}{ℏ} \sqrt{\frac{2 μ}{E}}

8. With E = ½μv² the square root is 2/v, which gives the Sommerfeld form of the exponent. identity

\sqrt{\frac{2 μ}{E}} = \frac{2}{v} \Rightarrow P \approx \exp (- \frac{2 π Z_{1} Z_{2} e^{2}}{ℏ v}) = e^{- 2 π η}, η \equiv \frac{Z_{1} Z_{2} e^{2}}{ℏ v}

9. Define the Gamow energy by writing the same exponent as √(E_G/E): square both sides and use α = e²/ħc. definition

\sqrt{\frac{E_{G}}{E}} = \frac{π Z_{1} Z_{2} e^{2}}{ℏ} \sqrt{\frac{2 μ}{E}} \Rightarrow E_{G} = \frac{2 μ (π Z_{1} Z_{2} e^{2})^{2}}{ℏ^{2}} = 2 μ c^{2} {(\frac{π Z_{1} Z_{2} e^{2}}{ℏ c})}^{2} = 2 μ c^{2} (π α Z_{1} Z_{2})^{2}

10. Evaluate the constant with m_uc² = 931.494 MeV and α = 1/137.036: 2 × 931.494 × (π/137.036)² = 1862.99 × 5.2558×10⁻⁴. numerical

E_{G} = 0.97913 MeV \times μ_{u} Z_{1}^{2} Z_{2}^{2}

Result: P ≈ exp(−√(E_G/E)) = e^−2πη with E_G = 2μc²(παZ₁Z₂)² = 0.979 μ_uZ₁²Z₂² MeV. Derived: the exponent and its dependence Z₁Z₂√(μ/E) — heavier and more highly charged pairs need proportionally higher energies, hence the order H, He, C, Ne, O, Si of the burning stages and the reason no star burns iron. Calculated: the constant 0.979 MeV from CODATA values. Imported into the S-factor: the finite-radius correction of step 6 and everything the nuclear force does inside R.

Step 3 — the S-factor: separating nuclear physics from the barrier

The cross-section contains a geometric factor (the de Broglie wavelength squared, ∝ 1/E) and the Gamow factor. Pull both out and what is left, the astrophysical S-factor, varies slowly with energy and is what experiments measure and extrapolate:

Definition of S(E)

σ (E) = \frac{S (E)}{E} \exp (- \sqrt{\frac{E_{G}}{E}})

Units: keV barn (or MeV b, eV b). p + p: S(0) = 4.01×10⁻²² keV b — the weak interaction makes it fantastically small, which is why the Sun lives 10 Gyr instead of exploding. ³He + ³He: 5.21×10³ keV b. ¹⁴N + p: 1.66 keV b. The values on this page come from the "Solar Fusion II" evaluation (Adelberger et al. 2011); each rate box in sections 5–6 lists its S(0), S′(0), S″(0) and the energy range over which the expansion is trusted.

Step 4 — average over the gas: the Gamow peak

The reaction rate per pair is ⟨σv⟩, the cross-section times speed averaged over the Maxwell–Boltzmann distribution:

Thermonuclear rate integral

⟨ σ v ⟩ = \sqrt{\frac{8}{π μ}} \frac{1}{(k T)^{3 / 2}} \int_{0}^{\infty} S (E) \exp (- \frac{E}{k T} - \sqrt{\frac{E_{G}}{E}}) d E

The integrand is the product of a falling exponential (fewer particles at high E) and a rising one (easier tunnelling at high E). The product is a narrow bump — the Gamow peak — at an energy far above kT but far below the barrier.

T = 15.7 MK log scale

Maxwell–Boltzmann factor e^−E/kTtunnelling factor e^{−√(E_G/E)}product (×10ⁿ, the Gamow peak)Gaussian approximation

The Gamow peak for the selected reaction: E₀ = keV, width Δ = keV, τ = 3E₀/kT = , so the rate scales locally as T^ν with ν = (τ − 2)/3 = . Rate N_A⟨σv⟩ = cm³ mol⁻¹ s⁻¹ (). Notice how a 10 % change in T shifts the whole peak.

Properties of the Gamow peak

E_{0} = {(\frac{\sqrt{E_{G}} k T}{2})}^{2 / 3} = 1.220 {(Z_{1}^{2} Z_{2}^{2} μ T_{6}^{2})}^{1 / 3} keV, Δ = 4 \sqrt{\frac{E_{0} k T}{3}} = 0.749 {(Z_{1}^{2} Z_{2}^{2} μ T_{6}^{5})}^{1 / 6} keV, τ = \frac{3 E_{0}}{k T}

T₆ = T/10⁶ K. In the Sun (T₆ = 15.7): p + p peaks at E₀ = 6.1 keV with Δ = 6.6 keV; ¹⁴N + p at 27 keV (Δ = 14 keV); ⁴He + ¹²C would peak at 58 keV, where its tunnelling factor is e⁻⁸⁵ — hopeless at solar temperatures, which is why helium burning waits for 10⁸ K, where the peak sits at 300 keV.

Derivation: saddle-point evaluation of the rate integral

Start from: the rate integral; Laplace's method for an integral dominated by the maximum of its exponent (expansions); the Gaussian integral ∫exp(−x²/a²)dx = a√π (the Gaussian). Symbols: μ the reduced mass, μ_u = μ/m_u; E₀ the peak energy; Δ the full width at 1/e; τ = 3E₀/kT; T₆ = T/10⁶ K; kT = 0.0861733 T₆ keV; E_G = 0.97913 μ_uZ₁²Z₂² MeV.

1. Name the exponent of the integrand. definition

f (E) = - \frac{E}{k T} - \sqrt{\frac{E_{G}}{E}}

2. Its maximum is where the derivative vanishes (derivatives): differentiate, set to zero and solve for E₀. identity

f^{'} (E) = - \frac{1}{k T} + \frac{1}{2} \sqrt{E_{G}} E^{- 3 / 2} = 0 \Rightarrow E_{0}^{3 / 2} = \frac{\sqrt{E_{G}} k T}{2} \Rightarrow E_{0} = {(\frac{\sqrt{E_{G}} k T}{2})}^{2 / 3}

3. Height of the peak: from step 2, √E_G = 2E₀^3/2/kT, so the tunnelling term at E₀ equals twice the thermal term, and f(E₀) is minus three times E₀/kT. identity

\sqrt{\frac{E_{G}}{E_{0}}} = \frac{2 E_{0}^{3 / 2}}{k T E_{0}^{1 / 2}} = \frac{2 E_{0}}{k T} \Rightarrow f (E_{0}) = - \frac{E_{0}}{k T} - \frac{2 E_{0}}{k T} = - \frac{3 E_{0}}{k T} \equiv - τ

4. Curvature at the peak: differentiate f′ once more and substitute the same relation for √E_G. identity

f^{''} (E) = - \frac{3}{4} \sqrt{E_{G}} E^{- 5 / 2} \Rightarrow f^{''} (E_{0}) = - \frac{3}{4} \frac{2 E_{0}^{3 / 2}}{k T} E_{0}^{- 5 / 2} = - \frac{3}{2 E_{0} k T}

5. Expand f to second order about E₀ (Taylor); the integrand becomes a Gaussian whose full width at 1/e of the maximum defines Δ. approximation

f (E) \approx - τ - \frac{3 (E - E_{0})^{2}}{4 E_{0} k T} \equiv - τ - \frac{(E - E_{0})^{2}}{(Δ / 2)^{2}} \Rightarrow {(\frac{Δ}{2})}^{2} = \frac{4 E_{0} k T}{3}, Δ = 4 \sqrt{\frac{E_{0} k T}{3}}

6. Put in the constants (E_G in keV: 979.13 μ_uZ₁²Z₂²; kT = 0.0861733 T₆ keV): E₀ = (½)^2/3E_G^1/3(kT)^2/3 = 0.62996 × 9.9298 × 0.19511 × (Z₁²Z₂²μ_u)^1/3T₆^2/3 keV; Δ = 4(E₀kT/3)^1/2 = 4 × (1.2204 × 0.0861733/3)^1/2 × (Z₁²Z₂²μ_u)^1/6T₆^5/6 keV; τ = 3E₀/kT = 3 × 1.2204/0.0861733 × (Z₁²Z₂²μ_u/T₆)^1/3. numerical

E_{0} = 1.220 {(Z_{1}^{2} Z_{2}^{2} μ_{u} T_{6}^{2})}^{1 / 3} keV, Δ = 0.749 {(Z_{1}^{2} Z_{2}^{2} μ_{u} T_{6}^{5})}^{1 / 6} keV, τ = \frac{3 E_{0}}{k T} = 42.49 {(\frac{Z_{1}^{2} Z_{2}^{2} μ_{u}}{T_{6}})}^{1 / 3}

7. Integrate the Gaussian with S(E) ≈ S(E₀) taken outside: ∫exp[−(E − E₀)²/(Δ/2)²]dE = (Δ/2)√π (the lower limit 0 is many widths away from E₀), then simplify √(8/π) × √π/2 = √2. identity

⟨ σ v ⟩ \approx \sqrt{\frac{8}{π μ}} \frac{1}{(k T)^{3 / 2}} S (E_{0}) e^{- τ} \frac{\sqrt{π}}{2} Δ = \sqrt{\frac{2}{μ}} \frac{Δ}{(k T)^{3 / 2}} S (E_{0}) e^{- τ}

8. Temperature dependence of the prefactor: Δ ∝ (E₀kT)^1/2 and E₀ ∝ (kT)^2/3, so Δ/(kT)^3/2 ∝ (kT)^−2/3 ∝ τ², and the rate is τ²e^−τ times constants. identity

\frac{Δ}{(k T)^{3 / 2}} \propto \frac{(k T)^{1 / 3} (k T)^{1 / 2}}{(k T)^{3 / 2}} = (k T)^{- 2 / 3} \propto τ^{2} \Rightarrow ⟨ σ v ⟩ \propto τ^{2} e^{- τ}, τ \propto T^{- 1 / 3}

9. What the page does with it: N_A⟨σv⟩ for p + p at T₆ = 15.7 from this formula with S_eff (the working formula) is cm³ mol⁻¹ s⁻¹; the classic Caughlan & Fowler (1988) fit gives 9.8×10⁻²⁰, the same within a few per cent. The local power law ν = (τ − 2)/3 follows in temperature exponent. numerical

Result: ⟨σv⟩ ≈ √(2/μ) Δ (kT)^−3/2 S(E₀) e^−τ, with E₀ = 1.220 (Z₁²Z₂²μ_uT₆²)^1/3 keV, Δ = 0.749 (Z₁²Z₂²μ_uT₆⁵)^1/6 keV and τ = 42.49 (Z₁²Z₂²μ_u/T₆)^1/3. Derived: the τ²e^−τ law and the T^2/3 growth of the peak; Calculated: the three constants from CODATA values; Measured or fitted: S(E₀).

Putting in numbers, the working formula used throughout this page (with S in MeV b, μ in u, T₉ = T/10⁹ K, rate per mole so that N_A⟨σv⟩ is in cm³ mol⁻¹ s⁻¹):

N_{A} ⟨ σ v ⟩ = 7.8324 \times 10^{9} {(\frac{Z_{1} Z_{2}}{μ})}^{1 / 3} S_{eff} (E_{0}) T_{9}^{- 2 / 3} \exp [- 4.2487 {(\frac{Z_{1}^{2} Z_{2}^{2} μ}{T_{9}})}^{1 / 3}]

S_{eff} = S (0) [1 + \frac{5 k T}{36 E_{0}}] + S^{'} (0) E_{0} [1 + \frac{35 k T}{36 E_{0}}] + \frac{1}{2} S^{''} (0) E_{0}^{2} [1 + \frac{89 k T}{36 E_{0}}]

The bracketed corrections account for the peak's asymmetry and for S(E) varying across it (Iliadis, Nuclear Physics of Stars, §3.2). For p + p at T₆ = 15.7 this gives N_A⟨σv⟩ = cm³ mol⁻¹ s⁻¹; the classic Caughlan & Fowler (1988) fit gives 9.8×10⁻²⁰ — the same within a few per cent.

Step 5 — from a rate to a power

Energy generation rate per gram

ε_{12} = \frac{r_{12} Q_{eff}}{ρ} = ρ N_{A} \frac{Y_{1} Y_{2}}{1 + δ_{12}} N_{A} ⟨ σ v ⟩ Q_{eff} [erg g^{- 1} s^{- 1}]

Y_i = X_i/A_i are molar abundances (mol g⁻¹), Q_eff the energy per reaction minus the neutrino's share, in erg. For a three-body reaction like the triple-alpha process the rate goes as ρ² and the identical-particle factor is 1/3! = 1/6.

Reaction	E_G	at T =	E₀	τ	ν (local T-exponent)

Live table at each fuel's characteristic burning temperature for your star. The exponent ν is why the pp chain is a gentle T⁴ heater, the CNO cycle a T¹⁸ furnace, and helium burning a T⁴⁰ bomb held in check only by the thermostat of section 3.

Graduate note 1: electron screening

Nuclei in a plasma are surrounded by a cloud of electrons that partly cancels their charge, lowering the barrier by U₀ = Z₁Z₂e²/R_D, with R_D the Debye radius. In the weak-screening limit (Salpeter 1954) the rate is multiplied by

f = \exp (H_{12}), H_{12} = \frac{U_{0}}{k T} = 0.188 Z_{1} Z_{2} \sqrt{ζ ρ} T_{6}^{- 3 / 2}, ζ = \sum_{i} \frac{(Z_{i}^{2} + Z_{i}) X_{i}}{A_{i}}

valid only for H₁₂ ≪ 1. At the Sun's centre f ≈ 1.05 for p + p and ≈ 1.4 for ¹⁴N + p. In the dense cores of very-low-mass stars and in degenerate helium cores the plasma is strongly coupled and this formula fails (intermediate/strong screening, Chabrier & Baraffe 2000); the page then reports "not computed". Toggle: apply weak screening where valid — current factors: .

Graduate note 2: resonances

If the compound nucleus has an excited state at just the right energy, the cross-section is not smooth but spikes (a Breit–Wigner resonance). For a narrow resonance at centre-of-mass energy E_r with strength ωγ the rate is

N_{A} ⟨ σ v ⟩_{res} = 1.5399 \times 10^{11} (μ T_{9})^{- 3 / 2} \sum_{i} (ω γ)_{i} \exp (- \frac{11.605 E_{r, i}}{T_{9}}) (E_{r}, ω γ in MeV)

Resonances matter for ¹⁷O(p,α)¹⁴N even at solar energies (a 65 keV resonance, LUNA 2016), for ¹⁴N(p,γ) above ~0.1 GK (259 keV), and — most famously — for the triple-alpha process, where the Hoyle state of ¹²C at 7.65 MeV sits almost exactly at the Gamow peak of ⁸Be + ⁴He at 10⁸ K (section 8). This page uses the smooth S-factor formula where the evaluations validate it and adds tabulated resonance terms where they matter; each rate box states its domain.

Background: three different "equilibria" you will meet

Thermal equilibrium: particles share energy so fast (collisions every 10⁻¹⁵ s) that one temperature describes them all — always true in stellar interiors. Steady state (used for ³He in the pp chain and for the CNO isotopes): a species is produced as fast as it is destroyed, so its abundance is constant even though nothing about the composition is "in equilibrium" — it is a bottleneck, not a balance. Chemical (statistical) equilibrium (silicon burning, section 9): every reaction runs forward and backward at the same rate, so the abundances are fixed by temperature, density and the proton-to-neutron ratio alone, regardless of the reaction paths. A large Q-value never makes a reaction fast; the barrier and the S-factor do. In the pp chain the first step releases the least energy and is a million times slower than the rest — the slowest step sets the pace of the whole chain.

