Compact Objects: White Dwarfs, Neutron Stars, and Black Holes

Compact objects are the dense remnants of dead stars: white dwarfs (Earth-sized, supported by electron degeneracy pressure, below the Chandrasekhar limit of about 1.4 M_☉), neutron stars (about 10 to 12 km in radius, supported by neutron degeneracy pressure and nuclear forces, up to about 2 to 2.3 M_☉), and black holes (regions from which not even light can escape, with an event horizon at the Schwarzschild radius R_s = 2GM ÷ c²). In brief: a white dwarf has a density of about 10⁹ kg/m³, a neutron star about 5 × 10¹⁷ kg/m³ (similar to the density of an atomic nucleus). The Schwarzschild radius is 2.95 km per solar mass. Black holes are detected by their effects on nearby matter: accretion discs emitting X-rays, orbits of stars, and gravitational waves from mergers.

When the nuclear fuel of a star runs out, gravity compresses the core until quantum effects or nuclear forces halt the collapse, or fail to halt it. This guide describes the three types of compact objects, their physical properties and how they are observed, compares them in a table, works through numerical examples of density, escape speed and the Schwarzschild radius, and corrects common mistakes.

What are white dwarfs?

A white dwarf is the exposed core of a star that has shed its envelope: a sphere of carbon and oxygen (or helium) with a mass of about 0.6 M_☉ and a radius similar to the Earth’s (about 6,400 km), so that a teaspoon of its matter would have a mass of several tonnes. It no longer produces energy, but shines because of stored heat, slowly cooling over billions of years. Gravity is balanced by electron degeneracy pressure, a consequence of the Pauli exclusion principle: no two electrons can occupy the same quantum state, so compressing them requires high energies and produces a pressure independent of temperature. The more massive a white dwarf, the smaller it is (radius ∝ M^−1/3). Above the Chandrasekhar limit of about 1.4 M_☉, degeneracy pressure cannot support the star. A white dwarf in a binary system that accretes matter up to this limit explodes as a type Ia supernova. Sirius B (1.0 M_☉, 0.0084 R_☉) is the best-known example.

What are neutron stars?

If the collapsing core of a massive star exceeds the limit, the electrons are forced into the protons (inverse beta decay, p + e⁻ → n + ν_e), forming a ball of neutrons. A typical neutron star has a mass of 1.4 M_☉ and a radius of about 11 to 12 km, a density of 5 × 10¹⁷ kg/m³: a sugar cube of this matter would have a mass of a few hundred million tonnes. The surface gravity is about 10¹¹ times that of the Earth; the escape speed is about 0.6c. Because angular momentum and magnetic flux are conserved in the collapse, neutron stars rotate rapidly (up to hundreds of times per second) and have enormous magnetic fields (10⁸ T, and up to 10¹¹ T in magnetars). A rotating neutron star with a magnetic axis not aligned with the rotation emits beams of radio waves that sweep past the Earth like a lighthouse: a pulsar (discovered by Jocelyn Bell Burnell in 1967), with periods from milliseconds to seconds, extremely regular. Neutron star mergers produce gravitational waves, gamma-ray bursts and heavy elements (gold, platinum), as observed in the event GW170817. Neutron degeneracy pressure and the nuclear force can support stars up to a maximum mass of about 2 to 2.3 M_☉ (the Tolman-Oppenheimer-Volkoff limit); the observed maximum is about 2.1 M_☉.

What are black holes?

If the remaining mass is too large for any known force to stop the collapse, the matter collapses to a black hole. In general relativity, a non-rotating black hole is described by the Schwarzschild solution: the event horizon is a spherical surface of radius R_s = 2GM ÷ c² (2.95 km for the Sun’s mass), from which nothing, not even light, can escape. A distant observer sees an object falling toward the horizon slow down and redshift, never quite reaching it; an infalling observer crosses the horizon without noticing it, but is then inevitably drawn to the central singularity, and tidal forces become extreme (spaghettification). Black holes are characterized only by mass, spin and charge (“no-hair” theorem). Types: stellar-mass black holes (3 to about 100 M_☉), supermassive black holes (10⁵ to 10¹⁰ M_☉) at the center of most galaxies, including Sagittarius A* in the Milky Way (4.3 × 10⁶ M_☉), and intermediate ones. Evidence: X-rays from gas heated in accretion discs (Cygnus X-1), stars orbiting Sagittarius A* at thousands of kilometers per second (Nobel Prize in Physics 2020 for Genzel and Ghez), gravitational waves from mergers (LIGO, from 2015), and the images of the shadow of M87* (2019) and Sagittarius A* (2022) by the Event Horizon Telescope. Stephen Hawking showed (1974) that black holes emit thermal radiation and slowly evaporate, but for stellar-mass black holes this is negligible (a temperature of about 6 × 10⁻⁸ K).

