Cataclysmic Events and the Final States of Stars

Explosions and collapse are a normal part of a star's life cycle, and they forge and scatter the elements from which later stars and planets form. The events near the end of a star's life, and the compact object left behind, are governed almost entirely by mass, chiefly the mass of the core.

Close binaries and novae

More than half of all stars belong to binary or larger groups. In a close binary, the two stars of masses $M_{1}$ and $M_{2}$ orbit their common center of mass closely enough to interact. An observer rotating with the system feels the sum of the two gravitational forces and the centrifugal pseudoforce. Along the line joining the stars there is a point where these cancel, a Lagrangian point $L$ ; the equipotential surface through it wraps each star in a teardrop Roche lobe.

Two stars share equipotential Roche lobes meeting at the Lagrangian point; matter overflowing star one spirals into an accretion disk around star two.

When a star fills its Roche lobe, its surface feels no gravity at $L$ , so gas pours through into the companion's lobe. Because the system rotates, the Coriolis effect makes the stream spiral into an accretion disk rather than fall straight in. If the companion $M_{2}$ is a white dwarf, cataclysm follows. Stored disk material dumped onto the dwarf's surface heats it and raises the luminosity by a factor of 10 to 100, a nova; recurrences range from weeks to millennia. A classical nova brightens by up to a factor of a million when enough hydrogen accumulates to trigger a surface thermonuclear explosion.

Supernovae

A supernova is the destruction of an entire star, more luminous at peak than its whole host galaxy. Supernovae are classified by spectrum and light curve.

Supernova classification: Type I lacks hydrogen lines and splits by silicon and helium features; Type II shows hydrogen and splits by light-curve shape.

The two types have entirely different origins.

Type Ia, a carbon detonation. In a binary, a companion pours mass onto a carbon-oxygen white dwarf. When the dwarf's mass reaches the Chandrasekhar limit, its central pressure ignites runaway carbon fusion, a thermonuclear carbon bomb that unbinds the star. The common trigger mass makes all Type Ia events nearly identical in peak brightness, which is why they serve as standard candles.
Type II, core collapse. In a star above about $8 M_{⊙}$ , gravity keeps driving fusion through oxygen, neon, and silicon until the core is iron. Iron has the highest binding energy per nucleon, so further fusion absorbs energy, and the core has no way to resist gravity.

Late in a massive star, fusion products settle in shells around an inert iron core, the layout that precedes a Type II core-collapse supernova.

Collapse compresses and heats the core past $1 0^{9} K$ , where photons photodisintegrate iron and then helium, each step absorbing energy and accelerating the fall:

^{56} Fe ⟶ 13^{4} He + 4 n,^{4} He ⟶ 2 p + 2 n .

In free fall the core crushes protons and electrons together by inverse beta decay, converting into neutrons and a flood of neutrinos:

p + e^{-} ⟶ n + ν_{e} .

The envelope outside the core is blown away in the explosion. The 1987 supernova SN1987A in the Large Magellanic Cloud, $170, 000 ly$ away, released the predicted neutrino burst, detected at the Kamiokande observatory, confirming the core-collapse picture and hinting at nonzero neutrino mass. The supernova of 1054, recorded by Chinese astronomers, is now the Crab Nebula.

White dwarfs

A star below about $6 M_{⊙}$ ends by shedding its outer layers as a planetary nebula, leaving a white dwarf of mass near $1 M_{⊙}$ packed into a radius of order $1 0^{7} m$ , comparable to Earth. Its density reaches about $5 \times 1 0^{5} g/cm^{3}$ . With fusion ended, nothing thermal holds it up; gravity compresses it until the exclusion principle stops the electrons, the same degeneracy pressure that resists compression of a Fermi gas. Balancing electron degeneracy pressure against gravity gives a nonrelativistic mass-radius relation

R = (3.1 \times 1 0^{17} m \cdot kg^{1/3}) \frac{Z ^{5/3}}{A} M^{- 1/3},

with $Z$ and $A$ the atomic number and mass number of the material. The radius shrinks as the mass grows: a heavier white dwarf is smaller.