SECTION 5

Hydrogen burning I — the proton–proton chains

The Sun turns 600 million tonnes of hydrogen into helium every second, and the very first step is so improbable that an average proton waits billions of years for it. That slowness is a feature: it is why the Sun is a steady lamp and not a bomb. Below, the three routes from four protons to one helium nucleus, with every energy and every branching ratio computed for the star and the moment you selected.

The pp chains at your star's centre (T_c = MK, ρ_c = g cm⁻³, X = , Y = ). Percentages are the fractions of ⁴He nuclei completed through each branch, from the instantaneous reaction fluxes of the live network (). Line thickness follows the flux.

The reactions

Reaction	Q (MeV)	ν energy (MeV)	S(0)	N_A⟨σv⟩ at T_c	Mean lifetime of the first nucleus	Validity

Q values computed from the AME2020 binding energies (positron annihilation included, as the star receives it); neutrino energies are the branch-weighted mean energies carried away (pp 0.265, pep 1.442, ⁷Be 0.862 (89.6 %) / 0.384 (10.4 %), ⁸B 6.71 mean, hep 9.63 mean). "Lifetime" is the mean time before the first-named nucleus undergoes that reaction under the current central conditions: $τ_{1} = 1 / (n_{2} ⟨ σ v ⟩)$ .

Why p + p is so slow

When two protons touch, they form a "diproton" that flies apart again in 10⁻²² s. Only if, during that instant, one proton converts into a neutron by the weak interaction (emitting e⁺ + ν) does a stable deuteron form. The weak force is ~10²⁵ times feebler than the strong force at these energies, so S(0) = 4.01×10⁻²² keV b: 25 orders of magnitude below a typical strong-interaction reaction. Result: the proton's mean life against fusion at your star's centre is . The next step, d + p, takes . Deuterium therefore never accumulates (its abundance is ~10⁻¹⁸ of hydrogen), and the pp reaction sets the pace of the whole chain.

Three ways to finish

After d + p, the star holds ³He. It can meet another ³He (pp-I, needs two full passes through p + p per ⁴He) or a ⁴He that is already there (pp-II and pp-III, which need only one). The ³He + ⁴He route makes ⁷Be, which either captures an electron (pp-II, neutrino of 0.862 MeV) or, rarely, a proton (pp-III, producing ⁸B whose decay neutrinos reach 15 MeV — the ones the Homestake, Kamiokande and SNO detectors saw). Hotter cores and more helium favour pp-II/III: in the present Sun's centre the ⁷Be branches carry more than half the flux; averaged over the whole core, pp-I still makes ~85 % of the Sun's helium.

Energy generation of the pp chains

ε_{p p} = ψ q_{H} \frac{ρ X^{2}}{m_{u}} ⟨ σ v ⟩_{p p}, q_{H} = \frac{Q_{eff}}{4 m_{u}}

Each p + p reaction starts a chain that ends in half a ⁴He (pp-I) or a whole one (pp-II, pp-III), so ψ runs from 1 (pure pp-I) to 2 (pure pp-II/III); q_H is the energy per gram of hydrogen after subtracting the neutrino losses of that branch: 26.20 MeV per ⁴He for pp-I (2.0 % lost), 25.66 MeV for pp-II (4.0 %), 19.76 MeV for pp-III (26 %). At your star's centre now: ε_pp = erg g⁻¹ s⁻¹ with ψ = , neutrino loss %. Textbook fit for comparison (Kippenhahn, Weigert & Weiss 2012): ε_pp = 2.57×10⁴ ψ f g₁₁ ρ X² T₉^−2/3 exp(−3.381/T₉^1/3), g₁₁ = 1 + 3.82 T₉ + 1.51 T₉² + 0.144 T₉³ − 0.0114 T₉⁴ → erg g⁻¹ s⁻¹ here.

Derivation: the ³He steady state and the branching ratios

Start from: the reaction rate per volume; the competition of lifetimes; the fitted ⁷Be electron-capture rate (⁷Be capture) and pep fraction (pep). Symbols: n₃, n₄, n_p the ³He, ⁴He and proton densities; ⟨σv⟩₃₃, ⟨σv⟩₃₄, ⟨σv⟩_hep, ⟨σv⟩₁₇ the rate coefficients of ³He + ³He, ³He + ⁴He, ³He + p and ⁷Be + p; r_pp, r_pep the initiation rates; H_b the ⁴He completion rate through branch b, f_b = H_b/H; λ_e the ⁷Be electron-capture rate per nucleus.

1. Balance ³He: it is produced once per p + p and once per pep reaction, destroyed two at a time by ³He + ³He (the reaction rate is ½n₃²⟨σv⟩₃₃ and each removes two, so the loss is n₃²⟨σv⟩₃₃) and one at a time by ³He + ⁴He and ³He + p. identity

r_{p p} + r_{p e p} = n_{3}^{2} ⟨ σ v ⟩_{33} + n_{3} n_{4} ⟨ σ v ⟩_{34} + n_{3} n_{p} ⟨ σ v ⟩_{hep}

2. Solve the quadratic for the positive root. identity

a n_{3}^{2} + b n_{3} - c = 0, a = ⟨ σ v ⟩_{33}, b = n_{4} ⟨ σ v ⟩_{34} + n_{p} ⟨ σ v ⟩_{hep}, c = r_{p p} + r_{p e p} \Rightarrow n_{3} = \frac{- b + \sqrt{b^{2} + 4 a c}}{2 a}

3. Two limits of the root: when the 33 term dominates (b² ≪ 4ac, cool stars) and when the 34 term dominates (b² ≫ 4ac, hot helium-rich cores). approximation

n_{3} \approx \sqrt{\frac{c}{a}} = \sqrt{\frac{r_{p p} + r_{p e p}}{⟨ σ v ⟩_{33}}} (b^{2} ≪ 4 a c), n_{3} \approx \frac{c}{b} = \frac{r_{p p} + r_{p e p}}{n_{4} ⟨ σ v ⟩_{34}} (b^{2} ≫ 4 a c)

4. Completions: each ³He + ³He reaction completes one ⁴He, and each ³He + ⁴He reaction leads, through ⁷Be, to one net ⁴He; the pp-I fraction of the helium production is the ratio of the first to the sum. definition

H_{I} = \frac{1}{2} n_{3}^{2} ⟨ σ v ⟩_{33}, H_{II} + H_{III} = n_{3} n_{4} ⟨ σ v ⟩_{34}, f_{I} = \frac{\frac{1}{2} n_{3}^{2} ⟨ σ v ⟩_{33}}{\frac{1}{2} n_{3}^{2} ⟨ σ v ⟩_{33} + n_{3} n_{4} ⟨ σ v ⟩_{34}}

5. Initiations per completion (pep and hep neglected): pp-I costs two p + p, the other branches one, so r_pp = (1 + f_I)H and the counting factor of the pp-energy entry follows; the energy-weighted ψ_E that the page displays multiplies it by Q̄_dep/Q_I. identity

r_{p p} = 2 H_{I} + H_{II} + H_{III} = (1 + f_{I}) H \Rightarrow ψ_{count} = \frac{2 H}{r_{p p}} = \frac{2}{1 + f_{I}}

6. The ⁷Be fork: electron capture at rate λ_e per ⁷Be nucleus competes with proton capture at rate n_p⟨σv⟩₁₇; the split is the ratio of the two rates. In the Sun it is about 10⁻³: the ⁸B branch is rare but its neutrinos are the most energetic. identity

\frac{f_{III}}{f_{II}} = \frac{n_{p} ⟨ σ v ⟩_{17}}{λ_{e} (^{7} Be)}

7. Time to reach the steady state: ³He relaxes on its own destruction time — about 10⁶ yr at the solar centre but longer than the age of the universe below ~8 MK, which is why the page follows ³He with its network instead of assuming the steady state. identity

τ_{3} = \frac{1}{n_{3} ⟨ σ v ⟩_{33} + n_{4} ⟨ σ v ⟩_{34}}

8. The electron-capture rate and the pep fraction come from the Solar Fusion II fits (λ_e(⁷Be), R(pep)/R(pp)). imported

Result: n₃ from the quadratic of step 2, f_I from step 4, f_III/f_II from step 6, ψ_count = 2/(1 + f_I). Derived: all of it under the steady-state assumption; Measured or fitted: the four rate coefficients and λ_e; Calculated: the live values from the page's network (which integrates the ³He equation rather than assuming steady state).

Electron capture on ⁷Be and the pep reaction (Solar Fusion II, eqs. 40 and 46)

λ_{e} (^{7} Be) = 5.60 \times 10^{- 9} \frac{ρ}{μ_{e}} T_{6}^{- 1 / 2} [1 + 0.004 (T_{6} - 16)] s^{- 1} (10 < T_{6} < 16)

\frac{R (p e p)}{R (p p)} = 1.130 \times 10^{- 4} \frac{ρ}{μ_{e}} T_{6}^{- 1 / 2} [1 + 0.02 (T_{6} - 16)], \frac{1}{μ_{e}} = \sum_{i} \frac{Z_{i} X_{i}}{A_{i}}

Continuum electron capture scales with the electron density ρ/μ_e and with T^−1/2 (slower electrons are captured more easily). The ⁷Be mean life at your star's centre: (the terrestrial half-life is 53.2 days; in the plasma the nucleus has no bound electrons to capture, so it lives longer). Outside 10–16 MK the same form is used and the value is marked .

Graduate note: neutrino losses and the solar neutrino spectrum

Each branch emits neutrinos with a characteristic spectrum: pp (continuous to 0.42 MeV), pep (line at 1.442 MeV), ⁷Be (lines at 0.862 and 0.384 MeV), ⁸B (continuous to ~15 MeV), hep (to 18.8 MeV), plus ¹³N, ¹⁵O and ¹⁷F from the CNO cycle (to 1.2–1.7 MeV). The luminosity constraint ties the total neutrino flux to the Sun's light: $L_{⊙} = \sum_{i} (\frac{Q}{2} - ⟨ E_{ν} ⟩_{i}) Φ_{i} \times 4 π d^{2}$ . Borexino measured the pp flux directly in 2014 and detected CNO neutrinos in 2020 (about 1 % of the Sun's energy), confirming the branching this page computes. Neutrinos leave the star at the speed of light; their energy never contributes to the pressure, which is why all energy budgets here subtract them first.

SECTION 6

Hydrogen burning II — the CNO cycles

Stars heavier than about 1.3 Suns burn hydrogen a different way: not by fusing protons with each other but by feeding them, one at a time, to carbon, nitrogen and oxygen nuclei that act as catalysts. The cycle returns the catalyst unchanged and hands back one helium nucleus. Because the protons must tunnel into nuclei of charge 6–8, the rate climbs as T¹⁸ near 20 MK — a furnace so temperature-sensitive that it forces massive stars to grow convective cores.

The CN cycle (inner ring) and the NO branch (outer loop) at your star's centre. Each arc is labelled with the mean lifetime of that nucleus before its next reaction at the current T_c; the animated dots run proportionally faster on short-lived species. The slowest link, ¹⁴N(p,γ)¹⁵O, sets the speed of the whole cycle, so almost all catalyst nuclei end up waiting in the ¹⁴N queue.

Reaction	Q (MeV)	S(0) (keV b)	N_A⟨σv⟩ at T_c	Lifetime

β⁺ decays: ¹³N (t_½ = 9.97 min, ⟨E_ν⟩ = 0.707 MeV), ¹⁵O (2.04 min, 0.997 MeV), ¹⁷F (64.5 s, 0.999 MeV). Total per cycle: 26.73 MeV of which % leaves as neutrinos, leaving Q_eff = 24.97 MeV to heat the star. The ¹⁵N + p reaction goes to ¹²C + α about 1000 times for every ¹⁶O + γ, so the outer loop is a slow leak that converts the star's original ¹⁶O into ¹⁴N over time.

Energy generation of the CNO cycle in steady state

ε_{CNO} = ρ N_{A} Y_{p} Y_{14} N_{A} ⟨ σ v ⟩_{14} Q_{eff} \Rightarrow ε_{CNO} \propto ρ X X_{CNO} T^{ν}, ν \approx 18 (T \sim 2 \times 10^{7} K)

In steady state every catalyst passes through the bottleneck ¹⁴N(p,γ) once per cycle, so the cycle rate is the ¹⁴N + p rate and the energy per pass is Q_eff. At your star's centre: ε_CNO = erg g⁻¹ s⁻¹ versus ε_pp = ; the CNO share is % of the central power (local ν = ). Textbook fit for comparison (Kippenhahn et al. 2012, pre-LUNA rate): ε_CNO = 8.24×10²⁵ g₁₄ X_CNO X ρ T₉^−2/3 exp(−15.231 T₉^−1/3 − (T₉/0.8)²), g₁₄ = 1 − 2.00 T₉ + 3.41 T₉² − 2.43 T₉³ → erg g⁻¹ s⁻¹ here — about 1.9× the modern value, because the 2004 LUNA measurement halved the ¹⁴N(p,γ) S-factor (3.2 → 1.66 keV b), which also made globular-cluster ages nearly a billion years older.

Energy generation versus temperature at your star's ρ_c, X and X_CNO · the dot is your star's centre · overlay textbook fits

ε_ppε_CNOε_3α (for Y = 1, helium burning)your star

Crossover. The pp curve rises as T⁴, the CNO curve as T¹⁸: they cross at MK for this composition (about 17–18 MK at solar composition, corresponding to a 1.3 M☉ star). Lower the metallicity slider and watch the CNO curve sink: with fewer catalysts a star of given central temperature makes less CNO power. In a real star the core then contracts and heats until the power matches the luminosity the star must radiate — so metal-poor stars run hotter, smaller and bluer. Here we hold the central conditions fixed so you can see the composition effect alone.

Derivation: CN-cycle equilibrium abundances and the ¹²C/¹³C ratio

Start from: the mean lifetime against a reaction and the reaction rate; conservation of the catalyst nuclei. Symbols: n_i the number density of catalyst species i (¹²C, ¹³C, ¹⁴N, ¹⁵N); τ_p(i) its lifetime against proton capture; n_p the proton density; N_CNO = Σn_i the total number of catalysts per volume.

1. Lifetime of species i against proton capture, and its destruction rate per volume. definition

τ_{p} (i) = \frac{1}{n_{p} ⟨ σ v ⟩_{i}}, {(\frac{d n_{i}}{d t})}_{loss} = - \frac{n_{i}}{τ_{p} (i)}

2. In a closed loop the production of species i is the destruction of species i − 1; in steady state every dn_i/dt vanishes, so all four destruction rates are equal, and each abundance is proportional to its lifetime. identity

\frac{d n_{i}}{d t} = \frac{n_{i - 1}}{τ_{p} (i - 1)} - \frac{n_{i}}{τ_{p} (i)} = 0 \Rightarrow \frac{n (^{12} C)}{τ_{p} (^{12} C)} = \frac{n (^{13} C)}{τ_{p} (^{13} C)} = \frac{n (^{14} N)}{τ_{p} (^{14} N)} = \frac{n (^{15} N)}{τ_{p} (^{15} N)} \Rightarrow n_{i} \propto τ_{p} (i)

3. Ratios: the proton density cancels, so the equilibrium ratio of two catalysts is the inverse ratio of their rate coefficients and depends only on temperature (weakly, because both have nearly the same Gamow exponent). identity

\frac{n (^{12} C)}{n (^{13} C)} = \frac{τ_{p} (^{12} C)}{τ_{p} (^{13} C)} = \frac{⟨ σ v ⟩_{13}}{⟨ σ v ⟩_{12}}

4. Conservation: each lap returns the ¹²C, so the total number of catalysts is constant; the share of each species is its lifetime over the sum of lifetimes. The page's network checks this conservation to one part in 10¹⁰. identity

N_{CNO} = \sum_{i} n_{i} = const \Rightarrow n_{i} = N_{CNO} \frac{τ_{p} (i)}{\sum_{j} τ_{p} (j)}, n (^{14} N) = N_{CNO} \frac{τ_{14}}{τ_{12} + τ_{13} + τ_{14} + τ_{15}}

5. Numbers for the example of ① (lifetime ratios 1 : 0.21 : 400 : 0.008). numerical

\frac{n (^{14} N)}{N_{CNO}} = \frac{400}{1 + 0.21 + 400 + 0.008} = 0.997, \frac{^{12} C}{^{13} C} = \frac{1}{0.21} = 4.8, \frac{^{15} N}{^{12} C} = 0.008

6. Approach to equilibrium: the slowest relaxation time is τ_p(¹⁴N) itself, while the ¹²C → ¹³C part equilibrates on τ_p(¹²C) ~ 10⁶ yr at 20 MK — which is why partially processed material shows the low ¹²C/¹³C long before ¹⁴N has reached its final share. identity

t_{relax} (^{12} C \to^{13} C) \sim τ_{p} (^{12} C) ≪ t_{relax} (^{14} N) \sim τ_{p} (^{14} N)

At your star's centre τ_p(¹²C) : τ_p(¹³C) : τ_p(¹⁴N) : τ_p(¹⁵N) = , giving ¹²C/¹³C ≈ and a ¹⁴N share of % of all CNO nuclei (live values from the rates at the current central conditions).

