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Physics 1931

The Maximum Mass of Ideal White Dwarfs

Subrahmanyan Chandrasekhar

No dead star can outweigh ~1.44 Suns.

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In depth · the introduction

A burnt-out star can only be so heavy. Cross a line of about 1.44 Suns, and nothing — not even matter squeezed to a teaspoon weighing tonnes — can stop it from collapsing.

The big idea

When a star like our Sun runs out of fuel, it doesn't explode. It sheds its outer layers and leaves behind a dense, glowing ember called a white dwarf — about the mass of the Sun crammed into the size of the Earth. What holds this ember up against its own crushing gravity isn't heat; it's a strange quantum rule that forbids its electrons from being squeezed too close together.

Chandrasekhar's discovery was that this support has a breaking point. As you pile on more mass, the electrons are forced to move faster and faster — eventually near the speed of light. Once they hit that wall, they simply can't push back any harder. He calculated the exact mass where the resistance gives out: about 1.44 times the Sun. Above it, no white dwarf can exist.

How it came about

In 1930 a nineteen-year-old Indian student, Subrahmanyan Chandrasekhar, was sailing from Madras to England to begin his studies at Cambridge. On the long voyage he worked through the physics of white dwarfs and realised something his elders had missed: in the densest stars, the electrons move so fast that Einstein's relativity has to be folded in. When he did, a maximum mass fell out of the equations.

The idea ran headlong into Arthur Eddington, the most famous astrophysicist of the age. At a 1935 meeting of the Royal Astronomical Society, Eddington stood up and mocked the result in front of the room, refusing to believe a star could ever collapse without limit. His authority cast a long shadow over the young Chandrasekhar, who was proved right only over the following decades — and awarded the Nobel Prize in 1983.

Why it mattered

This single number turned out to be one of the hinges of the universe. It decides the fate of every dying star: stay under the limit and you become a quiet white dwarf; cross it and you collapse into a neutron star or a black hole. It is the reason black holes can form at all.

It also gave astronomers a cosmic measuring stick. Because exploding white dwarfs always reach the same critical mass before they blow, they all shine with nearly the same brightness — letting us gauge distances across billions of light-years, and discover that the universe's expansion is speeding up.

A way to picture it

Imagine a crowd packed into an elevator. A few people stand comfortably; add more and they press shoulder to shoulder, pushing back against the walls. That outward push is the electron pressure holding a white dwarf up. But people can only brace so hard. Keep cramming bodies in and, past a certain number, no amount of pushing can stop the floor from giving way. The Chandrasekhar mass is exactly that number — the last person the elevator can hold before it drops.

An interactive mass–radius plot of a white dwarf: drag a mass slider and the star's radius shrinks as it gets heavier, the curve diving toward zero at the Chandrasekhar mass of about 1.44 Suns; an inset shows the star physically shrinking as you slide.

Where it sits

The white dwarf was first understood by R. H. Fowler in 1926 using the brand-new quantum mechanics; Chandrasekhar's leap was to add Einstein's relativity, the same physics behind E = mc². His limit opened the door for others — Lev Landau, Robert Oppenheimer and Fritz Zwicky — to predict neutron stars and, eventually, black holes. Every modern story about supernovae, gravitational waves from merging compact stars, and the death of stars begins at the line he drew in 1931.

The original document
Original source text
S. Chandrasekhar · The Astrophysical Journal 74 (1931): 81–82 · Trinity College, Cambridge, November 12, 1930
Abstract
The theory of the polytropic gas spheres in conjunction with the equation of state of a relativistically degenerate electron-gas leads to a unique value for the mass of a star built on this model. This mass (=0.91 ⊙) is interpreted as representing the upper limit to the mass of an ideal white dwarf.
The note begins from R. H. Fowler's account of the white dwarf as a body supported not by heat but by a degenerate electron gas — matter so compressed that the electrons are packed to the limit the Pauli exclusion principle allows, and resist further squeezing even at zero temperature.
Chandrasekhar's step is to insist that, in a sufficiently dense star, the electrons move so near the speed of light that they must be treated relativistically. With the relativistic equation of state, the pressure grows more slowly with density than Fowler's non-relativistic law; the gas behaves as a polytrope of index n = 3, whose total mass is fixed by the constants of nature alone and is independent of its radius.
He evaluates this mass and finds a single number. Beyond it, the degenerate electron pressure can no longer balance the star's own gravity, so no static white dwarf of larger mass can exist.
[ … ]
(Chandrasekhar's 1931 figure, 0.91 ⊙, used the constants and mean molecular weight then in hand. His fuller 1935 treatment, with the modern value of the mean molecular weight per electron μ_e = 2, gives the now-standard limit of about 1.44 ⊙.)
S. Chandrasekhar · Trinity College, Cambridge · November 12, 1930