Stellar Structure. Section 5: The Physics of Stellar Interiors Lecture 11 – Total pressure: final remarks Stellar energy sources Nuclear binding energy Charged particle reactions Important reactions in stars: … H to He … He to C. Other effects on the pressure.
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Stellar Structure Section 5: The Physics of Stellar Interiors Lecture 11 – Total pressure: final remarks Stellar energy sources Nuclear binding energy Charged particle reactions Important reactions in stars: … H to He … He to C
Other effects on the pressure • Have looked at pressure in zero-temperature degenerate gas • Do we need to make any corrections to this? • Relativistic effects in non-degenerate gases (see blackboard): • the pressure behaves like an ideal gas at all temperatures • the thermal energy depends on the kinetic energy of the particles (but is the same function of P in the NR and ER limits as for degenerate gases) • Thermal effects: produce a Maxwell-Boltzmann tail at high p. Total pressure does have temperature terms (see blackboard), but the thermal corrections to the degenerate pressure formula are small
Total pressure when electrons are degenerate • For (most) ionized gases, the electron density is larger than the ion density, so even the non-degenerate electron pressure is larger than the ion pressure: nekT > nikT • In the degenerate case, the electron pressure is much greater than nekT, so the ion pressure is negligible • The radiation pressure is generally smaller than the ion pressure, especially at high densities • Thus, to a good approximation, when electrons are degenerate, we have: (5.34)
Stellar energy sources:nuclear binding energy • How much nuclear energy is available in practice? • The binding energy of a compound nucleus is the energy equivalent of the mass difference between the mass of the nucleus and the sum of the masses of its components, Q(Z,N) (see blackboard) • More useful is the binding energy per nucleon (or “packing fraction”), defined as Q(Z,N)/(Z+N) (or Q/A) – and shown schematically and actually in Handout 5 • Energy can be released by fusion and fission • Fusion of H to He releases the most energy/nucleon (see blackboard) – about 80% of the total available (H → Fe)
Charged particle reactions • Main forces between charged nucleons are • repulsive electromagnetic (Coulomb) force • attractive nuclear force (strong interaction) • Weak interactions important if electrons or positrons involved • Gravitational forces (almost) always negligible • Nuclear force is short-range, Coulomb force is long-range • Overall effect: deep nuclear potential well, plus Coulomb potential barrier – Handout 6, top • Classically – reaction only if particle energy > barrier height (otherwise, particle pass in hyperbolic orbits)
Barrier penetration and resonances • Quantum mechanics allows barrier penetration even for low-energy particles (Handout 6 again) • Probability of penetration, and probability of reaction once inside nucleus, both depend on relative energies of colliding particles, so total cross-section is an integral over velocity, weighted by the velocity distribution (see blackboard) • Thinking of particle as a wave, some of the energy is reflected • For low energy particles, the reflection coefficient is high, except for energies corresponding to energy levels in the compound nucleus formed by the reaction • For these resonant reactions, the interaction cross-section is much higher
Reaction rates • Resonances are important for low energy particles with a normally low probability of interaction • The reflection coefficient is high because of the high Coulomb barrier or the virtual discontinuity at the edge of the nuclear potential well • For higher energy particles, these effects are weaker, and resonances are much less important (see blackboard) • For non-resonant interactions, we can write down a (rather complicated) expression for N, the number of reactions kg-1s-1 • The main uncertainty is the specific nuclear factor, S • The reaction rate rises with temperature, but falls again at very high temperature (see blackboard for N, and its T dependence)
Important reactions in stars • Most of the energy produced in stars comes from two reactions: • 4 H1→ He4 • 3 He4 → C12 • H to He • 1938-9: Bethe & von Weizsäcker proposed CNO cycle – see Handout 6 • C, N, O act as catalysts, with relative abundances of isotopes remaining fixed in equilibrium, while He/H increases • ratios of catalysts in equilibrium not same as on Earth; e.g. C12/C13 → 4 (~90 on Earth) • allows identification of processed material in stars or gas clouds
H to He reactions – the pp chain (Handout 6) • First step in pp chain is very improbable: one proton needs to decay into a neutron and a positron (plus a neutrino – weak interaction) while the other proton is nearby – i.e. in a timescale ~10-20 s, much shorter than the weak interaction timescale • Only ~ 1 in 1027 proton ‘collisions’ result in the formation of a D nucleus! Acts as bottleneck to whole process, and makes CNO cycle competitive. Not shown to be viable until 1952 (Salpeter) • Other -decays occur in side-chains; all release neutrinos • Neutrinos escape freely (‘solar neutrino problem’) • pp-chain dominates at low T, CNO takes over at higher T: pp T4 and CNO T17 (5.38)
He to C reactions (Handout 6) • He fusion can occur when the temperature is ~108 K • Triple-alpha process is also highly unusual (although no weak interactions involved) • Intermediate product, Be8, is highly unstable to decay back into two alpha particles: lifetime ~310-16 s • Also – reaction resonant, but slightly endothermic: requires energy • But mean time for collisions between alpha particles < Be8 lifetime • Second reaction, to form C, also resonant, forming excited state of C12, whose energy ~ combined energy of Be and He nuclei
The power of prediction! • Excited state in C12 predicted by Fred Hoyle in 1952-3 on visit to W A Fowler’s nuclear physics group in CalTech • Prediction based on observed abundances of C12 and O16 • C12 destroyed again by adding alpha particle to form O16 • Need C12 to be formed fast enough to avoid complete loss into oxygen: requires resonance at specific energy • Persuaded members of Fowler’s group to look for resonance at 7.68 MeV in C12 – discovery* converted Fowler into nuclear astrophysicist (for which he won 1983 Nobel Prize…) • Reaction so fast it is essentially 3-body: 3 2T40 (5.39) * Phys. Rev. 92, 649-50, 1953