Result: n_i ∝ τ_p(i), so ¹²C/¹³C = ⟨σv⟩₁₃/⟨σv⟩₁₂ and ¹⁴N holds τ₁₄/Στ of the catalysts. Derived: all of it from the steady state of a closed loop; Measured or fitted: the four rate coefficients; Calculated: the live ratios.

Background: what "catalyst" means here, and why the cycle needs seeds

A catalyst takes part in a reaction and is returned unchanged at the end. In the CNO cycle a ¹²C nucleus absorbs four protons in turn (two of which convert to neutrons by β⁺ decay) and finally spits out an α particle, becoming ¹²C again. The net reaction is the same as the pp chain — 4 p → ⁴He + 2 e⁺ + 2 ν — but the barrier is crossed by a proton against Z = 6–8 rather than Z = 1, which is why the cycle needs about 15–20 MK to compete. The catalysts must already exist: the very first stars, made only of hydrogen and helium from the Big Bang, could not run the CNO cycle until they had made their own carbon by the triple-alpha process (section 8). That is why the metallicity slider changes the CNO power.

Graduate note: why CNO burning creates convective cores, and the hot-CNO limit

With ε ∝ T¹⁸, the energy generation is concentrated within a few per cent of the stellar radius. The radiative temperature gradient needed to carry that flux, $\nabla_{rad} \propto κ l (r) / m (r)$ , exceeds the adiabatic gradient near the centre (Schwarzschild criterion), so the core convects and is mixed; its mass fraction grows from ~10 % at 1.5 M☉ to ~80 % at 100 M☉. Mixing keeps the central hydrogen from running out first, lengthens the main sequence, and produces a sharp end when the whole core runs dry at once. At T ≳ 10⁸ K (novae, X-ray bursts) proton captures on ¹³N and ¹⁷F outrun their β decays and the cycle becomes limited by the decay lifetimes (the "hot CNO cycle", β-limited to ~1000 s per cycle); the network on this page checks that condition and reports when its reduced form is no longer valid.

SECTION 7

The live reaction network: watching the core's composition change

Everything above gives a rate at one instant. Here the page integrates the rates forward in time for a parcel of gas at your star's centre — nine nuclear species, twelve reaction channels, the real Solar-Fusion-II rates, no fudge factors — and plots how hydrogen turns into helium while the CNO catalysts shuffle into nitrogen. Two clocks are shown, and the difference between them is itself a lesson.

stellar-age axis parcel-time axis log abundances · hover for values

¹H³He⁴He¹²C¹³C¹⁴N¹⁵N¹⁶O¹⁷O▮ now

Mass fractions through core hydrogen burning at the centre of your star. Notice ³He rising to its steady state and ¹²C collapsing into ¹⁴N within a few million years (the CN cycle reaching equilibrium) long before hydrogen is appreciably consumed, then the slow conversion of ¹⁶O into ¹⁴N through the NO branch.

ε deposited (erg g⁻¹ s⁻¹)ε lost to neutrinosT_c (MK, right axis)CNO share of ε

Energy generation and central temperature along the run. Central conditions follow .

Two clocks

Parcel clock τ_parcel
Tabulated core-H lifetime t_MS
Ratio t_MS / τ_parcel
Hydrogen at the end of the run
Integration

The parcel clock is what the physics gives for gas sitting at the centre: with no mixing it would exhaust its hydrogen in τ_parcel. The star as a whole lives longer because the centre is a tiny fraction of the burning region (Sun-like stars) or because convection keeps refilling it (massive stars). On the "stellar-age axis" the run is stretched so that hydrogen exhaustion coincides with the tabulated lifetime — a constant-luminosity, mixed-core age assignment, printed here so you know it is an assumption and not a result.

What the solver is doing (and how it checks itself)

The equations being integrated

\frac{d Y_{i}}{d t} = \sum_{r} ν_{i r} F_{r}, F_{a b} = \frac{ρ Y_{a} Y_{b} N_{A} ⟨ σ v ⟩_{a b}}{1 + δ_{a b}}, ε = N_{A} \sum_{r} F_{r} Q_{r, dep}

Y_i = X_i/A_i molar abundances; ν_ir the stoichiometric coefficients (e.g. p + p → …→ ³He consumes 3 protons); F_r reaction fluxes in mol g⁻¹ s⁻¹. The seven fast intermediates (²H, ⁷Be, ⁷Li, ⁸B, ¹³N, ¹⁵O, ¹⁷F) are folded into their producing reactions because they live for seconds to months while the retained species change over millions of years; the page verifies that condition at every run (table on the right) and would disable the result if it failed.

The system is stiff: p + p takes 10¹⁰ years, ¹⁵N + p takes hours. An explicit method would need 10¹⁴ steps. The page uses the backward Euler method — solve $Y_{n + 1} = Y_{n} + h f (Y_{n + 1})$ by Newton's method with a numerical Jacobian, partial pivoting and a positivity-preserving damping — with step-doubling error control, step rejection, and a cap of 10 % change per step for every major species. Backward Euler has a bonus: because Y_n+1 − Y_n = h f(Y_n+1) holds exactly, the reaction counts of each step are exactly consistent with the abundance changes, which makes the energy bookkeeping below a genuine independent test.

Abundance–energy closure test (independent check)

E_{dep} + E_{ν} \overset{?}{=} Δ E_{rest} = N_{A} c^{2} \sum_{i} (Y_{i, 0} - Y_{i, end}) {\tilde{m}}_{i}

Left side: energy accumulated from the reaction fluxes with their tabulated Q-values and neutrino energies. Right side: the change in rest mass of the parcel from the mass table alone (electron-inclusive masses

{\tilde{m}}_{i}

; the positron annihilation energy is inside Q and is not added again). They agree to — this catches wrong stoichiometry, double-counted annihilation, stray factors of N_A, or inconsistent Q assignments. Nucleon number is conserved to and the count of CNO nuclei to .

Folded species	Lifetime τ	τ / t_evolution	Competing capture

A folded species is valid when its lifetime is under 10⁻³ of the shortest evolution time of the retained species it feeds or drains (³He and ⁴He for the pp intermediates; the parent and daughter catalysts for the β⁺ emitters) and any competing capture is under 1 % of its decay rate (the "hot CNO" condition). Status: .

Graduate note: what the parcel cannot capture, and how detailed codes do it

A full stellar-evolution code (MESA, KEPLER, GENEC) solves the structure equations on ~10³ mass shells coupled to a network of 20–200 isotopes, with convective mixing treated as a diffusion process, mass loss at the surface, and the equation of state including degeneracy and radiation pressure. The central temperature is then an output of the thermal balance, not an input. This page's parcel uses two labelled prescriptions for T_c(X): a solar-calibrated μ scaling for pp-dominated stars (T_c ∝ μ^0.44, ρ_c ∝ μ^1.85, which reproduces the change from the ZAMS Sun to today's Sun), and a "thermostat" for CNO-dominated stars — the central temperature rises just enough to keep the deposited CNO power constant as the fuel drains (with ρ_c ∝ μ). Both stop at X = 0.02 because the real star readjusts its whole structure near hydrogen exhaustion (the "hook" in the HR diagram), which no single parcel can follow.

SECTION 8

Helium burning: three alphas, one lucky resonance, and the origin of carbon and oxygen

When the core runs out of hydrogen it contracts and heats (section 3). At about 10⁸ K helium ignites — but only by way of a nucleus that does not exist. ⁸Be falls apart in 10⁻¹⁶ s, so three helium nuclei must effectively meet at once, and the reaction would still be hopelessly slow if the ¹²C nucleus did not happen to have an excited state at exactly the right energy. Fred Hoyle predicted that state in 1953 from the fact that carbon-based astronomers exist.

⁴He + ⁴He ⇌ ⁸BeQ = MeV (endothermic; ⁸Be lives 8×10⁻¹⁷ s)

⁸Be + ⁴He → ¹²C* (7.654 MeV) → ¹²C + γQ = MeV

net: 3 ⁴He → ¹²C + γQ = MeV, 5.85×10¹⁷ erg per gram

¹²C + ⁴He → ¹⁶O + γQ = MeV

¹⁶O + ⁴He → ²⁰Ne + γQ = MeV (slow at He-burning temperatures)

At your star's helium-burning conditions (T = , ρ = g cm⁻³) the ⁸Be/⁴He steady-state ratio is : for every billion helium nuclei about one is transiently ⁸Be. The Gamow peak of ⁸Be + α sits at E₀ = keV with width keV — the Hoyle state lies 287 keV above the ⁸Be + α threshold, inside the tail of the peak. The triple-alpha rate scales as T^ν with ν = here (≈ 40 at 10⁸ K).

⁴He¹²C¹⁶O²⁰Ne

Helium-burning parcel at your star's core helium-burning conditions, integrated for the tabulated duration (). Final mixture: ¹²C , ¹⁶O (C/O by mass = ), ²⁰Ne ; energy closure . The parcel burns at fixed T and ρ; in a real convective core the temperature rises as helium runs low, which favours the ¹²C(α,γ) step and lowers the final C/O — so treat the ratio as illustrative.

Triple-alpha rate and energy generation

ε_{3 α} = N_{A} Q_{3 α} \frac{ρ^{2} Y_{α}^{3} N_{A}^{2} ⟨ σ v ⟩_{3 α}}{3!}, N_{A}^{2} ⟨ σ v ⟩_{3 α} = 2.79 \times 10^{- 8} T_{9}^{- 3} e^{- 4.4027 / T_{9}} + 1.35 \times 10^{- 8} T_{9}^{- 3 / 2} e^{- 24.811 / T_{9}} {cm}^{6} {mol}^{- 2} s^{- 1}

Y_α = X₄/4 in mol g⁻¹; the 1/3! counts each triple once. Exponent 4.4027/T₉ is the 380 keV total resonance energy (⁸Be ground state 92 keV + Hoyle state 287 keV) divided by kT: the rate is dominated by the resonance, so it is ∝ ρ² and violently temperature-dependent. The same formula in the textbook form ε_3α = 5.1×10⁸ ρ² Y³ T₉⁻³ e^{−4.4027/T₉} erg g⁻¹ s⁻¹ (Kippenhahn & Weigert) agrees to . ¹²C(α,γ)¹⁶O uses the 2017 deBoer et al. evaluation; its S(300 keV) = 140 ± 21 keV b is still the most consequential uncertainty in nuclear astrophysics because it fixes the carbon-to-oxygen ratio of the universe, the white-dwarf composition and the fate of massive stars.

Derivation: why the rate is ∝ ρ² and where 4.4027/T₉ comes from

Start from: the ⁸Be Saha equilibrium; the narrow-resonance rate; the three-body energy generation bookkeeping; the counting of triples (counting). Symbols: n_α, n₈ the ⁴He and ⁸Be number densities; μ₁ = m_α/2 the reduced mass of α + α, μ₂ = 2m_α/3 that of α + ⁸Be (both reduced masses, not molecular weights); E₁ = 91.84 keV the energy of ⁸Be above two α particles and E₂ = 287.6 keV the Hoyle state above ⁸Be + α (measured); Γ_α = 8.5 eV, Γ_rad = 3.7 meV the widths (measured); ω = 1 (all spins zero); ⟨σv⟩_3α the effective three-body coefficient (cm⁶ s⁻¹), defined so that r_3α = n_α³⟨σv⟩_3α/3!.

1. Step one, α + α ⇌ ⁸Be: ⁸Be is unbound and breaks up in 10⁻¹⁶ s, so its abundance is the Saha equilibrium value. imported

n_{8} = n_{α}^{2} {(\frac{2 π ℏ^{2}}{μ_{1} k T})}^{3 / 2} e^{- E_{1} / k T}, μ_{1} = \frac{m_{α}}{2}

2. Step two, ⁸Be + α → ¹²C* → ¹²C + γ through the Hoyle state: the narrow-resonance rate per ⁸Be nucleus, with the strength ωγ = ωΓ_αΓ_rad/Γ ≈ Γ_rad because the state almost always falls apart again (Γ ≈ Γ_α) and only one decay in 2300 reaches ¹²C. imported

rate per^{8} Be = n_{α} {(\frac{2 π}{μ_{2} k T})}^{3 / 2} ℏ^{2} ω γ e^{- E_{2} / k T}, μ_{2} = \frac{2 m_{α}}{3}, ω γ \approx Γ_{rad}

3. Multiply the ⁸Be abundance by the capture rate per ⁸Be: the two Boltzmann factors combine and (2πħ²)^3/2(2π)^3/2ħ² Γ_rad = (2πħ²)³ (Γ_rad/ħ). identity

r_{3 α} = n_{8} \times rate per^{8} Be = n_{α}^{3} {(\frac{2 π ℏ^{2}}{k T})}^{3} (μ_{1} μ_{2})^{- 3 / 2} \frac{Γ_{rad}}{ℏ} e^{- (E_{1} + E_{2}) / k T}

4. Insert the reduced masses: μ₁μ₂ = (m_α/2)(2m_α/3) = m_α²/3, so (μ₁μ₂)^−3/2 = 3^3/2/m_α³ — the classic form. identity

r_{3 α} = 3^{3 / 2} n_{α}^{3} {(\frac{2 π ℏ^{2}}{m_{α} k T})}^{3} \frac{Γ_{rad}}{ℏ} e^{- (E_{1} + E_{2}) / k T}, E_{1} + E_{2} = 379.5 keV

5. Numbers: 2πħ²/(m_αk × 10⁹ K) = 7.617×10⁻²⁴ cm², cubed 4.42×10⁻⁷⁰ T₉⁻³ cm⁶, × 3^3/2 = 2.30×10⁻⁶⁹; Γ_rad/ħ = 3.7×10⁻³ eV/6.58×10⁻¹⁶ eV s = 5.6×10¹² s⁻¹; (E₁ + E₂)/k = 379.5 keV/(8.617×10⁻⁵ eV K⁻¹) = 4.404×10⁹ K — the 4.4027 comes from the measured energies. numerical

r_{3 α} = n_{α}^{3} \times 2.30 \times 10^{- 69} \times 5.6 \times 10^{12} T_{9}^{- 3} e^{- 4.40 / T_{9}} {cm}^{- 3} s^{- 1}

6. Convert to the tabulated form: with r_3α = n_α³⟨σv⟩_3α/3! (each triple counted once), N_A²⟨σv⟩_3α = 6N_A² × 2.30×10⁻⁶⁹ × 5.6×10¹² T₉⁻³e^−4.40/T₉ — the tabulated 2.79×10⁻⁸ T₉⁻³ e^{−4.4027/T₉} cm⁶ mol⁻² s⁻¹. numerical

N_{A}^{2} ⟨ σ v ⟩_{3 α} = 6 \times (6.022 \times 10^{23})^{2} \times 2.30 \times 10^{- 69} \times 5.6 \times 10^{12} T_{9}^{- 3} e^{- 4.40 / T_{9}} = 2.8 \times 10^{- 8} T_{9}^{- 3} e^{- 4.40 / T_{9}}

7. Energy per gram with the three-body bookkeeping and Q_3α = 7.275 MeV. identity

ε_{3 α} = N_{A} Q_{3 α} \frac{ρ^{2} Y_{α}^{3} N_{A}^{2} ⟨ σ v ⟩_{3 α}}{3!}

8. Temperature exponent (logarithmic slope): 41 at 0.1 GK, 19 at 0.2 GK. The second term of the tabulated rate, 1.35×10⁻⁸T₉^−3/2e^{−24.811/T₉}, is a higher-lying resonance kept from the same evaluation (Caughlan & Fowler 1988). identity

ν = \frac{d \ln (T_{9}^{- 3} e^{- 4.4027 / T_{9}})}{d \ln T} = \frac{4.4027}{T_{9}} - 3

9. The example of ① from step 7 (T₉ = 0.1, ρ = 10⁵, Y_α = 0.25). numerical

ε_{3 α} = \frac{6.02 \times 10^{23} \times 1.166 \times 10^{- 5} \times 10^{10} \times 0.0156 \times 2.1 \times 10^{- 24}}{6} erg g^{- 1} s^{- 1} = 3.8 \times 10^{2} erg g^{- 1} s^{- 1}

Result: r_3α = 3^3/2n_α³(2πħ²/m_αkT)³(Γ_rad/ħ)e^{−(E₁+E₂)/kT}, i.e. N_A²⟨σv⟩_3α = 2.79×10⁻⁸ T₉⁻³e^{−4.4027/T₉}. Derived: the sequential equilibrium-plus-resonance form, the T⁻³ and the ρ² dependence; Measured: E₁, E₂, Γ_rad, Γ_α; Calculated from them: 4.4027 and 2.79×10⁻⁸ (and the example). The textbook form with the mass fraction Y is the triple-alpha fit.

