1. 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

2. 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)

3. Post-He-burning – 1 (no WD remnant) Main Sequence mass > 8 M  Nuclear burning continues beyond C, mainly by addition of He nuclei to form O, Ne, Mg, Si etc, as far as Fe: limit of ‘free’ energy Core partially supported by degenerate electrons – some electrons in high-energy states may be captured by Ne or Mg nuclei Pressure drops, core cannot support itself, collapses catastrophically (timescale: 10s of milliseconds!) 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 (~90% of mass of star) – Type II supernova (may leave compact core → NS or BH – see later)

4. Explosive nucleosynthesis (formation of elements heavier than iron) Very high densities favour neutronisation: e - + p + → n + (Normally, neutron is unstable, timescale ~900 s) Neutrino flux helps to accelerate shock Neutron flux allows rapid neutron addition to Fe and heavier elements, forming n-rich nuclei Addition very fast compared to  -decay timescale – elements produced called r-process elements (r for rapid) – seen in supernova remnants (AGB evolution: much smaller neutron flux available, n-addition occurs on timescale long compared to  -decay timescale – forms n-poor nuclei by s-process (s for slow) – s-process elements seen in atmospheres of red giants and supergiants)

5. Post-He-burning – 2 (produces WD) Main Sequence mass < 8 M  ; repeat Neutrino processes cool centre, inhibiting C ignition (needs T ~ 5  10 8 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)

6. 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

7. 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 ~10 9 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)

8. Structure of a zero-temperature black dwarf As before, use as variable x = p 0 /m e c (p 0 = 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)

9. 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

10. 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