White dwarfs and the Chandrasekhar limit

Stellar embers

A white dwarf is the dense remnant core of a low- or intermediate-mass star. After the star sheds its outer layers, the leftover carbon-oxygen core remains hot and compact. It no longer produces significant fusion. Instead, it slowly cools over time.

A typical white dwarf has a mass comparable to the Sun but a radius comparable to Earth. Its density is enormous by everyday standards.

Degeneracy pressure

White dwarfs are supported by electron degeneracy pressure. This pressure comes from quantum mechanics, specifically the Pauli exclusion principle. Electrons are fermions, and no two identical electrons can occupy the same quantum state.

When matter is compressed strongly, electrons are forced into higher momentum states. This creates pressure even at zero temperature. Unlike ordinary gas pressure, degeneracy pressure does not depend mainly on heat.

Gravity versus quantum pressure

Hydrostatic equilibrium still applies:

ho(r)}{r^2}$$ But the pressure source is electron degeneracy rather than thermal motion. For nonrelativistic degenerate electrons, pressure scales approximately as $$Ppropto ho^{5/3}$$ This can support a white dwarf against gravity. ## Mass-radius relation White dwarfs have an unusual mass-radius relation: more massive white dwarfs are smaller. Adding mass increases gravity, compresses the star, and raises degeneracy pressure. A rough trend for nonrelativistic white dwarfs is $$Rpropto M^{-1/3}$$ This is opposite to ordinary objects, where adding mass often increases size. ## Chandrasekhar limit As a white dwarf gains mass, electron speeds become relativistic. Relativistic degeneracy pressure scales differently, roughly $$Ppropto ho^{4/3}$$ This is not stiff enough to support arbitrarily large mass. The maximum mass is the Chandrasekhar limit: $$M_{Ch}approx1.4M_odot$$ Above this mass, a cold nonrotating white dwarf cannot remain stable. ## Type Ia supernovae In binary systems, a white dwarf can gain mass from a companion. If conditions trigger runaway carbon fusion, the white dwarf may explode as a Type Ia supernova. These events are bright and useful for measuring cosmic distances because their peak luminosities can be standardized. The exact explosion pathways are an active area of astrophysical research. ## Cooling white dwarfs A lone white dwarf gradually radiates stored thermal energy and fades. Since it has no major continuing energy source, its luminosity declines with time. White dwarf cooling can be used to estimate ages of stellar populations. ## The big idea White dwarfs are supported by electron degeneracy pressure, a quantum effect from the Pauli exclusion principle. Their mass-radius relation is inverse, and their maximum stable mass is about $1.4M_odot$. White dwarfs connect quantum mechanics, stellar evolution, and cosmological distance measurement.