Where helium burning leads, by stellar mass

Below ~2 M☉: the helium flash

The contracting helium core becomes electron-degenerate (ρ ~ 10⁶ g cm⁻³) before it reaches 10⁸ K. Degenerate pressure does not care about temperature, so when the triple-alpha process ignites there is no thermostat (section 3): the temperature runs away, the core briefly produces ~10¹⁰ L☉ (all absorbed by the envelope), until the gas becomes non-degenerate and expands. The star then settles on the horizontal branch burning helium quietly at ~10⁸ K in a 0.5 M☉ core, with a hydrogen shell around it supplying most of the luminosity.

2–8 M☉: quiet ignition, then a white dwarf

Helium ignites non-degenerately. After core helium exhaustion the C/O core contracts and does become degenerate; hydrogen and helium burn in alternating shells (the thermally pulsing AGB), the envelope is lost as a planetary nebula, and the C/O core is left as a white dwarf of 0.5–1.1 M☉. Carbon never ignites because a degenerate C/O core lighter than ~1.06 M☉ never gets hot enough — the contraction ladder (section 3) hits the degeneracy wall.

Above ~8 M☉: on to carbon

The core stays non-degenerate, contracts to 5–8×10⁸ K and ignites carbon (section 9). During helium burning these stars also convert the ¹⁴N left by the CNO cycle into ²²Ne via ¹⁴N(α,γ)¹⁸F(β⁺ν)¹⁸O(α,γ)²²Ne — the origin of the "neutron excess" (more neutrons than protons) that later decides that the star ends with ⁵⁶Fe/⁵⁴Fe rather than ⁵⁶Ni, and the neutron source ²²Ne(α,n)²⁵Mg of the weak s-process.

Background: what "degenerate" means and why it removes the thermostat

Electrons are fermions: no two can share the same quantum state. Compress a gas enough and the low-energy states fill up; extra electrons must sit in high-energy states even at zero temperature. Their momentum then produces a pressure that depends only on density (P ∝ ρ^5/3, or ρ^4/3 when relativistic), not on temperature. This pressure supports white dwarfs and the cores of low-mass giants. Heating a degenerate gas does not raise its pressure, so it does not expand and cool: a nuclear fuel ignited under degeneracy burns in a runaway (flash) until the temperature is high enough that the electrons are no longer degenerate. Electrons are degenerate when ρ/μ_e ≳ 2.4×10⁻⁸ T^3/2 g cm⁻³ — the wall in section 3's T–ρ diagram.

SECTION 9

Carbon, neon, oxygen and silicon burning: the neutrino-cooled sprint to iron

Past helium, each fuel is heavier, needs a higher temperature, releases less energy per gram — and, from carbon burning on, the core's energy leaves not as light but as neutrinos, which escape instantly and force the core to burn faster and faster to keep up. A 25 M☉ star spends 7 million years on hydrogen, 800 000 on helium, 500 on carbon, a year on neon, four months on oxygen and about a day on silicon.

Stage	Fuel → products	T_c	ρ_c	Duration	Energy per gram of fuel	ε_nuc at T_c (model)	ε_ν pair (approx.)	Source

Conditions and durations are interpolated between the published rows of Woosley, Heger & Weaver (2002) for 13, 15, 20, 25 and 75 M☉ (the 75 M☉ row for Z = 10⁻⁴ Z☉ is listed separately in section 14); rows marked "estimated" lie outside those masses. ε_nuc uses the reaction rates of this page with the adopted product mixtures; the neutrino column is the non-degenerate pair-annihilation asymptote only, for orientation.

HHeCNeOSi

Stage durations versus initial mass (points: published models; lines: the interpolation used here; vertical line: your star). Fourteen orders of magnitude between the first and last stage.

ε_C (¹²C + ¹²C, X₁₂ = 0.2)ε_O (¹⁶O + ¹⁶O, X₁₆ = 0.7)ε_ν pair annihilation

Why the clock accelerates. Thermal neutrino losses (pair annihilation e⁺e⁻ → νν̄ above ~5×10⁸ K, plus photo- and plasmon-neutrinos) grow as T⁹ or steeper and are not blocked by the star's opacity. From carbon burning on, L_ν ≫ L_γ: the core must generate energy at the neutrino-loss rate, and it does so by contracting and heating until nuclear burning keeps up. Drawn at ρ = 2×10⁵ g cm⁻³ (carbon) and 3×10⁶ g cm⁻³ (oxygen).

The reactions

Carbon burning · 6–9×10⁸ K

¹²C + ¹²C → ²⁰Ne + ⁴HeQ = MeV (~56 %)

¹²C + ¹²C → ²³Na + pQ = MeV (~44 %)

¹²C + ¹²C → ²³Mg + nQ = MeV (rare, endothermic)

The released protons and alphas are captured at once (²³Na(p,α)²⁰Ne, ²⁰Ne(α,γ)²⁴Mg, ¹²C(p,γ)¹³N…), so the effective outcome adopted here is 7 ¹²C → 3 ²⁰Ne + ²⁴Mg, 5.0 MeV per ¹²C consumed (4.0×10¹⁷ erg g⁻¹), with ¹⁶O surviving as a spectator. Ashes: ~70 % ¹⁶O, 25 % ²⁰Ne, 5 % ²⁴Mg. The Gamow energy for ¹²C + ¹²C is 2.0 GeV; the rate exponent ν ≈ 30 at 8×10⁸ K.

Neon burning · 1.2–1.7×10⁹ K

²⁰Ne + γ → ¹⁶O + ⁴HeQ = MeV (photodisintegration)

²⁰Ne + ⁴He → ²⁴Mg + γQ = MeV

net: 2 ²⁰Ne → ¹⁶O + ²⁴MgQ = MeV, 1.1×10¹⁷ erg g⁻¹

Not a fusion of two neons: at 10⁹ K the photon gas has enough MeV photons to knock alphas off the relatively fragile ²⁰Ne (its last α is bound by only 4.73 MeV), and those alphas are captured by other ²⁰Ne. Neon burning is the first stage governed by photodisintegration, a preview of silicon burning.

Oxygen burning · 1.5–2.7×10⁹ K

¹⁶O + ¹⁶O → ²⁸Si + ⁴HeQ = MeV (~60 %)

¹⁶O + ¹⁶O → ³¹P + pQ = MeV (~40 %)

¹⁶O + ¹⁶O → ³¹S + nQ = MeV

Secondary captures turn the ashes into mostly ²⁸Si and ³²S (adopted here as 60/40 by mass, 4.7×10¹⁷ erg g⁻¹ of oxygen). Several side reactions involve β⁺ decays and electron captures, which lower the electron fraction Y_e below 0.5 — the neutron excess grows.

Silicon burning · 2.7–3.7×10⁹ K

²⁸Si + γ → ²⁴Mg + α, ²⁴Mg + γ → ²⁰Ne + α, … → 7 α"melting" of silicon

²⁸Si + α → ³²S → ³⁶Ar → ⁴⁰Ca → ⁴⁴Ti → ⁴⁸Cr → ⁵²Fe → ⁵⁶Nithe alpha ladder, Q per step below

²⁸Si + ²⁸Si never happens (the Coulomb barrier is prohibitive). Instead photons dismantle part of the silicon into alphas, and the rest captures those alphas up the ladder. Most links are in equilibrium in both directions, so the abundances are set by temperature, density and Y_e rather than by any one reaction path — a state of quasi-statistical equilibrium that becomes full nuclear statistical equilibrium (NSE) above ~4×10⁹ K. The energy release is only 1.9×10¹⁷ erg g⁻¹ (to ⁵⁶Ni) or 2.6×10¹⁷ (to ⁵⁶Fe): a day's worth against the neutrino drain.

The alpha ladder from ⁴He to ⁵⁶Ni: binding energy per nucleon of each rung and the energy released by adding an α (computed from the mass table). The steps get smaller and smaller; after ⁵⁶Ni the next rung, ⁶⁰Zn, is bound less tightly per nucleon — the ladder has reached the top of the binding-energy curve.

Why the core ends as iron — and which iron

Silicon burning makes ⁵⁶Ni first because it is the most tightly bound nucleus with Z = N — and the alpha ladder can only make Z = N nuclei. But the core's Y_e has already dropped to 0.46–0.49 through electron captures and β⁺ decays during oxygen and silicon burning, so NSE shifts the composition toward neutron-richer nuclei: at Y_e ≈ 0.48 it favours ⁵⁴Fe and ⁵⁸Ni, at Y_e ≈ 0.46 ⁵⁶Fe (Y_e = 26/56 = 0.464), and the presupernova core is a mixture of ⁵⁴Fe, ⁵⁶Fe, ⁵⁸Ni with traces of ⁵⁵Fe and ⁵⁷Co (Woosley's 25 M☉ model after core silicon burning: 49 % ⁵⁴Fe, 15 % ⁵⁸Ni, 14 % ⁵⁶Fe, Y_e = 0.4775). ⁵⁶Ni is made — in the explosive silicon burning of the supernova shock, where there is no time for electron captures — and its decay ⁵⁶Ni → ⁵⁶Co → ⁵⁶Fe powers the supernova's light curve (section 11).

Nuclear statistical equilibrium in one line

forward rate = reverse rate \Leftrightarrow μ (A, Z) = Z μ_{p} + (A - Z) μ_{n}

Nuclei keep reacting but the abundances stop changing. Given T, ρ and Y_e, the Saha-type relation below fixes every abundance; the mixture is dominated by the most tightly bound nuclei compatible with the available neutron-to-proton ratio. NSE does not need the weak interactions to be in equilibrium — Y_e is an input that electron captures slowly change.

Graduate note: the NSE abundance relation and the Y_e constraint

Y (A, Z) = G (A, Z) {(\frac{ρ N_{A}}{θ})}^{A - 1} \frac{A^{3 / 2}}{2^{A}} Y_{p}^{Z} Y_{n}^{A - Z} \exp (\frac{B (A, Z)}{k T}), θ = 5.943 \times 10^{33} T_{9}^{3 / 2} {cm}^{- 3}

with G the partition function and B the binding energy (Woosley's lecture notes; Clayton ch. 7). Two constraints close the system: mass conservation Σ A Y = 1 and charge conservation Σ Z Y = Y_e. The neutron excess η = 1 − 2Y_e is 0 for pure ⁵⁶Ni, 0.037 for ⁵⁴Fe and 0.071 for ⁵⁶Fe; a mixture of ⁵⁴Fe, ⁵⁶Fe and ⁵⁸Ni can only represent Y_e ≥ 0.464 — for the still lower Y_e ≈ 0.42–0.45 reached at the onset of collapse, NSE moves on to ⁵⁸Fe, ⁶²Ni and heavier neutron-rich isotopes. At T ≳ 10¹⁰ K the exponential no longer wins and NSE dissolves the nuclei into α particles and nucleons: that is the photodisintegration of section 11.

Background: hydrostatic versus explosive burning, and the two kinds of neutrinos

Everything in this section is hydrostatic burning: slow, in a core held up by pressure, with time for electron captures to lower Y_e. Explosive burning happens in the supernova shock in under a second at 4–5×10⁹ K: it also reaches NSE but freezes out with Y_e ≈ 0.5, which is why the ejecta carry ⁵⁶Ni while the collapsed core kept the ⁵⁴Fe/⁵⁶Fe. Two neutrino sources appear on this page: reaction neutrinos from β⁺ decays and electron captures in the burning chains (subtracted from each Q), and thermal neutrinos made by the hot plasma itself (pair annihilation, photo- and plasmon-neutrinos) — a separate energy sink that dominates the late stages and is never part of any Q-value.

SECTION 10

The onion star: how the burning shells stack up

Each time the centre exhausts a fuel, burning does not stop — it moves outward to a shell where that fuel still exists, while the core contracts and ignites the next one. By the time silicon burns in the centre, a massive star is a nested set of shells: an iron core, then silicon, oxygen/neon, carbon/oxygen, helium and hydrogen, each layer the ash of the burning that happens just outside it. The cross-section at the top of the page draws this for your star at the moment you scrubbed to; here is the same structure as numbers.

Layer (centre outward)	Composition	Outer mass coordinate	Typical radius	Burning at its base

Reading the two views

Mass-coordinate view: the radius of each ring is proportional to the enclosed mass. This shows how the material is distributed — the iron core is 10 % of the star, the hydrogen envelope more than half.

Log-radius view: rings are spaced by the logarithm of the physical radius. This shows how absurdly compact the interior is: the iron core is ~2000 km across, the whole nuclear-burning region fits inside a few solar radii, and the red-supergiant envelope reaches out 1000 solar radii — a ratio of a million between the iron core and the surface. The core would fit inside the Earth many times over, yet it decides the fate of a star wider than the orbit of Mars.

Mass loss matters: by the time it explodes, a 25 M☉ star has blown away half its mass in winds; the shell masses above are the final values from the published models.

Why burning moves outward into shells

When the core exhausts a fuel it contracts (virial theorem) and heats — but so does the shell immediately above it, which still holds unburned fuel. That shell reaches the ignition temperature first, so burning restarts there. The inner core, now inert, keeps contracting until it reaches the ignition temperature of its own ashes, at which point it burns again with a shell source above it. Each shell source also pushes the layers outside it to expand (the mirror principle: contraction inside a shell source causes expansion outside it), which is what turns a compact blue star into a red supergiant of 1000 solar radii while its core shrinks to the size of the Earth. Convection appears wherever the energy flux is too large for radiation to carry (the CNO cycle's T¹⁸ and helium burning's T⁴⁰ concentrate the power), so most shells are partially mixed, which blurs the boundaries in real models.

