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