What Is Degeneracy Pressure, and What Holds Up a White Dwarf?

When a star like the Sun runs out of nuclear fuel, its core contracts into a white dwarf: roughly the mass of the Sun packed into something about the size of Earth. There is no fusion inside to push back against gravity. Something else does the holding. That something is a purely quantum effect, and it is the job of our character Exclusor. This post builds on our explainer Why Can’t Two Electrons Share the Same State?.

What is a white dwarf like?

NASA’s description is a good start: a typical white dwarf is about half as massive as the Sun yet only slightly bigger than Earth, with a density around 1 × 10⁹ kg/m³ — some 200,000 times Earth’s average (NASA Imagine the Universe^[1]). The nearest one, Sirius B, was weighed with the Hubble Space Telescope by measuring how its gravity shifts the wavelengths of its light: about 98% of the Sun’s mass, in a body smaller than Earth (ESA/Hubble 2005^[2]).

What is degeneracy pressure?

Electrons are fermions, spin-½ particles that obey the Pauli exclusion principle: no two identical fermions can occupy the same quantum state. Wolfgang Pauli received the 1945 Nobel Prize in Physics for discovering it (NobelPrize.org, Pauli^[3]). In an atom this fills shells. In a dense star it fills momentum space.

Squeeze a gas of electrons into a smaller volume and the low-momentum states fill up first, two electrons (spin up and spin down) per momentum state. The remaining electrons must take higher and higher momenta. Even at zero temperature, then, a dense electron gas has large kinetic energy and exerts pressure. NASA puts it bluntly: for gravity to compress the white dwarf further, it must force electrons “where they cannot go” (NASA^[1]). This is electron degeneracy pressure. It barely depends on temperature, which is why a white dwarf can cool for billions of years without collapsing.

How the pressure scales

For a degenerate electron gas, the momentum of the most energetic electrons (the Fermi momentum) is

p_F = ħ (3π² n_e)^1/3

where ħ is the reduced Planck constant and n_e the number of electrons per cubic metre. Two limits follow, written for mass density ρ and constants K₁, K₂ set by fundamental constants and composition (cross-check^[4]):

• Non-relativistic electrons (speeds well below c): P = K₁ ρ^5/3
• Ultra-relativistic electrons (speeds near c): P = K₂ ρ^4/3

That change of exponent, from 5/3 to 4/3, is the whole story of the white-dwarf mass limit.

Fowler, 1926: the first explanation

In the 1920s white dwarfs posed a puzzle that Arthur Eddington had raised: how could such dense matter ever cool? Chandrasekhar’s Nobel lecture recounts how R. H. Fowler reformulated that paradox and resolved it in 1926 by applying the new Fermi–Dirac statistics: at white-dwarf densities the electrons are highly degenerate, and their zero-point energy is so large that a star can settle into a state in which all its energy is of this kind (Chandrasekhar, Nobel Lecture^[5]). Fowler’s paper is “On Dense Matter”, Mon. Not. R. Astron. Soc. 87, 114 (1926; Crossref DOI 10.1093/mnras/87.2.114).

In Fowler’s non-relativistic theory a white dwarf is a polytrope of index 3/2, and its radius is inversely proportional to the cube root of its mass: R ∝ M^−1/3. Heavier white dwarfs are smaller. On that theory, equilibrium exists for every mass (Nobel Lecture^[5]).

Adding relativity: why there is a maximum mass

As the mass rises, the central density rises, and the most energetic electrons approach the speed of light. Chandrasekhar’s lecture notes that already for a one-solar-mass white dwarf the threshold electron velocity at the centre is about 0.63c, so special relativity cannot be ignored (Nobel Lecture^[5]). In the ultra-relativistic limit the pressure stiffens only as ρ^4/3. A star built on P = K₂ ρ^4/3 is a polytrope of index 3, and such a configuration has one mass only, fixed by K₂. Below it, degeneracy wins; above it, no equilibrium exists and the radius of the idealised model goes to zero.