SECTION 11

The iron catastrophe: why the core collapses, and how it explodes

Iron is the end of the road because it is the top of the binding-energy curve: fusing it absorbs energy instead of releasing it. So an iron core has no furnace. It is held up only by the pressure of its electrons, and that support has a limit — the Chandrasekhar mass. As silicon shells keep adding iron, the core edges toward that limit while two processes actively remove its support: photons start breaking iron nuclei apart (absorbing energy), and electrons are captured by protons (removing pressure). Within a fraction of a second the core falls in on itself at a quarter of the speed of light.

The Chandrasekhar mass

M_{Ch} = 1.457 (2 Y_{e})^{2} M_{⊙}, M_{Ch,eff} \approx M_{Ch} [1 + {(\frac{s_{e}}{π Y_{e}})}^{2}]

Y_e is the number of electrons per nucleon (0.5 for ⁵⁶Ni, 0.464 for ⁵⁶Fe, 0.42–0.46 in real presupernova cores after electron captures); s_e is the electronic entropy per baryon in units of k (≈ 0.7–1 in a hot core), which raises the limit above the cold value. Set them here to see the effect: Y_e = , s_e = → M_Ch = M☉ (cold), M_Ch,eff = M☉. The iron core that your star's silicon burning delivers is about M☉ (from detailed models). This panel illustrates the stability condition; the timeline's collapse simply follows the end of silicon burning in the published models.

Derivation: where the Chandrasekhar mass comes from

Start from: the ultra-relativistic degeneracy pressure; hydrostatic equilibrium with the mass-continuity equation; the Sommerfeld expansion of a degenerate Fermi gas (imported; Pauli); dimensional analysis. Symbols: n_e = Y_eρ/m_u the electron density; K the polytropic constant; n the polytropic index (3 here); θ(ξ) the Lane–Emden function with ρ = ρ_cθⁿ, r = aξ; ξ₁ its first zero; E_F the Fermi energy; s_e the electron entropy per baryon in units of k; (ħc/G)^1/2 = 2.176×10⁻⁵ g the Planck mass.

1. Equation of state of relativistic degenerate electrons, written as a polytrope of index 3. imported

P = \frac{ℏ c}{4} (3 π^{2})^{1 / 3} n_{e}^{4 / 3} = K ρ^{4 / 3}, K = \frac{ℏ c}{4} (3 π^{2})^{1 / 3} {(\frac{Y_{e}}{m_{u}})}^{4 / 3}

2. Combine hydrostatic equilibrium with mass continuity to eliminate m(r). identity

\frac{d P}{d r} = - \frac{G m ρ}{r^{2}}, \frac{d m}{d r} = 4 π r^{2} ρ \Rightarrow \frac{1}{r^{2}} \frac{d}{d r} (\frac{r^{2}}{ρ} \frac{d P}{d r}) = - \frac{G}{r^{2}} \frac{d m}{d r} = - 4 π G ρ

3. Insert a general polytrope P = Kρ^1+1/n with ρ = ρ_cθⁿ: the pressure term becomes a derivative of θ. identity

\frac{1}{ρ} \frac{d P}{d r} = K \frac{n + 1}{n} ρ^{1 / n - 1} \frac{d ρ}{d r} = K (n + 1) \frac{d (ρ^{1 / n})}{d r} = K (n + 1) ρ_{c}^{1 / n} \frac{d θ}{d r} \Rightarrow \frac{1}{r^{2}} \frac{d}{d r} (r^{2} \frac{d θ}{d r}) = - \frac{4 π G ρ_{c}^{1 - 1 / n}}{(n + 1) K} θ^{n}

4. Scale the radius, r = aξ, choosing a so that the constants disappear: the Lane–Emden equation, with θ(0) = 1, θ′(0) = 0, and the surface at the first zero ξ₁. definition

a^{2} \equiv \frac{(n + 1) K ρ_{c}^{1 / n - 1}}{4 π G} \Rightarrow \frac{1}{ξ^{2}} \frac{d}{d ξ} (ξ^{2} \frac{d θ}{d ξ}) = - θ^{n}

5. The mass: integrate the density over the sphere and use the Lane–Emden equation itself to do the integral. identity

M = \int_{0}^{R} 4 π r^{2} ρ d r = 4 π a^{3} ρ_{c} \int_{0}^{ξ_{1}} ξ^{2} θ^{n} d ξ = 4 π a^{3} ρ_{c} {[- ξ^{2} \frac{d θ}{d ξ}]}_{0}^{ξ_{1}} = 4 π a^{3} ρ_{c} ξ_{1}^{2} | θ^{'} (ξ_{1}) |

6. The special case n = 3: a² = Kρ_c^−2/3/(πG), so a³ρ_c = (K/πG)^3/2ρ_c⁻¹ρ_c — the central density cancels. Whatever its central density, a relativistic degenerate star has this one mass: for P ∝ ρ^4/3 the pressure force and gravity scale identically under compression, so compression cannot restore a balance that mass has broken. identity

a^{3} ρ_{c} = {(\frac{K}{π G})}^{3 / 2} ρ_{c}^{- 1} ρ_{c} \Rightarrow M = 4 π {(\frac{K}{π G})}^{3 / 2} ξ_{1}^{2} | θ^{'} (ξ_{1}) |

7. The n = 3 numbers: the page integrates the Lane–Emden equation itself (fourth-order Runge–Kutta, step h = 0.01 in ξ, series start θ = 1 − ξ²/6 at ξ = 10⁻⁴, stopped at the first θ ≤ 0 without root interpolation) and obtains ξ₁ = 6.897 and ξ₁²|θ′(ξ₁)| = 2.018, the tabulated values 6.8968 and 2.0182 to this precision. numerical

ξ_{1} = 6.897, ξ_{1}^{2} | θ^{'} (ξ_{1}) | = 2.018

8. Insert K: gather the powers of 4, 3π² and π, then multiply by the 4π in front: 4π × √3/(8√π) = √(3π)/2. identity

{(\frac{K}{π G})}^{3 / 2} = {(\frac{ℏ c}{G})}^{3 / 2} {(\frac{1}{4})}^{3 / 2} \frac{(3 π^{2})^{1 / 2}}{π^{3 / 2}} \frac{Y_{e}^{2}}{m_{u}^{2}} = {(\frac{ℏ c}{G})}^{3 / 2} \frac{\sqrt{3}}{8 \sqrt{π}} \frac{Y_{e}^{2}}{m_{u}^{2}} \Rightarrow M_{Ch} = \frac{\sqrt{3 π}}{2} ξ_{1}^{2} | θ^{'} (ξ_{1}) | {(\frac{ℏ c}{G})}^{3 / 2} \frac{Y_{e}^{2}}{m_{u}^{2}}

9. Numbers: (ħc/G)^3/2 = (2.176×10⁻⁵ g)³ = 1.031×10⁻¹⁴ g³; divided by m_u² = 2.757×10⁻⁴⁸ g² gives 3.74×10³³ g; × 1.535 × 2.018 = 1.16×10³⁴ g = 5.83 M☉ (times Y_e²). numerical

M_{Ch} = 5.83 Y_{e}^{2} M_{⊙} = 1.457 (2 Y_{e})^{2} M_{⊙}

10. Thermal correction: at finite temperature the relativistic Fermi gas has P = P₀[1 + (2π²/3)(kT/E_F)² + …] (Sommerfeld expansion); since M ∝ K^3/2 the limit rises by the 3/2 power of that bracket, ≈ 1 + π²(kT/E_F)². The electron entropy per baryon of a degenerate relativistic gas is s_e = π²Y_ekT/E_F, so kT/E_F = s_e/(π²Y_e). approximation

M_{Ch,eff} \approx M_{Ch} [1 + π^{2} {(\frac{k T}{E_{F}})}^{2}] = M_{Ch} [1 + {(\frac{s_{e}}{π Y_{e}})}^{2}]

11. The examples of ①: Y_e = 0.5 and 0.464 cold, and a hot core with Y_e = 0.46, s_e = 0.8. In a presupernova core Y_e falls (electron captures) and s_e falls (neutrino losses), both lowering M_Ch,eff toward the growing iron core — collapse begins when they meet, at about 1.3–1.5 M☉ for the models on this page. numerical

5.83 \times 0.25 = 1.457 M_{⊙}, 5.83 \times 0.2153 = 1.255 M_{⊙}, 1.234 [1 + {(\frac{0.8}{π \times 0.46})}^{2}] = 1.234 \times 1.31 = 1.61 M_{⊙}

Result: M_Ch = (√(3π)/2) ξ₁²|θ′(ξ₁)| (ħc/G)^3/2Y_e²/m_u² = 5.83 Y_e² M☉, and M_Ch,eff ≈ M_Ch[1 + (s_e/πY_e)²]. Derived: the mass formula and its independence of ρ_c; Calculated: ξ₁ and ξ₁²|θ′₁| by the page's integrator, and the constant 5.83; Imported: the Sommerfeld expansion and the entropy relation of step 10.

The two triggers

⁵⁶Fe + γ → 13 ⁴He + 4 nQ = MeV (absorbs 2.2 MeV per nucleon)

⁴He + γ → 2 p + 2 nQ = MeV (7.1 MeV per nucleon)

p + e⁻ → n + ν_e (Z,A) + e⁻ → (Z−1,A) + ν_eelectron capture: Y_e ↓, pressure ↓, ν escape

At 10¹⁰ K the photons average ~1 MeV and their high-energy tail dismantles iron (a Saha-type equilibrium, like ionising hydrogen but at a million times the energy); undoing all the fusion of the star's lifetime costs erg for a 1.4 M☉ core (complete dissociation to nucleons would cost erg). Meanwhile the electron Fermi energy, ε_F ≈ 11.1 (ρ₁₀Y_e)^1/3 MeV at ρ = 10¹⁰ρ₁₀ g cm⁻³, exceeds the capture thresholds of the iron-group nuclei, so electrons — the only thing holding the core up — are swallowed. Both processes push the adiabatic index below the critical 4/3: the core is dynamically unstable.

How fast? The free-fall time

t_{ff} = \sqrt{\frac{3 π}{32 G ρ}} \approx 0.021 s {(\frac{ρ}{10^{10} g {cm}^{- 3}})}^{- 1 / 2}

At the mean density of the presupernova iron core (~10⁹–10¹⁰ g cm⁻³) that is 20–70 ms; the inner ~0.5–0.7 M☉ collapses homologously (v ∝ r) and the outer core falls supersonically. From the onset of collapse to nuclear density takes about 0.2–0.3 s. Infall speeds reach 50 000–70 000 km s⁻¹.

t = −0.20 s · log-time scrubber from 0.2 s before bounce to 3 s after · the hero cross-section at the top follows the same clock when the time slider is at "Collapse"

Core collapse and neutrino-driven explosion (schematic radii and velocities scaled from Janka 2012 and Pols ch. 13). Phases: (1) infall as electron capture and photodisintegration remove the support; (2) at ρ ≈ 2–3×10¹⁴ g cm⁻³ the nuclear force turns repulsive and the inner core rebounds — bounce; (3) the bounce launches a shock that stalls within ~10 ms at 100–200 km, exhausted by dissociating infalling iron (~9 MeV per nucleon) and by neutrino losses; (4) the proto-neutron star radiates ~3×10⁵³ erg of neutrinos over ~10 s, of which about 1 % is absorbed behind the shock; (5) after 0.1–1 s the heated region re-launches the shock (aided by turbulence) and the star explodes with ~10⁵¹ erg of kinetic energy.

The energy budget

Item	Energy	How it is computed
Gravitational binding energy released forming a , 12 km neutron star	erg	E ≈ 0.6 GM²/R (uniform sphere); real equations of state give 2–4×10⁵³ erg. The published model's own value, (M_b − M_g)c², is (section 11b)
Carried away by neutrinos (all six flavours, ⟨E⟩ ≈ 10–15 MeV, ~10 s)	≈ 99 %	SN 1987A: 24 events in three detectors within 13 s — total ≈ 3×10⁵³ erg inferred
Kinetic energy of the ejecta	≈ 10⁵¹ erg (1 "Bethe")	~10 M☉ of ejecta at ~3000–10 000 km s⁻¹ (typical successful explosion; ranges 0.3–3×10⁵¹)
Light radiated over months	≈ 10⁴⁹ erg	L ≈ 10⁸–10⁹ L☉ for ~100 days; powered mostly by ⁵⁶Ni → ⁵⁶Co → ⁵⁶Fe
Radioactive energy of the ejected ⁵⁶Ni (0.07 M☉ typical)	erg	Q(⁵⁶Ni→⁵⁶Co) + Q(⁵⁶Co→⁵⁶Fe) = MeV per nucleus, from the mass table; half-lives 6.08 d and 77.2 d
Energy the star released by fusion over its whole life	erg	Σ (burned mass × energy per gram) from the timeline of section 12 — about a tenth of one second of the neutrino burst

What is left, by initial mass

Your star ( M☉): .

≲ 8 M☉ · white dwarf

No collapse. The C/O (or He) core is left as a white dwarf supported by electron degeneracy, cooling for ever unless a companion feeds it to the Chandrasekhar mass (a Type Ia supernova, which burns the whole star to ⁵⁶Ni instead of collapsing).

8–9 M☉ · the transition band

Carbon ignites off-centre; the O/Ne/Mg core may grow to 1.37 M☉ where electron captures on ²⁴Mg and ²⁰Ne trigger a collapse before neon ever burns — an electron-capture supernova (the Crab Nebula's progenitor is a candidate) — or be left as an O/Ne/Mg white dwarf. The published core-collapse models used below begin at 9 M☉.

≳ 9 M☉ · core collapse

All fuels to iron, then collapse. In the published models the outcome is not a simple function of mass: most stars below ~15 M☉ explode and leave a 1.2–1.7 M☉ neutron star (radius ~12 km, 10¹⁴ g cm⁻³ — a stellar-mass nucleus); between 15 and 30 M☉ islands of successful and failed explosions alternate with the core's compactness, and failures leave black holes; the most massive stars lose their envelopes and the compact helium stars that remain often explode again. Sufficiently massive helium cores can hit the pair instability instead (electron–positron pairs soak up the pressure; pulsations, complete disruption or collapse), outside this model. Section 11b reads the model nearest your star.

Graduate note: why the prompt shock fails and neutrinos are needed

The bounce shock starts with ~10⁵¹ erg but must plough through ~0.5–1 M☉ of infalling iron-group material, dissociating it at ~9 MeV per nucleon (~2×10⁵¹ erg per 0.1 M☉) while electron captures on the freed protons radiate neutrinos from the shock-heated matter once the density drops below ~10¹¹–10¹² g cm⁻³. The shock stalls into an accretion shock at 100–200 km. Behind it, neutrinos streaming from the proto-neutron star (ν_e + n → p + e⁻, ν̄_e + p → n + e⁺) deposit energy in the "gain region"; heating wins over cooling when the neutrino luminosity is high enough for the given accretion rate (the critical-luminosity condition of Burrows & Goshy 1993), and multidimensional instabilities (convection, the standing accretion shock instability) lengthen the time matter spends in the gain region. Whether a given star explodes depends on its presupernova structure — the density gradient outside the iron core (compactness) — which is why explodability is not a monotonic function of mass (O'Connor & Ott 2011; Sukhbold et al. 2016).

SECTION 11b

Your star's explosion, read from a published model

Whether a collapsing core explodes cannot be computed from a formula: it needs a full simulation of the neutrino-heated, turbulent, stalled shock. This section therefore does something different from the rest of the page. It takes the nearest of 33 published solar-metallicity models (Sukhbold, Ertl, Woosley, Brown & Janka 2016) whose presupernova structure, explosion outcome and, for the exploders, bolometric light curve are all embedded here from the same archive, prints the published numbers, and then adds only elementary, stated calculations on top of them: the remnant's gravitational mass and the energy the neutrinos carry away, the peak temperatures the shock produces in each layer, a schematic light curve, and the neutrino signal a detector on Earth would see.

