5.4 Compact objects: white dwarfs, neutron stars, and black holes

Formation and Properties of Compact Objects

Compact objects are the dense remnants left behind when stars exhaust their nuclear fuel. White dwarfs, neutron stars, and black holes each represent a different endpoint of stellar evolution, determined primarily by the progenitor star's mass. Together, they form a sequence of increasing density and gravitational extremity, and studying them connects core concepts in quantum mechanics, general relativity, and nuclear physics.

Formation of white dwarf stars

Stars with initial masses between roughly 0.6 and 8 solar masses end their lives as white dwarfs. Once helium fusion in the core ceases, the core contracts while the outer layers are expelled as a planetary nebula (the Ring Nebula is a well-known example). What remains is the exposed, degenerate core.

A typical white dwarf has a mass of 0.6 to 1.4 solar masses packed into an Earth-sized radius of about 6,000–8,000 km. That gives densities of $10^6$ to $10^9 \; \text{kg/m}^3$. For perspective, a teaspoon of white dwarf material would weigh roughly a ton.

White dwarfs are composed primarily of carbon and oxygen and are supported against further collapse by electron degeneracy pressure, a quantum mechanical effect arising from the Pauli exclusion principle. Because no fusion is occurring, a white dwarf simply radiates its residual thermal energy and cools over billions of years.

The Chandrasekhar limit sets the maximum mass a white dwarf can sustain at $\approx 1.44 \; M_\odot$. If a white dwarf accretes enough matter to exceed this limit, electron degeneracy pressure can no longer support it. The result is a Type Ia supernova, which completely destroys the star rather than leaving a remnant.

Structure of neutron stars

Stars with initial masses of roughly 8–20 solar masses undergo a core-collapse supernova at the end of their lives. During the collapse, the core's density rises so high that electrons are forced into protons via inverse beta decay, producing neutrons. The result is a neutron star.

Neutron stars pack 1.4 to about 3 solar masses into a radius of only 10–20 km. Their density reaches $\sim 10^{17} \; \text{kg/m}^3$, comparable to the density of an atomic nucleus. A sugar-cube-sized sample would have a mass of roughly a billion tons.

The internal structure has distinct layers:

1. Outer crust: neutron-rich nuclei arranged in a crystalline lattice, with a sea of electrons.
2. Inner crust: increasingly neutron-rich nuclei coexisting with free neutrons.
3. Core: a dense fluid of mostly neutrons (with some protons, electrons, and possibly exotic particles). The exact state of matter here remains an open question in nuclear physics.

The star is supported by neutron degeneracy pressure along with the strong nuclear force. The upper mass limit for neutron stars (the Tolman–Oppenheimer–Volkoff limit) sits around 2–3 solar masses, though the precise value depends on the still-uncertain nuclear equation of state.

Two additional properties make neutron stars remarkable:

• Rapid rotation. Conservation of angular momentum during collapse spins the neutron star up to periods ranging from milliseconds to seconds. These are observed as pulsars, which sweep beams of radiation across our line of sight like a lighthouse.
• Extreme magnetic fields. Magnetic flux conservation during collapse amplifies the field to $10^8$–$10^{15}$ Gauss (compare Earth's field at $\sim 0.5$ Gauss). Neutron stars at the upper end of this range are called magnetars.

Unique Properties of Black Holes and Comparison of Compact Objects

Concept of black holes

When a star with an initial mass above roughly 20–25 solar masses exhausts its fuel, the core collapse may produce an object so dense that no known force can halt the contraction. The result is a black hole.

The defining feature is the event horizon, the boundary beyond which the escape velocity exceeds the speed of light. For a non-rotating (Schwarzschild) black hole, this radius is:

$R_s = \frac{2GM}{c^2}$

For a 10 solar-mass black hole, $R_s \approx 30 \; \text{km}$. Everything inside the event horizon is causally disconnected from the outside universe.

At the center lies the singularity, a point (or ring, for rotating black holes) where density formally diverges and general relativity breaks down. Most physicists expect a future theory of quantum gravity to resolve this singularity.

Black holes come in several mass classes:

• Stellar-mass ($\sim 3$–$100 \; M_\odot$): formed from individual massive stars. Cygnus X-1 was the first strong candidate, identified by X-ray emission from its accretion disk.
• Supermassive ($10^6$–$10^{10} \; M_\odot$): found at the centers of galaxies. The Event Horizon Telescope imaged the shadow of the supermassive black hole in M87* in 2019.
• Intermediate-mass ($10^2$–$10^5 \; M_\odot$): evidence is growing but their formation channel is still debated.

Material spiraling into a black hole forms an accretion disk that heats up through friction and emits strongly in X-rays. This is the primary way stellar-mass black holes are detected in binary systems.

Hawking radiation is a theoretical prediction that quantum effects near the event horizon cause black holes to emit thermal radiation and slowly lose mass. For stellar-mass and supermassive black holes, this process is extraordinarily slow, with evaporation timescales far exceeding the current age of the universe.

Types of stellar remnants

The table below summarizes how the three compact objects compare:

• White dwarfs are detected as faint, hot stars that slowly cool and dim. Sirius B, the companion to the brightest star in the night sky, is the classic example.
• Neutron stars are most commonly observed as pulsars (radio, X-ray, or gamma-ray) or as X-ray sources in binary systems. The Crab Pulsar, embedded in the Crab Nebula supernova remnant, pulses about 30 times per second.
• Black holes cannot be observed directly. They are inferred from gravitational effects on companion stars, X-ray emission from accretion disks, and gravitational wave signals from mergers (detected by LIGO/Virgo).

Long-term fates:

• White dwarfs radiate away their thermal energy and are expected to eventually become hypothetical black dwarfs, though the universe is not yet old enough for any to have formed.
• Neutron stars remain stable unless they accrete enough mass to exceed the TOV limit, at which point they would collapse into a black hole.
• Black holes persist essentially indefinitely on astrophysical timescales but would theoretically evaporate via Hawking radiation over $\sim 10^{67}$ years (stellar-mass) or longer.