For the idealised model the limit is

M_Ch = (ω₃√(3π) / 2) (ħc/G)^3/2 / (μ_e m_H)²

where c is the speed of light, G the gravitational constant, m_H the mass of a hydrogen atom, μ_e the mean number of nucleons per electron, and ω₃ ≈ 2.018 a number from the Lane–Emden equation (the lecture quotes 2.018; Wikipedia, Chandrasekhar limit^[4]). For carbon or oxygen, μ_e = 2 and the limit is about 1.44 solar masses, usually rounded to 1.4 (NASA^[1]). Chandrasekhar’s 1931 paper assumed μ_e = 2.5 and obtained 0.91. Real white dwarfs top out a little lower once electrostatic and other effects left out of the ideal model are included.

Who found the limit?

The limit carries Subrahmanyan Chandrasekhar’s name, and his 1931 paper “The Maximum Mass of Ideal White Dwarfs” (Astrophys. J. 74, 81; Crossref DOI 10.1086/143324) gave the version for a star in proper hydrostatic equilibrium. But he was not first to find a relativistic limit. Eric Blackman’s note sets the record straight: Wilhelm Anderson (1929) and Edmund Stoner (1930) had already combined relativity with electron degeneracy and found maximum masses for uniform-density stars; Chandrasekhar cited Stoner, and his own polytrope limit came out about 20% below Stoner’s (Blackman 2011^[6]). Chandrasekhar shared the 1983 Nobel Prize in Physics with William Fowler (not R. H. Fowler) “for his theoretical studies of the physical processes of importance to the structure and evolution of the stars” (NobelPrize.org 1983^[7]). Our post on physics credit myths tells the longer story, including Eddington’s public rejection of the idea in 1935.

What happens above the limit?

A core heavier than the limit cannot be held up by electrons. In massive stars it collapses until neutrons, packed by their own degeneracy and by nuclear forces, can stop it — a neutron star — or until nothing can, and a black hole forms. A white dwarf that gains mass from a companion and approaches the limit can ignite carbon and explode as a Type Ia supernova. How most Type Ia progenitors reach explosion — steady accretion, or the merger of two white dwarfs — is still debated, and some unusually bright events may involve white dwarfs above the standard limit (Wikipedia, super-Chandrasekhar events^[4]). We label that open.

In the shop

Exclusor is the rule that refuses to let electrons double up. The Chandrasekhar sticker (“No Seconds”) is the ceiling, and the Pauli sticker is the man behind the rule. More physicist portraits are in Stickers. For the spin side of the story, see Why Do Electrons Have Spin ½?.

FAQ

Is degeneracy pressure a force between electrons?

No. It is not electric repulsion. It comes from the exclusion principle forcing electrons into higher-momentum states, which raises their kinetic energy as the gas is compressed.

Why are heavier white dwarfs smaller?

More mass needs more pressure to support it, and degenerate electrons supply more pressure only when packed more densely. In Fowler’s theory, R ∝ M^−1/3.

Is the Chandrasekhar limit exactly 1.4 solar masses?

No. About 1.44 in the idealised carbon–oxygen model, a little less with realistic corrections, and different for other compositions.

References

1. NASA GSFC Imagine the Universe, “White Dwarf Stars”. https://imagine.gsfc.nasa.gov/science/objects/dwarfs2.html
2. ESA/Hubble (2005), heic0516: Hubble measures the mass of the white dwarf Sirius B. https://esahubble.org/news/heic0516/
3. NobelPrize.org, Wolfgang Pauli, Physics 1945, facts. https://www.nobelprize.org/prizes/physics/1945/pauli/facts/
4. Wikipedia, “Chandrasekhar limit” (cross-check of μₑ dependence and super-Chandrasekhar debate). https://en.wikipedia.org/wiki/Chandrasekhar_limit
5. S. Chandrasekhar (1983), Nobel Lecture “On Stars, Their Evolution and Their Stability” (PDF, NobelPrize.org). https://www.nobelprize.org/uploads/2018/06/chandrasekhar-lecture.pdf
6. E. G. Blackman (2011), “Anderson and Stoner Published White Dwarf Mass Limits Before Chandrasekhar,” arXiv:1103.1342. https://arxiv.org/abs/1103.1342
7. NobelPrize.org, The Nobel Prize in Physics 1983 (Chandrasekhar, Fowler). https://www.nobelprize.org/prizes/physics/1983/summary/

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.