The selected model:

Calibration set: (the Z9.6 set is calibrated to the Crab supernova for 9–12 M☉, the W18 set to SN 1987A above 12 M☉). Solar-metallicity models: the metallicity slider does not change this panel.

Presupernova star (published)	value
Initial mass of the model	M☉
Mass at collapse (after mass loss)	M☉
Helium core · C/O core	· M☉
Iron core (Y_e < 0.49)	M☉
Radius	R☉
Compactness ξ_2.5 = (2.5 M☉) / (r(2.5 M☉)/1000 km)
M₄ (mass inside entropy s = 4) · μ₄ (its density gradient)	M☉ ·
Outcome (published)
Does the shock get revived?
Explosion energy E
Baryonic remnant M_b (mass cut after fallback)	M☉
Fallback mass (already included in M_b)	M☉
Ejected ⁵⁶Ni · tracer (matter of uncertain composition)	· M☉
Derived here (stated calculations)
Remnant
Gravitational mass M_g (β = GM_g/Rc² = )	M☉
Energy radiated in neutrinos, (M_b − M_g)c²	erg
Ejected mass M_ej = M_preSN − M_b	M☉
Bookkeeping check M_ej + M_b = M_preSN

Neighbouring models on the archive's 0.1 M☉ grid:

Gravitational mass of the neutron star

\frac{M_{b} - M_{g}}{M_{g}} = \frac{0.6 β}{1 - 0.5 β}, β = \frac{G M_{g}}{R c^{2}}, R = 12 km

The binding energy of the neutron star is radiated as neutrinos while it forms, so the mass that gravity feels afterwards (M_g) is less than the mass of the baryons that fell in (M_b). This radius-dependent fit (Lattimer & Prakash 2001) is the one Sukhbold et al. use; it is solved numerically here. E_ν ≈ (M_b − M_g)c² then follows from mass–energy equivalence. No conversion is made for black holes.

Explosion energy — ×10⁵¹ erg sensitivity experiment: same progenitor, calibrated fallback and remnant retained

Peak temperature behind the shock, on the model's own r(m). The solid curve is the approximate diagnostic T_peak(r) = (3E/4πar³)^1/4: it assumes the explosion energy E stands in for the internal energy of a radiation-dominated fireball filling the sphere of radius r (E_rad = (4π/3) r³ a T⁴), which it does only roughly. The dashed curve is the temperature before collapse, for comparison; the dotted pink line is the mass cut. The horizontal lines mark qualitative processing regions; the table gives the ejecta mass exposed to each — masses only, never isotopes: assigning ⁵⁶Ni or any other product to a band would need a full network with the local Y_e. The archive's own ejected ⁵⁶Ni is printed on the left and feeds the light curve. The energy slider shows how the bands move with E (T_peak ∝ E^1/4).

Peak temperature of a radiation-dominated fireball

E = \frac{4 π}{3} r^{3} a T^{4} \Rightarrow T_{peak} (r) = {(\frac{3 E}{4 π a r^{3}})}^{1 / 4}

a = 7.566×10⁻¹⁵ erg cm⁻³ K⁻⁴ is the radiation constant. At r = 3.7×10⁸ cm (3700 km) and E = 10⁵¹ erg this gives 5×10⁹ K — hot enough to rearrange silicon into the iron group; at 10⁹ cm it is 2.4×10⁹ K (explosive oxygen and neon burning) and at 2×10⁹ cm, 1.4×10⁹ K, the composition survives. Woosley & Weaver (1995) used this scaling to organise explosive nucleosynthesis.

bolometric light curve · the model curve is schematic; the reference curve is the archive's own radiation-hydrodynamics result for the same model

schematic model (this page)radioactive deposition alonepublished light curve of the model

Light curve.

Plateau of a hydrogen-rich supernova (Popov scaling, normalised by Sukhbold et al. 2016)

L_{50} = 1.85 \times 10^{42} E_{51}^{5 / 6} M_{10}^{- 1 / 2} R_{500}^{2 / 3} erg s^{- 1}, t_{p} = 88 E_{51}^{- 1 / 6} M_{10}^{1 / 2} R_{500}^{1 / 6} d

E₅₁ is the explosion energy in 10⁵¹ erg, M₁₀ the hydrogen-envelope mass in units of 10 M☉ (here M_env = max[0, M_preSN − max(M_He, M_b)]), R₅₀₀ the presupernova radius in units of 500 R☉. A recombination wave moving inward through the ionised envelope releases the shock-deposited energy at a nearly constant rate: that is the plateau. Applied only to extended hydrogen-rich stars (R > 100 R☉ and M_env > 1 M☉). The join between plateau and tail is drawn schematically; the ⁵⁶Ni mass is never adjusted to make them meet.

Radioactive tail and its trapping

L_{rad} = M_{Ni} [ε_{Ni} e^{- t / τ_{Ni}} f_{γ} + ε_{Co} \frac{τ_{Co}}{τ_{Co} - τ_{Ni}} (e^{- t / τ_{Co}} - e^{- t / τ_{Ni}}) (0.965 f_{γ} + 0.035)], f_{γ} = 1 - e^{- (t_{0} / t)^{2}}, t_{0} = \sqrt{\frac{3 κ_{γ} M_{ej}}{4 π v^{2}}}

τ_Ni = 8.77 d and τ_Co = 111.4 d are the e-folding times (half-lives 6.075 and 77.24 d divided by ln 2); ε_Ni = and ε_Co = erg g⁻¹ s⁻¹ follow from 1.72 MeV of γ-rays per ⁵⁶Ni decay and 3.61 MeV of γ-rays plus 0.12 MeV of positron kinetic energy per ⁵⁶Co decay. The τ_Co/(τ_Co − τ_Ni) factor is the exact solution of the two-step decay chain (Bateman). γ-rays escape once the ejecta thin out: f_γ is the fraction absorbed by a uniformly expanding sphere of optical depth (t₀/t)² with κ_γ = 0.03 cm² g⁻¹; the positrons (3.5 % of the cobalt energy) are assumed to deposit fully. For stripped stars the same heating is fed through the Arnett (1982) diffusion solution L(t) = ∫₀ᵗ (2t′/τ_m²) e^{(t′²−t²)/τ_m²} Q(t′) dt′ with τ_m = (2κM_ej/βcv)^1/2, κ = 0.1 cm² g⁻¹, β = 13.8 and v from E = (3/10)M_ejv².

Neutrinos: the energy that leaves unseen

Almost all of the energy released by the collapse leaves as neutrinos: erg for the selected model, from (M_b − M_g)c² above — about a hundred times the kinetic energy of the ejecta. Shared equally among the six species (ν_e, ν̄_e and the four heavy-flavour states; an illustrative assumption), the electron antineutrinos carry erg. With a mean energy of 12 MeV that is antineutrinos.

Water detectors see them through inverse beta decay, ν̄_e + p → n + e⁺. At a distance of kpc the fluence at Earth is cm⁻²; a Super-Kamiokande-sized target of 1.5×10³³ free protons (22.5 kt of water) then registers about interactions (spectrum-averaged cross-section cm²), before detector efficiency and thresholds. Scaled to the 50 kpc of SN 1987A the same model would give in such a detector; the 1987 detectors were smaller and recorded 24 events in total (Kamiokande-II, IMB and Baksan) — context, not a validation of this idealised count.

Inverse beta decay

σ (E_{ν}) ≃ 9.52 \times 10^{- 44} \frac{E_{e} p_{e}}{{MeV}^{2}} {cm}^{2}, E_{e} = E_{ν} - 1.293 MeV, N = N_{p} \frac{E_{{\bar{ν}}_{e}} / ⟨ E ⟩}{4 π d^{2}} \int f (E) σ (E) d E

Vogel & Beacom (1999) at leading order; the threshold is 1.806 MeV. f(E) is the normalised Fermi–Dirac spectrum E²/(e^E/T + 1) with zero chemical potential and T = ⟨E⟩/3.15 = MeV, so that ⟨E⟩ = 12 MeV. Because σ grows as E², the detected spectrum peaks well above the emitted one (right).

Emitted and detected spectra. The emitted Fermi–Dirac spectrum (green) and the same spectrum weighted by the inverse-beta-decay cross-section (orange), each normalised to its maximum. Neutrino flavour oscillations, detector response and the time structure of the burst (≈ 10 s) are not modelled.

Why some stars explode and others do not: the Ertl two-parameter diagnostic for all 200 models

Ertl et al. (2016) found that the fate of a presupernova star is organised by two numbers of its structure: M₄, the mass inside the point where the entropy per baryon reaches s = 4 (roughly the edge of the silicon core), and μ₄, the mass gradient dm/dr just outside it (in 0.3 M☉ per 1000 km). A steep gradient (small μ₄) means the mass accretion onto the stalled shock falls quickly, which favours revival. Models below the line μ₄ = k₁(M₄μ₄) + k₂ explode in the calibrated simulations, with k₁ = 0.283 and k₂ = 0.043 for the W18 calibration. The criterion has exceptions and is a diagnostic; the archive outcome is what the page uses.

explodes (archive outcome)no explosionno outcome rowselected model

Each dot is one of the 200 presupernova models (M₄μ₄ against μ₄, from the archive's properties table); the dashed line is the W18 separation line. Hover for the model.

SECTION 12

A life in one line: the timeline and the energy budget

The stages of your star, drawn to a logarithmic scale so that the year of neon burning and the day of silicon burning are visible next to the millions of years of hydrogen burning. Drag the time slider at the top of the page and the marker moves along this bar.

Stage durations on a log-time axis (the width of each block is proportional to log duration; the numbers are the durations themselves).

HHeCNeOSi

Energy released per stage = mass of fuel burned × energy per gram (section 2), with the burned masses taken from the fuel-access fraction q_c for hydrogen and from the schematic shell structure for the later stages (approximate). Hydrogen burning dominates by far; the whole post-helium sprint releases less than a per cent of the total.

Stage	Starts at age	Lasts	Fuel (approx.)	Energy (erg)	Mean power (L☉)

Mean power = energy / duration. Compare the last column with the star's photon luminosity (L = L☉ on the main sequence): from carbon burning on, the nuclear power exceeds the light output by orders of magnitude — the surplus is the neutrino luminosity.

Background: how to read a logarithmic timeline

On the bar above, equal widths mean equal ratios, not equal differences: the block for 10 000 years is as wide as the block for 10 years to 10 000 years. This is the only way to draw a sequence whose durations span from 10⁷ years to 1 day (a ratio of 10¹⁰) on one screen. Where the page says "the star spends 90 % of its life on the main sequence", that is on a linear clock; the log bar deliberately exaggerates the late stages so you can see them at all.

SECTION 13

Where the elements come from — and why the story does not end at iron

Every element on this page up to the iron group was made by the burning stages you have just followed, in a star of some mass, and returned to space by winds or by the explosion. Hover the table: the colour is the dominant process; for many elements several processes contribute. Beyond iron the page changes gear again: it shows the published solar pattern and the published s-/r-process split, and then runs two demonstrators — the classical s-process as an exposure experiment, and the r-process equilibrium path — that make the physics of neutron capture visible without pretending to compute the yields of a real event.

overlay: s-process share of each element, Arlandini et al. 1999 (stellar model; grey = not in their table)

Which star made which element (simplified)

H, most He, traces of Li: the Big Bang, three minutes in.
Li, Be, B: mostly cosmic rays shattering C, N and O nuclei in interstellar space (they are destroyed inside stars).
C and most N: helium burning and the CNO cycle in stars of 1–8 M☉, dredged up and blown off on the asymptotic giant branch — plus massive stars.
O, Ne, Mg, Na, Al: helium, carbon and neon burning in massive stars (oxygen is the most abundant element made in stars).
Si, S, Ar, Ca, P, Cl, K: oxygen burning, hydrostatic and explosive.
Ti, V, Cr, Mn, Fe, Co, Ni: silicon burning to ⁵⁶Ni and NSE — roughly half of the Galaxy's iron from core-collapse supernovae, half from Type Ia (exploding white dwarfs).
Cu, Zn, Ga, Ge, Sr, Y, Zr, Ba, La, Ce, Pb…: the s-process — slow neutron capture in helium-burning shells of AGB stars (neutrons from ¹³C(α,n)¹⁶O) and in massive stars (²²Ne(α,n)²⁵Mg).
Ag, Te, I, Xe, Eu, Pt, Au, U, Th…: the r-process — rapid neutron capture in neutron-star mergers (GW170817, 2017) and possibly rare magnetorotational supernovae.

Beyond iron: why fusion cannot continue, and what does

Adding an alpha particle to ⁵⁶Ni gives ⁶⁰Zn, which is bound less tightly per nucleon; every charged-particle step past the iron group costs energy and must fight a Coulomb barrier of Z ≈ 28. No hydrostatic star can do that. Heavier elements are built instead by neutron capture, which has no Coulomb barrier at all:

Neutron capture and β-decay: the two-step dance

(Z, A) + n \to (Z, A + 1) + γ, (Z, A + 1) \to (Z + 1, A + 1) + e^{-} + {\bar{ν}}_{e}

s-process (slow): neutron captures every ~10 years, so unstable isotopes decay before capturing again — the path hugs the valley of stable nuclei and ends at ²⁰⁹Bi (the next products decay by α). Neutron density ~10⁷–10¹⁰ cm⁻³. r-process (rapid): captures every millisecond at neutron densities above 10²⁰ cm⁻³ pile neutrons on until the nucleus can hold no more (the "waiting points"); the wildly neutron-rich nuclei then β-decay back toward stability. Only the r-process makes the elements heavier than bismuth — thorium, uranium, plutonium.

The energy released by neutron capture is a few MeV per capture, but the heavy elements are made in such small quantities (a solar-mass star contains about 10⁻⁷ M☉ of gold) that they play no role in the star's energy budget. Their importance is different: the r-process needs conditions found only in the most violent events in the universe, so every gold atom on Earth records a neutron-star merger or an exotic supernova that happened before the Sun formed.

The solar system's heavy elements, as published

solar system, isotope by isotope (Lodders 2010 elements × Wallet-card isotope fractions; Si = 10⁶)s-process main component N_s (Arlandini et al. 1999 stellar model)r-process residual N_r = N_⊙ − N_s (their Table 2)

Abundance against mass number. Two normalisations are shown deliberately side by side: the grey curve converts Lodders' logarithmic scale, N/N_H = 10^A(X)−12 × (isotope fraction), to the Si = 10⁶ scale of meteoritic work (A(Si) = 7.53); the coloured points are Arlandini's own numbers on their own solar reference (Anders & Grevesse 1989), never rescaled to Lodders. Their decomposition covers 63 ≤ A ≤ 209 and treats the main s-component only: below A ≈ 90 the "weak" s-process of massive stars is missing from N_s, so the residual there overstates the r-process. Hover for values. The three peaks of the r-residual near A ≈ 80, 130 and 195 and the s-peaks near 88, 138 and 208 are the fingerprints of the closed neutron shells N = 50, 82, 126 — reached far from stability by the r-process and at stability by the s-process.

