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Stellar Structure. Section 6: Introduction to Stellar Evolution Lecture 17 – AGB evolution: … MS mass > 8 solar masses … explosive nucleosynthesis … MS mass < 8 solar masses Formation of compact remnants White dwarf stars Structure equations for black dwarfs Chandrasekhar’s results.
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Stellar Structure Section 6: Introduction to Stellar Evolution Lecture 17 – AGB evolution: … MS mass > 8 solar masses … explosive nucleosynthesis … MS mass < 8 solar masses Formation of compact remnants White dwarf stars Structure equations for black dwarfs Chandrasekhar’s results
Post-He-burning – 1 (no WD remnant)Main Sequence mass > 8 M; review • Nuclear burning as far as Fe: limit of ‘free’ energy • Core collapses catastrophically to nuclear densities, and bounces, leading to outward-travelling shock wave • Shock also accelerated by pressure of neutrinos, produced in explosive nucleosynthesis generated by energy of collapse • Leads to ejection of outer layers – Type II supernova • Explosive nucleosynthesis: neutronisation (e- + p+→ n + )→ high fluxes of neutrinos and neutrons • Neutrons added faster than-decay timescale produce r-process nuclei (neutron-rich), seen in SNR • (s-process nuclei seen in AGB star envelopes)
Post-He-burning – 2 (produces WD)Main Sequence mass < 8 M; repeat • Neutrino processes cool centre, inhibiting C ignition (needs T ~ 5108 K) • Degenerate core: • pure helium (low initial mass) • He, C, O mixture (higher initial mass) • On AGB, substantial mass loss by stellar winds (and possibly thermal pulses) – helps to prevent core heating to C ignition • Finally, a “superwind” (observed, not understood) ejects entire outer envelope as coherent shell, revealing hot interior • Hot remnant ionizes shell → planetary nebula • Star then cools and fades → white dwarf star (Handout 15)
Final stages of evolution: compact remnants • After all nuclear fuels exhausted, star must become a compact remnant – follows from energy balance and virial theorem: • While surface is hot, star must radiate: only source of energy is now gravitational => contraction to maintain energy balance • But virial theorem => mean T then rises (see blackboard) • Star cannot cool as long as remains ideal gas => contraction until Pauli exclusion principle important and degeneracy sets in • Then (see blackboard) no constraint on mean temperature, and star can cool and ‘die quietly’ • Also (see blackboard) pressure force increases faster than gravity and new equilibrium possible
White dwarf stars • Expect some stars in hydrostatic equilibrium, with degenerate equation of state, and slowly cooling • Obvious candidates: white dwarf stars (see sketch on blackboard) • With observed parameters, the mean density ~109 kg m-3 • At these densities, electrons certainly degenerate (unless T implausibly high – unlikely, because L small) • Identify white dwarfs as cooling, degenerate stars • Now consider structure of a zero temperature model (a black dwarf)
Structure of a zero-temperature black dwarf • As before, use as variable x = p0/mec (p0 = Fermi momentum) • Then we have, from Section 5: • The last two combine to give (x) and this, with the first equation for P(x), defines the equation of state P() via the parameter x • The usual equations of hydrostatic equilibrium and mass conservation complete the set of structure equations (see blackboard)
Equations in scaled variables (see blackboard for equations) • Equations involve composition of star, through e = 2/(1+X) • Useful to remove composition from equations by a suitable scaling (actually a homology transformation) • One integration then gives the structure for all compositions • Unlike the general equations of stellar structure, these equations are numerically stable, and we can choose an arbitrary central density and integrate outwards until the density goes to zero • This point is the surface, and defines the radius and mass of the black dwarf • Varying the central density gives a 1-parameter family of models, and a corresponding mass-radius relation
Chandrasekhar’s results • First calculations by Chandrasekhar, late 1920s, found two curious results (see sketches on blackboard): • as the total mass increases, the total radius decreases • the total radius tends to zero for a finite total mass • There is a critical mass, above which no solution can be found (see blackboard) – the Chandrasekhar limiting mass • In the absence of hydrogen, the limiting mass is 1.44 M • Hard to measure masses and radii of white dwarfs – but available observations lie close to model relationship (Handout 16) • Chandrasekhar’s model now fully accepted