A lone white dwarf radiates its heat away with no furnace to replace it, cooling toward a black dwarf in equilibrium with the universe, a stage none has yet reached. A white dwarf in a binary that accretes to the Chandrasekhar limit instead implodes and detonates as a Type Ia supernova.

Neutron stars

If the collapsed core exceeds the Chandrasekhar limit, electron degeneracy fails and the core becomes neutrons. Treating those neutrons as an ideal Fermi gas gives a mass-radius relation of the same form,

R = (1.6 \times 1 0^{14} m \cdot kg^{1/3}) M^{- 1/3},

so a $1.0 M_{⊙}$ core has $R \approx 1.27 \times 1 0^{4} m = 12.7 km$ . Its density, about $1.2 \times 1 0^{14} g/cm^{3}$ , is nearly that of a bare neutron. Here the exclusion-principle repulsion of the strong nuclear force between neutrons balances gravity. This support also has a ceiling: current theory places the maximum neutron star mass between $1.7 M_{⊙}$ and $3 M_{⊙}$ .

On a log-log mass-radius plot, degenerate stars shrink as mass grows; neutron stars lie about three orders of magnitude smaller than white dwarfs.

Rapidly rotating neutron stars appear as pulsars, regular radio (and sometimes optical) sources first found in 1967 in supernova remnants such as the Crab Nebula. The neutron star inherits much of the original star's angular momentum and magnetic field, so it spins fast while dragging a tilted magnetosphere. Charged particles accelerated along the magnetic axis radiate in a narrow cone; as the tilted axis sweeps past Earth, the source pulses like a lighthouse. The Crab pulsar radiates $3 \times 1 0^{31} W$ with a period of $0.033 s$ .

A pulsar's magnetic axis is tilted from its spin axis, so its radiation cones sweep the sky and pulse each time a beam crosses Earth.

Black holes

When the remnant core exceeds the neutron-star ceiling, no known pressure stops the collapse. The escape velocity from a mass $M$ at radius $R$ , from equating kinetic and gravitational potential energy, is

v_{e} = (\frac{2 GM}{R})^{1/2} .

Setting $v_{e} = c$ defines the Schwarzschild radius.

Radiation is trapped as surely as matter. Light of rest wavelength $λ_{0}$ emitted at radius $R$ suffers the gravitational redshift

\frac{λ}{λ _{0}} = (1 - \frac{R _{S}}{R})^{- 1/2} .

As $R \to R_{S}$ the wavelength diverges and the photon energy $E = h c / λ$ falls to zero, so no energy escapes: the object neither emits nor reflects, appearing perfectly black.

Inside the Schwarzschild radius neither matter nor light escapes; passing light rays bend toward the hole, and those crossing the horizon are captured.

Black holes are inferred from their gravity. At the center of the Milky Way, stars tracked orbiting the compact radio source Sagittarius A* imply an unseen mass of about $3 \times 1 0^{6} M_{⊙}$ confined to a tiny volume. Unlike white dwarfs and neutron stars, black holes do not cool toward equilibrium.

The compact remnants also power gamma-ray bursts, the brightest gamma-ray flashes in the sky, lasting from under a second to minutes and distributed uniformly across the sky, which places them at cosmological distances. Many appear to accompany the collapse of very massive stars into neutron stars or black holes.

Remnant	Support against gravity	Typical mass	Typical radius
White dwarf	electron degeneracy pressure	$\sim 1 M_{⊙}$	$\sim 1 0^{4} km$
Neutron star	neutron degeneracy / strong force	$1.4$ - $3 M_{⊙}$	$\sim 12 km$
Black hole	none (collapse continues)	$> 3 M_{⊙}$	$R_{S} = 2 GM / c^{2}$