The capture paths on the chart of nuclides

r-process temperature T = 1.0 GK neutron density n_n = 10²⁴ cm⁻³ · hover an isotope for its numbers

stable isotope (NUBASE2020)classical s-process path from ⁵⁶Fe (unbranched)branch point on the s-path (half-life 1 d – 10 yr)r-process waiting point, S_n from measured AME2020 masseswaiting point from AME2020 extrapolated (#) masseswaiting point from FRDM95 theoretical masseschain reaches the edge of the mass table

Neutron number against proton number. The s-path is computed from the data: from ⁵⁶Fe, each capture goes to (Z, A+1); if that nucleus is unstable with a half-life under 10 years it decays at once along its NUBASE2020 mode, and the walk continues from the product — so the path stays on the stable isotopes and ends at ²⁰⁹Bi. Where an unstable nucleus lives more than a day, capture and decay really compete (the orange points); the single path is a simplification there and the affected isotopes are excluded from the comparison below. The r-process points are the equilibrium path: at the chosen T and n_n the (n,γ) ⇌ (γ,n) balance, a Saha equation, fixes which isotope of each element holds most of the abundance — that isotope waits for a β-decay, which is why the path sits at nearly constant neutron separation energy S_n ≈ 2–3 MeV rather than at the valley of stability. Masses beyond the measured range come from AME2020 extrapolations and then the FRDM95 mass model, flagged by colour; the vertical dotted lines are the closed shells N = 50, 82, 126 where the path piles up. The path is drawn up to Z = 95; beyond it fission returns matter to lighter nuclei and is not modelled.

The waiting-point (Saha) condition

\frac{Y (Z, A + 1)}{Y (Z, A)} = n_{n} {(\frac{2 π ℏ^{2}}{m_{u} k T})}^{3 / 2} \frac{G_{A + 1}}{2 G_{A}} {(\frac{A + 1}{A})}^{3 / 2} e^{S_{n} (Z, A + 1) / k T}, S_{n} (Z, A + 1) = Δ (Z, A) + Δ_{n} - Δ (Z, A + 1)

The ratio of two neighbouring isotopes in (n,γ) ⇌ (γ,n) equilibrium, exactly as the Saha equation balances ionisation against recombination: the neutron's thermal phase-space factor (m_ukT/2πħ²)^3/2 = 5.94×10³³ T₉^3/2 cm⁻³ against the Boltzmann factor of the neutron separation energy S_n, computed from mass excesses Δ. The partition functions G are set to 1 (a stated approximation; they include spin degeneracy and thermally excited states); the ratios are evaluated in logarithms and each isotopic chain is normalised separately, so nothing overflows and no abundance is zero. What the demonstrator leaves out is what a real calculation adds: β-decay rates that move matter to the next element, the freeze-out as n_n and T fall, β-delayed neutron emission, and fission cycling (Cowan et al. 2021; Mumpower et al. 2016).

The classical s-process: σN along an exposure distribution

σ_{A} N_{A} = \frac{f N_{56}}{τ_{0}} \prod_{i = 56}^{A} {(1 + \frac{1}{τ_{0} σ_{i}})}^{- 1}, ρ (τ) = \frac{f N_{56}}{τ_{0}} e^{- τ / τ_{0}}

A fraction f of the iron seeds is exposed to neutron fluences τ = ∫n_nv_Tdt distributed exponentially with mean τ₀ (in mb⁻¹; σ in mb at kT = 30 keV, Maxwellian-averaged, from KADoNiS). For a steady flow the product σN would be constant; the exponential distribution makes it fall gently with A and drop at the closed shells, where σ is tiny and the flow bottlenecks (Clayton & Ward 1974; Käppeler, Beer & Wisshak 1989). Seed check: the mean surviving seed is ⟨N₅₆⟩ = fN₅₆/(1 + τ₀σ₅₆) = fN₅₆ at the current τ₀. The product index follows the drawn path, not consecutive table rows.

The classical s-process as an exposure experiment

mean exposure τ₀ = 0.30 mb⁻¹ seed fraction f = 0.06 % of ⁵⁶Fe · hover for the isotope

σN of this exposure experiment along the pathσ × N_s of Arlandini's stellar model, s-dominated isotopes (N_s/N_⊙ ≥ 0.85)branching-sensitive isotopes (shown, excluded from the comparison)

σN against mass number. The yellow curve is what one exponential exposure distribution does to iron seeds (N₅₆ from Lodders 2010 on the Si = 10⁶ scale; the comparison points use Arlandini's own solar scale, a few per cent different for iron). The blue points, σ₃₀ × N_s from the published stellar AGB model, are the empirical σN curve of the main component. Move τ₀ and f: a single exposure reproduces the flat stretches and the shell drops between them but not both the light (A < 90) and heavy ends at once — which is why the real s-process needs at least two components (a "weak" one in massive stars and a "main" one in AGB stars) and a range of exposures. Nothing here is fitted to anything.

Kilonova: the light of an r-process event

When neutron-star mergers eject ~10⁻² M☉ of neutron-rich matter, the r-process runs to completion in about a second and leaves hundreds of radioactive isotopes whose decays heat the debris — a kilonova. Because so many nuclides with a spread of half-lives contribute, the heating rate is a power law rather than the two exponentials of ⁵⁶Ni: ε ≈ 2×10¹⁰ t_d^−1.3 erg g⁻¹ s⁻¹ (Metzger et al. 2010). Only part of it is absorbed: neutrinos escape, and the γ-rays, electrons and α-particles thermalise with an efficiency that falls with time as the ejecta thin (Barnes et al. 2016; their analytic fit, interpolated in mass and velocity, is used here). The light emerges once the diffusion time through the ejecta drops to the age of the explosion, t_peak ≈ (3κM_ej/4πβvc)^1/2 with β ≈ 3 — a characteristic timescale, not a computed maximum — and afterwards the luminosity tracks the instantaneous deposited heating.

ejecta mass 0.020 M☉ velocity 0.15 c opacity κ = 3.0 cm² g⁻¹

deposited heating M_ejε(t)f_th(t), drawn from t_peak on (approximate light curve)total decay heating M_ejε(t)GW170817 comparison point

An r-process light curve. The lanthanide-rich ejecta of a merger have opacities of 1–10 cm² g⁻¹ (against 0.1 for iron-group supernova ejecta), which is why kilonovae are faint, red and fast. The blue point is the observed counterpart of GW170817: a bolometric luminosity of about 5×10⁴¹ erg s⁻¹ at 0.6 days (Cowperthwaite et al. 2017), from which ≳ 0.05 M☉ of heavy elements were inferred (Drout et al. 2017) — a comparison, not a validation of this one-zone estimate, which has no colours, no lanthanide fraction and no separate ejecta components.

SECTION 14

Model notes, self-check and references

What is computed and what is looked up

Computed live from physics

All Q-values and binding energies from AME2020 mass excesses (100 nuclides).
Reaction rates from Solar Fusion II S-factors (pp chains, CNO), the LUNA 2016 evaluated ¹⁷O(p,α) rate, deBoer et al. 2017 for ¹²C(α,γ), and Caughlan & Fowler 1988 analytic fits for 3α, ¹⁶O(α,γ), ²⁰Ne(α,γ), ¹²C+¹²C, ¹⁶O+¹⁶O; each rate carries a validity range shown in the tables.
Gamow peaks, temperature exponents, weak screening (only where H₁₂ < 0.3), ⁷Be electron capture and pep ratio (SFII eqs. 40, 46), pp-chain branching, CNO equilibrium abundances and lifetimes, energy generation rates.
The one-zone networks (H burning with 9 retained + 7 folded species; He burning with 4 species): backward Euler, Newton, step-doubling error control; conservation and energy-closure checks every run.
Energy per gram of each fuel from the mass table with the adopted product mixtures; stage energies from burned masses.
Chandrasekhar mass, free-fall time, gravitational binding energy, photodisintegration and ⁵⁶Ni decay energies.
Section 11b, on top of the published model: the remnant's gravitational mass and neutrino energy (Lattimer & Prakash relation, R = 12 km), the peak-temperature bands T_peak = (3E/4πar³)^1/4 on the model's own r(m), the schematic light curve (Popov plateau normalised by Sukhbold et al. 2016; ⁵⁶Ni → ⁵⁶Co → ⁵⁶Fe deposition with the Bateman factor and γ-ray trapping; Arnett diffusion for stripped stars), and the inverse-beta-decay count for a Fermi–Dirac spectrum.
Section 13: solar abundances converted from A(X) to the Si = 10⁶ scale; the classical s-process path and σN product along it; the (n,γ) ⇌ (γ,n) equilibrium path from neutron separation energies (Saha relation, G = 1); the kilonova heating, thermalisation and diffusion time.

Looked up from published models (and labelled)

Central T, ρ, luminosity, radius and duration of each burning stage: Woosley, Heger & Weaver (2002) Table I for 13, 15, 20, 25, 75 M☉ (solar metallicity), interpolated in log M between rows. Outside 13–75 M☉ the nearest rows are used and the values are marked "estimated".
Zero-age main-sequence central conditions at 1, 2, 5, 10 M☉: Hansen & Kawaler (1994); the present Sun (Bahcall et al. 2001). Below 1 M☉: scalings from Chabrier & Baraffe (2000) — approximate.
Surface luminosity and effective temperature on the ZAMS: standard textbook tables; radius derived from L = 4πR²σT⁴.
Effective fuel-access fractions q_c(M), helium-burning conditions below 13 M☉, iron-core masses, remnant bands: from the same sources and the reviews cited below — schematic.
Presupernova composition boundaries (He, C/O, O/Ne/Mg, Si, Fe regions) of the cross-section and the onion table: Rauscher et al. 2002, Table 6 (models S15 and S25), interpolated in mass between 13 and 30 M☉ and scaled schematically outside that range. Region extents during earlier stages are illustrative, not model profiles. White-dwarf masses: the Kalirai et al. 2008 initial–final mass relation.
The hydrogen-shell (subgiant / red-giant) and asymptotic-giant-branch phases appear on the timeline as qualitative segments: their durations and central conditions are not modelled here, and the ages and total lifetime printed for later phases say so.
Main-sequence central evolution (μ scaling / CNO thermostat) and the stellar-age mapping of the network: labelled illustrations, not structure solutions.
Explosion outcomes, remnant masses, ejected ⁵⁶Ni, presupernova structures and bolometric light curves: the Sukhbold et al. 2016 archive (200 solar-metallicity models; Z9.6 calibration for 9–12 M☉, W18 above; 33 structures and 21 light curves embedded). The fate shown for every star of 9 M☉ or more is the outcome of the nearest embedded model — a published simulation result, not a computation of this page — and it is not varied with the metallicity slider. The Ertl et al. 2016 line is a diagnostic only.
Nuclear data for the heavy elements: AME2020 mass excesses (measured and extrapolated, flagged), FRDM95 theoretical masses, NUBASE2020 half-lives and decay modes, KADoNiS 1.0 Maxwellian-averaged cross-sections, Lodders 2010 elemental and Wallet-card isotopic abundances, Arlandini et al. 1999 Table 2 (s-process main component and residual, on their own solar scale), Barnes et al. 2016 thermalisation fits.

Registered limits: no time-dependent r-process (β-decay rates, freeze-out, delayed neutrons and fission are described, not computed; a published β-rate table was unreachable when this was built); no pair-instability, magnetar, collapsar or binary channels; the light curves are one-zone scalings, the neutrino count ignores oscillations and detector response, the kilonova has no colours or ejecta components; the r-process demonstrator stops at Z = 95.

Metallicity enters through the composition only (X_CNO = 0.68 Z, Y = 0.24 + 2.1 Z, μ, the hydrogen-burning limit). The central temperature, luminosity and radius are held at their Z = 0.02 values: a composition-sensitivity experiment at fixed conditions, not a metallicity-dependent stellar model. Rotation, magnetic fields, binaries and the details of mass loss are not modelled.

Size and load time

This build is a single file of about 2,145 KB (uncompressed; the embedded nuclear-data and model tables are most of it). In this browser it took … s from navigation to the first fully rendered frame (measured live by the page). Cold load in headless Edge on the build machine: 1.1 s to the load event and 1.2 s to the first rendered frame (measured 2026-09-14 on the 1.36 MB build with the derivation drawer, headless Edge 153, 1440 px wide, local file).

Layout

The page was reorganised in round 6 after measuring the delivered layout. The control bar's height is now measured by a ResizeObserver and written to a CSS variable that positions the sticky rail, the scroll padding and the panels, instead of an assumed 96 px offset that left the rail clipped behind a 192 px bar. The bar has two states — a one-line summary (mass, metallicity, stage and age, the section in view, Play, and the Contents and Dashboard buttons) and the full panel — open by default only where the viewport is at least 1180 × 760 px and the panel leaves a reading area of at least 320 px, and remembered only when toggled by hand; the summary's section name is held to one line so that its length cannot change the bar's height, and a landing re-aligns itself once the bar has settled (a defect the battery found at 375 px on the final build). The star's cross-section and evolution canvases sit at the top of the main column beside the dashboard, two-up only when that column is at least 880 px wide. The dashboard lives in the right rail from 1180 px and moves into a non-modal Dashboard panel below that width; its values wrap, their qualifications sit on their own line, and none overflows its cell at any mass or stage. A non-modal Contents panel and a scroll spy name the section under the bar; the styles are mobile-first with one desktop breakpoint and no cap on the prose width. Verified in headless Edge by a layout battery of 66–70 checks per viewport at ten viewport configurations (1920, 1600, 1366, 1180, 1179, 1000 and 375 px wide; 1366 × 768; 1366 × 600 with a stored open preference; a hash landing on section 9): the measured bar height and scroll padding, the rail stuck below the bar and above the viewport bottom, no dashboard value overflowing its cell for a 1 M☉ star and a 60 M☉ remnant, the panels' position, focus return and mutual exclusion, the tab order of the collapsed and expanded bar, both Play buttons agreeing with the playback state, and the cross-section canvas within the first 468 px at 1366 × 768; the 1600 × 1000, 1366 × 768 and 375 × 1000 runs were repeated on the final build. Printing is unaffected: the bar, the rail and the panels are suppressed and the page prints in one column (A4, 217 pages with every entry expanded).

The derivation drawer

Every displayed equation (46 of them) and 17 inline relations carry an Explain control that opens a three-tier entry: ① the idea in words with a worked number (open by default); ② the derivation written out line by line, closed by default so that it does not crowd the panel — a Start-from line naming the assumptions, the imported results (each linked to its own entry) and the symbols; numbered steps, each labelled with what it does (identity, definition, approximation, imported, assumption or numerical) and each identity, definition or approximation followed by the displayed equation it produces; and a Result line separating what was derived from what was measured or fitted and what the page calculates; ③ where it applies and where it comes from. There are 78 entries — 18 school-level prerequisites (dimensional analysis, exponentials and logarithms, derivatives and integrals, probability, the Boltzmann factor, state counting, the Pauli principle, …) and 60 physics entries from the size of a nucleus to the kilonova — cross-linked so that every derivation can be followed down to its prerequisites, with a breadcrumb trail, Back, and a control that returns to the equation. The build refuses to emit a page in which a displayed equation lacks an entry, an entry links to a missing one, or a ② tier departs from that structure (one start line, sequentially numbered steps with exactly one label each from the fixed vocabulary, an equation after every identity, definition or approximation, one result line) — all 78 entries pass, with 795 labelled steps, and the LaTeX-to-MathML converter runs its own self-test at every build; eight of the section derivations are the entries' own step-by-step text included at build time, so there is one source of truth, and the live numbers inside the entries are the page's own values. Printing the page appends all 78 entries with every tier expanded (217 A4 pages in the headless test, the every-line derivations included). Verified in headless Edge by a 36-check DOM battery: opening from buttons, from the equations themselves, from inline controls and from links inside included derivations; navigation with restored scroll and tier state; focus return to the launcher; live values updating inside an open entry after a mass change. Also verified in headless Edge with trusted key events sent through the DevTools protocol (the dialog's keyboard handling is the browser's own, so a synthetic page event would prove nothing): Enter on a focused Explain control opens the entry with focus on its heading, and Escape closes it and returns focus to the launcher. The accessibility tree that screen readers read exposes the open drawer as a modal dialog named after the entry, every button and link by name, and each equation as MathML with its structure intact (fraction, root, sub- and superscript, identifier, operator and number nodes) — the form that NVDA (through MathCAT, bundled since NVDA 2026.1) and JAWS 2026 speak and braille. Not verified: actual speech output, since no screen reader is installed on the build machine, and Narrator's MathML support, which its documentation does not describe. Registered simplifications inside the entries: partition functions are set to 1 in the Saha-type relations; the S_eff coefficients and the plateau prefactors are quoted from their sources rather than re-derived; the ⁷Be electron-capture and pep coefficients are explained by scaling arguments to within a factor 1.5 and quoted from Solar Fusion II.