Compact objects compared

Worked example: densities and Schwarzschild radii

1. White dwarf density. M = 0.6 M_☉ = 1.19 × 10³⁰ kg, R = 6.4 × 10⁶ m: ρ = M ÷ (4/3 πR³) = 1.19 × 10³⁰ ÷ 1.10 × 10²¹ = 1.1 × 10⁹ kg/m³, about a million times the density of water.
2. Neutron star density. M = 1.4 M_☉ = 2.79 × 10³⁰ kg and R = 11 km: ρ = 2.79 × 10³⁰ ÷ (4/3 π (1.1 × 10⁴)³) = 2.79 × 10³⁰ ÷ 5.58 × 10¹² = 5.0 × 10¹⁷ kg/m³. A volume of 1 cm³ has a mass of 5 × 10¹¹ kg (500 million tonnes).
3. Escape speed from a neutron star: v = √(2GM ÷ R) = √(2 × 6.674 × 10⁻¹¹ × 2.79 × 10³⁰ ÷ 1.1 × 10⁴) = 1.84 × 10⁸ m/s = 0.61c.
4. Schwarzschild radii. R_s = 2GM ÷ c²: for the Sun 2.95 km; for 10 M_☉, 29.5 km; for the Earth, 8.9 mm; for Sagittarius A* (4.3 × 10⁶ M_☉), 1.27 × 10¹⁰ m = 0.085 AU (about 12.7 million km, smaller than the orbit of Mercury).
5. Density of a black hole (mass over the volume inside the horizon): for 1 M_☉, ρ = 1.989 × 10³⁰ ÷ (4/3 π (2,950)³) = 1.8 × 10¹⁹ kg/m³; for Sagittarius A*, 1.8 × 10¹⁹ ÷ (4.3 × 10⁶)² = 1.0 × 10⁶ kg/m³, a thousand times the density of water. Supermassive black holes are not especially dense.
6. Pulsar. A neutron star spinning once every 1.5 ms (667 Hz) has an equatorial speed v = 2πR ÷ T = 2π × 1.2 × 10⁴ ÷ 1.5 × 10⁻³ = 5.0 × 10⁷ m/s = 0.17c.

How were they discovered?

The first white dwarf, Sirius B, was seen by Alvan Clark in 1862 and recognized as dense by Walter Adams in 1915. Chandrasekhar calculated the mass limit in 1930. Zwicky and Baade proposed neutron stars in 1934; the first pulsar, found by Jocelyn Bell Burnell and Antony Hewish in 1967, proved that they exist (Hewish received the Nobel Prize in Physics in 1974). Karl Schwarzschild found the black hole solution in 1916, only weeks after Einstein published general relativity. The first strong candidate, Cygnus X-1, was identified in 1971 to 1972. LIGO detected gravitational waves from the merger of two black holes on 14 September 2015, announced in 2016.

Which mistakes are common?

Mistake 1: Thinking that black holes suck everything in

Far from the horizon, a black hole has the same gravity as any object of the same mass. If the Sun were replaced by a black hole of the same mass, the planets would continue to orbit as before.

Mistake 2: Confusing the event horizon with the singularity

The horizon is the boundary of no return. The singularity is the center, where the theory predicts infinite density.

Mistake 3: Saying that white dwarfs and neutron stars are supported by fusion

They are supported by degeneracy pressure, a quantum effect, and no longer fuse.

Mistake 4: Believing that the Sun will become a black hole

Its mass is too low. It will end as a white dwarf.