Chandrasekhar limit

The Chandrasekhar limit represents the maximum mass at which a white dwarf star can remain in hydrostatic equilibrium. It marks the threshold beyond which the inward pull of gravity overwhelms the outward electron degeneracy pressure that supports the star. The accepted modern value is approximately 1.4 times the mass of the Sun, although refinements depending on chemical composition may vary this slightly. Named after Subrahmanyan Chandrasekhar, the limit serves as a fundamental boundary in astrophysics, determining the final evolutionary states of stars and governing the conditions leading to neutron stars, black holes, and certain types of supernovae.
White dwarfs differ from main sequence stars in that their structural support does not come from thermal gas pressure generated by nuclear fusion. Instead, they are stabilised by electron degeneracy pressure, an effect rooted in quantum mechanics. Once a star exhausts its nuclear fuel, its core contracts until the electrons are packed so densely that further compression is resisted by the Pauli exclusion principle. For stars of sufficiently low mass this degeneracy pressure halts the collapse, creating a stable white dwarf. However, when the mass exceeds the Chandrasekhar limit, degeneracy pressure becomes insufficient, leading inevitably to further collapse.

Physical basis of the limit

The Chandrasekhar limit arises from the interplay between gravity and electron degeneracy pressure. Electron degeneracy pressure originates from the arrangement of electrons within a confined volume. Electrons, being fermions, are forbidden from occupying identical quantum states. As the density of the stellar core increases, electrons fill higher and higher energy levels, raising the overall pressure even in the absence of thermal motions. This behaviour explains why white dwarfs do not require ongoing fusion to maintain their structure.
At lower densities the electrons behave non-relativistically, resulting in an equation of state that links pressure to density through a polytropic relationship with an index of 3/23/23/2. In this regime the radius of a model white dwarf decreases as mass increases, with the radius inversely proportional to the cube root of the mass. Continued compression, however, forces the electrons to attain energies comparable with their rest energy, making relativistic corrections essential. In the relativistic limit, the equation of state corresponds to a polytrope of index 333, producing a situation in which the total allowable mass becomes independent of central density. Solving the hydrostatic equilibrium equations for this state yields the limiting mass.
When an equation of state that transitions smoothly between the non-relativistic and relativistic regimes is adopted, the theoretical white dwarf radius still decreases with increasing mass but falls to zero at the critical mass: the Chandrasekhar limit. Beyond this point gravitational attraction cannot be balanced, and the star collapses. Depending on total mass and other physical conditions, the collapse produces either a neutron star or, if sufficiently massive, a black hole.

Stellar evolution and consequences

During the life of a star, nuclear fusion in the core produces heat that counteracts gravitational collapse. As hydrogen is converted into helium, the core grows denser and hotter, allowing fusion of progressively heavier nuclei until iron is produced. Elements heavier than iron cannot yield net energy through fusion, and once iron accumulates the energy-generating processes cease. The next step in stellar evolution depends critically on the mass of the remnant core.
Stars with masses below the Chandrasekhar limit eventually shed outer layers and leave behind white dwarfs that cool slowly over billions of years. They remain stable unless disturbed by accretion from a companion star. In binary systems a white dwarf may accrete matter until it nears the Chandrasekhar limit. Exceeding the limit triggers runaway carbon fusion, leading to a Type Ia supernova, a crucial event for cosmological distance measurement.
Stars with initial masses producing remnants above the limit undergo further collapse, forming neutron stars or, if the mass is sufficiently high, black holes. These outcomes highlight the Chandrasekhar limit’s central role in determining the fate of stellar remnants and the nature of compact objects throughout the universe.

Mathematical modelling: polytropes and degeneracy

Modelling white dwarfs employs the theory of polytropes, in which pressure and density follow a power-law relation. For non-relativistic degeneracy the polytropic index is 3/23/23/2, yielding mass–radius relationships consistent with observations of low-mass white dwarfs. In the relativistic limit, the polytropic index becomes 333, producing the distinct mass threshold independent of central density.
Hydrostatic equilibrium is described by balancing the inward gravitational force with the outward degeneracy pressure gradient. Solving this equation with the appropriate equation of state reveals that increasing mass forces the electrons into relativistic motion, reducing the effectiveness of degeneracy pressure. Consequently, a white dwarf approaching the limit becomes smaller and denser, with central densities so high that electrons begin to be captured by protons in the nuclei, forming neutrons and reducing electron pressure further. This process signals the onset of core collapse.

Historical development of the concept

Understanding the physics of degenerate stellar matter emerged during the early twentieth century. In 1926 Ralph Fowler applied Fermi–Dirac statistics to white dwarfs, establishing their behaviour as a degenerate electron gas. Edmund Stoner subsequently investigated the mass–radius relations for homogeneous models, incorporating relativistic effects and identifying a maximum mass for such stars. Wilhelm Anderson also applied relativistic corrections and obtained a comparable mass limit.
Yakov Frenkel published relevant equations of state in 1928, though his work was largely overlooked at the time. Between 1931 and 1935 Subrahmanyan Chandrasekhar developed a comprehensive treatment using polytropic models and relativistic physics. Working during a voyage to England in 1930, Chandrasekhar calculated the relativistic degeneracy pressure and combined it with stellar structure equations, deriving the limit now associated with his name. His Nobel Prize lecture later summarised these foundational contributions.
The discovery sparked debate, particularly with Arthur Eddington, who objected to the implications of the limit—namely, that stars above it could collapse into objects with extreme densities or even black holes. Eddington challenged the theoretical framework and proposed modifications to relativistic mechanics, but the broader physics community accepted Chandrasekhar’s analysis. Notable physicists such as Wolfgang Pauli and Niels Bohr supported the validity of combining relativity with Fermi–Dirac statistics.
The topic also intersects with work by Lev Landau, who independently performed calculations in 1932 but did not relate them to white dwarfs. Discussions of priority have since noted that Stoner and Anderson obtained mass limits before Chandrasekhar, though Chandrasekhar provided the first rigorous modelling using realistic stellar structures.

Significance in modern astrophysics

The Chandrasekhar limit remains a cornerstone of stellar astrophysics. It governs the endpoints of stellar evolution, the mechanism behind Type Ia supernovae, and the formation of compact objects. As Type Ia supernovae serve as standard candles in cosmology, the limit indirectly influences measurements of cosmic expansion. In addition, rigorous mathematical treatments of the limit, including derivations from many-body quantum mechanics, continue to refine understanding of dense matter.