Stellar fusion core - 2D cross-section

Hydrostatic balance and the polytrope

A star stays put when pressure balances gravity, dP/dr = -G m ρ / r², with dm/dr = 4π r² ρ. For P = K ρ^1+1/n these combine into the Lane-Emden equation (1/ξ²) d/dξ(ξ² dθ/dξ) = -θⁿ, with ρ = ρ_c θⁿ. For n = 3 (Eddington's standard model) the surface is at ξ₁ = 6.8968 and ρ_c = 54.18 mean densities. With an ideal gas T = T_c θ, and the core temperature is T_c = 0.854 μ m_H G M / (k R), where the mean molecular weight is μ = 1/(2X + 3Y/4 + Z/2) for fully ionised gas.

Burning rates

The proton-proton chain gives ε_pp = 2.57×10⁴ ρ X² T₉^-2/3 exp(-3.38 T₉^-1/3) erg/g/s and the CNO cycle ε_CNO = 8.24×10²⁵ ρ X X_CNO T₉^-2/3 exp(-15.23 T₉^-1/3), with T₉ = T/10⁹ K (Kippenhahn and Weigert). The exponents encode the Gamow factor: a reaction needs the thermal tail exp(-E/kT) times the tunnelling probability exp(-√(E_G/E)), E_G = 2 m_rc²(π α Z₁Z₂)², whose product peaks at E₀ = (√E_G kT / 2)^2/3, about 6 keV in the Sun's core, although kT is only 1.3 keV. CNO has a much steeper temperature dependence, so it takes over in hotter, more massive stars.

Photons and neutrinos

A photon made in the core is scattered by free electrons after only about a centimetre, so it random-walks outwards: t ≈ R² κρ̄ / c with κ = 0.2(1+X) cm²/g (electron scattering only), tens of thousands of years. Neutrinos fly straight out in a couple of seconds. The animation uses a much longer visual step so you can see the walk.

Honest simplifications: composition is uniform (the real Sun has burnt its core to X about 0.35), the radius is a fixed power law instead of a solved stellar model, radiation pressure and convection are ignored (a real 1 M_sun core is radiative but above about 1.3 M_sun it is convective), screening and the CNO equilibrium abundances are crudely fixed (X_CNO = 0.7 Z), and the polytrope gives a core density of about half the real value, so luminosities are only good to a factor of a few. Read the trends, not the absolute numbers.

• Step the mass from 0.5 to 3 solar masses and watch the core temperature, the luminosity and the CNO fraction climb.
• Lower the hydrogen fraction to mimic an ageing core: μ rises, the core heats up and compensates for the fuel loss.
• Raise the metallicity with a 2 solar-mass star and see CNO burning gain weight.
• Look at the Gamow graph: the two exponential factors multiply into a narrow peak far above kT.