**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

**Chandrasekhar’s 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) • Chandrasekhar’s model now fully accepted

**Virial theorem argument to explain Chandrasekhar’s 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

**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

**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, ~few106 K • Implies XWD≈ 0, and LWD comes from cooling of core

**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 MMS < ~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

**Properties of neutron stars** • Neutron stars have masses not much more than the Chandra mass, but radii much smaller than white dwarfs: RNS ~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

**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)

**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

**That’s all, folks**