Review record

The physics assumptions and the implementation design were reviewed adversarially by an independent AI reviewer before the engine was written; two rounds, 21 numbered assumptions. Corrections adopted included: the triple-alpha energy normalisation, the pp-III branch energy, continuous resonance terms, the LUNA evaluated ¹⁷O(p,α) rate, deBoer's ¹²C(α,γ) parameterisation, separate parcel and stellar clocks, the CNO thermostat normalisation, the folded-species validity criterion, the abundance–energy closure test, the use of the full WHW02 table instead of extrapolation, and non-universal fate/pair-instability wording. Registered residuals: the ¹³N(p,γ) competing-capture check uses an approximate rate; the thermostat and age mapping are illustrations; global stellar properties are interpolated, not computed. The folded-species criterion compares each intermediate with the retained species it exchanges with, a refinement of the consulted min-over-all-species form, which flagged massive stars spuriously while ¹²C burns down at the start of the main sequence.

Round 4 (the bridge from hydrogen to iron, the explosion and the elements beyond iron, and the derivation drawer) was reviewed the same way, two rounds per unit. Adopted corrections included: the fallback mass is already inside the published mass cut; the gravitational mass follows the Lattimer–Prakash relation rather than a binding-energy fit; A(X) is logarithmic; no nickel share is read from temperature bands; Arlandini's decomposition is displayed on its own normalisation; the r-process is an equilibrium-path demonstrator only; the inverse-beta-decay cross-section is σ ∝ E_ep_e integrated over a stated spectrum; the tour advances only its own scenes and independent selections are retained. Residual design points are listed as registered limits above.

Round 5 reviewed the derivation content itself. The step-by-step tiers of the 20 entries judged least certain were sent to an independent AI reviewer with a mandate to refute them; the algebra of every entry was agreed, 5 entries were agreed outright and 15 drew corrections of numbers, labels or validity wording, all adopted: the uniform-sphere gravitational energy of the Sun (2.3×10⁴⁸ erg, a 9 Myr Kelvin–Helmholtz time), the 2^−1/3 and (4/√3)(½)^1/3 factors in the reduction of the rate constants, the mass–luminosity toy normalisation (2000 L☉, not 100), the screening parameter ζ = 2X + 1.5Y, the reduced mass of ¹⁴N + p from actual masses (15.23 in the CNO fit), the textbook pp fit running 4–6 % below the live rate, the thermalisation efficiencies of the kilonova example, the zeroth-order status of the inverse-beta-decay cross-section, the empirical (not derived) status of the Lattimer–Prakash binding-energy relation, the waiting point as the maximum of a chain rather than the last rising ratio, and the composition generality of μ_e. A separate recheck run in parallel found six of the same and four more (the solar-centre lifetimes of ¹³N, ¹⁴N, ¹⁵N and ⁷Be against proton capture; the CNO example conditions; the γ-ray trapping factor in the radioactive-tail example) and two wrong static numbers in the sections themselves (the Gamow-peak energies quoted in section 4 and the fireball temperatures quoted in section 11b), all corrected. A delta-only second round agreed 16 of the 20 items and asked for five further wording fixes — the inverse-beta-decay constant contains only the inner radiative correction; the CNO fit exceeds the live rate by 1.65 at 21 MK, not by the S-factor ratio 1.9; the NSE crossover between ⁵⁶Ni and ⁵⁶Fe is an outcome of the density-dependent constrained solution; a neutron star's internal energy stays equation-of-state-dependent even when cold; and the lifetime examples now use one composition, the present solar centre with X = 0.34, so that pp-III : pp-II ≈ 1 : 380 there — all adopted. No third round was opened (two-round limit).

Round 6 reviewed two units separately, each with a mandate to refute: two rounds with one AI reviewer and one with a second, independent one. The layout unit started from measurements of the delivered page — a sticky offset of 96 px against a 192 px control bar, dashboard values that could not wrap, the rail dropping to the page bottom below 1100 px, the controls far below the star's visuals — and closed on twelve decisions: the control bar's height measured rather than assumed, a two-state bar with a summary row, the star's visuals beside the dashboard, the Contents and the Dashboard as non-modal panels, mobile-first styles with one desktop breakpoint, no cap on the prose width; the four points on which the reviewers disagreed (two-up visuals, the prose cap, the height budget, a second inline wrapper) were decided by the author. The derivation unit fixed the every-line form described above — the label vocabulary, the closed-by-default rule, one symbol convention for all entries (the reduced mass, the mean molecular weight and the chemical potential named apart; molar abundances Y_i apart from the helium fraction Y; kT = 86.17 T₉ keV), and the rule that a rate written as an integral must say whether the page integrates it (it does not: the rate function evaluates the saddle-point expression; the page's only quadratures are the Arnett light curve and the neutrino-spectrum average) — and found five errors in the previous drawer text, all corrected: the sign of the Bateman prefactor, N_Am_u treated as exactly 1 g, kT in the Gamow-peak numbers, an extra Avogadro factor in the triple-alpha energy rate, and the pp branching factor, which the page displays energy-weighted and which must not be inverted for the pp-I fraction. Re-deriving every entry line by line then turned up more than a dozen further wrong numbers or statements in the old text (among them a 1000-fold error in the mass–energy example, the electron Fermi energy at the solar centre, the total atomic binding energy of uranium and the degeneracy ratio at the solar centre), each corrected. Registered: partition functions still set to 1 in the Saha-type relations; the S_eff coefficients, the plateau prefactors and the kilonova fits still quoted rather than derived; the pp neutrino mean energy kept at the page's 0.265 MeV against the literature 0.267 MeV.

Self-check (runs when the page loads)

The 75 M☉ rows of Woosley, Heger & Weaver (2002) at two metallicities

At Z = 10⁻⁴ Z☉ a 75 M☉ star has almost no CNO catalysts, so it must run its hydrogen burning at 76 million K instead of 43, and loses hardly any mass (74 M☉ at the end instead of 6.4). Same physics, different composition.

References

Adelberger, E. G., et al. 2011, "Solar fusion cross sections. II. The pp chain and CNO cycles", Rev. Mod. Phys. 83, 195 (S-factors, ⁷Be and pep rates).
Woosley, S. E., Heger, A., & Weaver, T. A. 2002, "The evolution and explosion of massive stars", Rev. Mod. Phys. 74, 1015 (Table I burning stages; presupernova structure).
Wang, M., et al. 2021, "The AME 2020 atomic mass evaluation", Chinese Phys. C 45, 030003 (mass excesses).
Caughlan, G. R., & Fowler, W. A. 1988, "Thermonuclear reaction rates V", At. Data Nucl. Data Tables 40, 283.
deBoer, R. J., et al. 2017, "The ¹²C(α,γ)¹⁶O reaction and its implications for stellar helium burning", Rev. Mod. Phys. 89, 035007.
Bruno, C. G., et al. (LUNA) 2016, "Improved direct measurement of the 64.5 keV resonance strength in the ¹⁷O(p,α)¹⁴N reaction", Phys. Rev. Lett. 117, 142502 (Table III rate).
Pols, O. R., "Stellar Structure and Evolution", lecture notes, Radboud University / Utrecht (chapters 5–13: nuclear reactions, main sequence, massive stars, core collapse).
Woosley, S. E., AY220 lecture notes (UC Santa Cruz, 2019), lectures 11–13: neutrino losses, advanced burning, silicon burning and NSE, presupernova models.
Kippenhahn, R., Weigert, A., & Weiss, A. 2012, "Stellar Structure and Evolution", 2nd ed., Springer (textbook energy-generation fits, eqs. 18.63–18.67).
Iliadis, C. 2015, "Nuclear Physics of Stars", 2nd ed., Wiley (non-resonant and narrow-resonance rate formulas).
Clayton, D. D. 1983, "Principles of Stellar Evolution and Nucleosynthesis", U. Chicago Press (screening, neutrino losses, NSE).
Hansen, C. J., & Kawaler, S. D. 1994, "Stellar Interiors", Springer (ZAMS models).
Chabrier, G., & Baraffe, I. 2000, "Theory of low-mass stars and substellar objects", Ann. Rev. Astron. Astrophys. 38, 337.
Bahcall, J. N., Pinsonneault, M. H., & Basu, S. 2001, ApJ 555, 990 (standard solar model); Borexino Collaboration 2020, Nature 587, 577 (CNO neutrinos).
Janka, H.-T. 2012, "Explosion mechanisms of core-collapse supernovae", Ann. Rev. Nucl. Part. Sci. 62, 407; O'Connor, E., & Ott, C. D. 2011, ApJ 730, 70; Sukhbold, T., et al. 2016, ApJ 821, 38.
Heger, A., & Woosley, S. E. 2002, ApJ 567, 532 (pair-instability supernovae); Timmes, F. X., Woosley, S. E., & Weaver, T. A. 1996, ApJ 457, 834 (iron-core masses, Chandrasekhar corrections).
Rohlf, J. W. 1994, "Modern Physics from α to Z⁰", Wiley (semi-empirical mass formula coefficients).
Rauscher, T., Heger, A., Hoffman, R. D. & Woosley, S. E. 2002, ApJ 576, 323 (presupernova composition boundaries, Table 6).
Kalirai, J. S. et al. 2008, ApJ 676, 594 (initial–final mass relation of white dwarfs).
Pols, O. R. 2011, "Stellar Structure and Evolution" lecture notes, Utrecht (post-main-sequence phases, chs. 10–11).
Sukhbold, T., Ertl, T., Woosley, S. E., Brown, J. M., & Janka, H.-T. 2016, "Core-collapse supernovae from 9 to 120 solar masses based on neutrino-powered explosions", ApJ 821, 38 (presupernova properties, outcomes, structures, light curves; eqs. 9–10, 15, 17); Ertl, T., et al. 2016, ApJ 818, 124 (two-parameter explodability criterion).
Lattimer, J. M., & Prakash, M. 2001, ApJ 550, 426 (neutron-star binding energy); Vogel, P., & Beacom, J. F. 1999, Phys. Rev. D 60, 053003 (inverse beta decay); Nadyozhin, D. K. 1994, ApJS 92, 527 (⁵⁶Ni and ⁵⁶Co decay energies); Arnett, W. D. 1982, ApJ 253, 785 (diffusion light curves); Popov, D. V. 1993, ApJ 414, 712, and Kasen, D., & Woosley, S. E. 2009, ApJ 703, 2205 (plateau scalings); Woosley, S. E., & Weaver, T. A. 1995, ApJS 101, 181 (explosive burning by peak temperature).
Huang, W. J., et al. 2021, and Wang, M., et al. 2021, Chinese Phys. C 45, 030002/030003 (AME2020); Kondev, F. G., et al. 2021, Chinese Phys. C 45, 030001 (NUBASE2020); Möller, P., Nix, J. R., Myers, W. D., & Swiatecki, W. J. 1995, At. Data Nucl. Data Tables 59, 185 (FRDM); Dillmann, I., et al., KADoNiS 1.0 (Maxwellian-averaged cross-sections); Lodders, K. 2010, "Solar system abundances of the elements", in Principles and Perspectives in Cosmochemistry (Table 3); Arlandini, C., et al. 1999, ApJ 525, 886 (Table 2).
Clayton, D. D., & Ward, R. A. 1974, ApJ 193, 397, and Käppeler, F., Beer, H., & Wisshak, K. 1989, Rep. Prog. Phys. 52, 945 (classical s-process); Cowan, J. J., et al. 2021, Rev. Mod. Phys. 93, 015002, and Mumpower, M. R., et al. 2016, Prog. Part. Nucl. Phys. 86, 86 (r-process); Metzger, B. D., et al. 2010, MNRAS 406, 2650, Barnes, J., et al. 2016, ApJ 829, 110, and Metzger, B. D. 2019, Living Rev. Relativ. 23, 1 (kilonovae); Cowperthwaite, P. S., et al. 2017, ApJ 848, L17, and Drout, M. R., et al. 2017, Science 358, 1570 (GW170817).

Built 2026-09-14. Single self-contained file; no network access; all computation runs in your browser.

Bridge from high school: the twelve ideas everything else is built on

1 · Nuclei and the notation AX

2 · Energy units: eV, MeV, erg, joule

3 · E = mc² and the mass defect

4 · Exponentials, logarithms and log axes

5 · Temperature is motion

6 · Electric repulsion: the Coulomb barrier

7 · Particles are waves, so they can tunnel

8 · Pressure, ideal gas and mean molecular weight

9 · Gravity and hydrostatic balance

10 · Cross-sections and rates

11 · Conservation laws decide what can happen

12 · Reading a reaction

Why fusion releases energy — the binding-energy curve

What the numbers mean

Why the curve has this shape: the semi-empirical mass formula

Energy released along the road from hydrogen to iron

Stars: a fight between gravity and pressure that gravity slowly wins

The virial theorem: why losing energy makes a star hotter

The thermostat: why main-sequence stars burn steadily

Thermal (ideal gas)

Electron degeneracy

Radiation

Mass sets everything: luminosity and lifetime

The contraction ladder in one picture

Why the centre alone cannot tell you the luminosity, the lifetime or the shells

How nuclei tunnel: the Gamow peak, the S-factor and the reaction rate

Step 1 — the classical dead end

Step 2 — tunnelling through the Coulomb barrier

Step 3 — the S-factor: separating nuclear physics from the barrier

Step 4 — average over the gas: the Gamow peak

Step 5 — from a rate to a power

Hydrogen burning I — the proton–proton chains

The reactions

Why p + p is so slow

Three ways to finish

Hydrogen burning II — the CNO cycles

The live reaction network: watching the core's composition change

Two clocks

What the solver is doing (and how it checks itself)

Helium burning: three alphas, one lucky resonance, and the origin of carbon and oxygen

Where helium burning leads, by stellar mass

Below ~2 M☉: the helium flash

2–8 M☉: quiet ignition, then a white dwarf

Above ~8 M☉: on to carbon

Carbon, neon, oxygen and silicon burning: the neutrino-cooled sprint to iron

The reactions

Carbon burning · 6–9×10⁸ K

Neon burning · 1.2–1.7×10⁹ K

Oxygen burning · 1.5–2.7×10⁹ K

Silicon burning · 2.7–3.7×10⁹ K

Why the core ends as iron — and which iron

The onion star: how the burning shells stack up

Reading the two views

The iron catastrophe: why the core collapses, and how it explodes

The energy budget

What is left, by initial mass

≲ 8 M☉ · white dwarf

8–9 M☉ · the transition band

≳ 9 M☉ · core collapse

Your star's explosion, read from a published model

The selected model:

Neutrinos: the energy that leaves unseen

A life in one line: the timeline and the energy budget

Where the elements come from — and why the story does not end at iron

Which star made which element (simplified)

Beyond iron: why fusion cannot continue, and what does

The solar system's heavy elements, as published

The capture paths on the chart of nuclides

The classical s-process as an exposure experiment

Kilonova: the light of an r-process event

Model notes, self-check and references

What is computed and what is looked up

Computed live from physics

Looked up from published models (and labelled)

Size and load time

Layout

The derivation drawer

Review record

Self-check (runs when the page loads)

1 · Nuclei and the notation ^AX