1. Stellar Structure Section 6: Introduction to Stellar Evolution Lecture 18 – Mass-radius relation for black dwarfs Chandrasekhar limiting mass Comparison with observation Virial theorem explanation of mass limit Thermal effects (approximate model) Final fate of more massive remnants: … mass loss, neutron stars, black holes … observational evidence for ns, bh

2. Chandrasekhars results – repeat 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) Chandrasekhars model now fully accepted

3. Virial theorem argument to explain Chandrasekhars model Low-mass white dwarfs, with low central density, will be non- relativistic: P 5/3 Applying the virial theorem, and just looking at the scaling, gives a balance between two terms, and yields a mass-radius relation (see blackboard) As the central density increases, so does the mass (using the mass-radius relation) – see blackboard Higher-mass white dwarfs therefore have higher central densities and relativistic effects become important: P 4/3 In this extreme case (see blackboard), there is balance for only one mass: the Chandrasekhar mass

4. Thermal effects in surface layers – a simple model Realistically: degree of degeneracy decreases towards surface, with surface layers having low enough density to be completely non-degenerate; smooth transition Model: fully degenerate core, ideal gas envelope, sharp boundary between them Non-degenerate envelope, ideal gas equation of state Black dwarf degenerate core, non-relativistic degenerate equation of state

5. Temperature at transition layer (see blackboard for mathematics) Observed effective temperatures => radiative envelopes Observed mean density in envelopes => bound-free opacity dominates – take Kramers law Neglect radiation pressure in envelope Take surface values for M, L, Derive P-T relation in envelope Equate pressures at transition between core and envelope, and use to eliminate the density at that radius Solve for the temperature at the transition radius Core ~isothermal – so this is ~core temperature, ~few 10 6 K Implies X WD 0, and L WD comes from cooling of core

6. Final fate of stars more massive than Chandrasekhar mass May lose enough mass via winds and superwinds to produce white dwarf and planetary nebula: needs M MS < ~8 M More massive stars develop core with mass above Chandrasekhar limit, and undergo core collapse in Type II supernova explosion Collapse (implosion of core) very high core densities, and neutronisation, producing degenerate neutron gas Neutron degeneracy pressure can support core against gravity Remnant of SN explosion may be neutron star

7. Properties of neutron stars Neutron stars have masses not much more than the Chandra mass, but radii much smaller than white dwarfs: R NS ~10 km Neutrons also Fermi particles, so equation of state similar to that of white dwarfs, except that effects of special and general relativity now important, especially in structure equations Expect maximum mass, as for WD Relativistic effects alone Oppenheimer-Volkoff mass: ~0.72 M Must also include particle-particle interactions – poorly understood at nuclear densities, so can only say that maximum mass is likely to be between 2 and 3 M Some models shown on Handout 17

8. Core masses above NS limit If core mass above NS limit, nothing can halt collapse under gravity Quantum effects probably prevent collapse to singularity with infinite density, but unobservable: Remnant vanishes through its event horizon once escape speed from surface exceeds speed of light (see blackboard) Event horizon occurs at Schwarzschild radius – remnant within that radius is a black hole, detectable only by its (long-range) gravitational field: no light can escape Black holes have only mass, angular momentum and charge (Quantum effects do allow Hawking radiation)

9. Observations of extreme remnants? Neutron stars: Detected in pulsars, low-mass X-ray binaries May have been directly detected by thermal x-ray emission from hot surface in some X-ray binaries Masses from binaries all ~ Chandra mass (Handout 18), suggesting formation by accretion onto a white dwarf Black holes: Cannot be observed directly More than a dozen high-mass X-ray binaries contain compact remnants with masses above any possible neutron star mass => detected gravitationally

10. Thats all, folks