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§ Phase xv · Stellar Quanta · Chapter 02

White dwarfs

A sun-like star spends most of its life as a steady fusion engine, but the fuel runs out. The core squeezes inward and there is nothing classical that can stop it. The thing that catches the collapse is a quantum accounting rule, written down for chemistry in a Hamburg office in 1924, and a relativistic correction worked out by a nineteen-year-old on a slow boat from Bombay to Southampton.

A star is a balance. For ten billion years our Sun has been pushing outward with the pressure of hot gas and pulling inward with its own weight, and the two have very nearly cancelled. The push comes from fusion. Four hydrogen nuclei in the core slowly stitch themselves into helium, and the energy released keeps the core hot enough to keep fusing. Cool the core and the rate of fusion drops; squeeze the core and the rate rises. The thermostat has held for nearly five billion years and has another five to go.

Then the hydrogen in the core runs out. Helium ash builds up. The thermostat stalls. Gravity does not, and the core begins to contract. The contraction heats the core until helium itself starts to burn, three nuclei at a time, into carbon and oxygen. This is a more violent business than hydrogen fusion and it does not last long. A few hundred million years and the helium is gone too. Now there is a core of carbon and oxygen, no fuel left that the star is heavy enough to ignite, and gravity still pulling.

The question is plain. What holds the core up after the fire goes out?

For a long time nobody had a good answer. Arthur Eddington, the most respected stellar theorist of his generation, modelled stars as balls of ordinary gas and found that an ordinary gas at the densities expected inside a stellar remnant would not behave like an ordinary gas at all. It would be so compressed that the electrons in it could no longer be thought of as a thermal cloud. They would be jammed into their lowest available quantum states, all of them at once, and they would push back not because they were hot but because they had nowhere else to go.

That refusal of electrons to share a state is the quantum accounting rule we met in Phase ix. Two electrons cannot have the same complete quantum address. The address has four labels: a principal number n, an orbital number ℓ, a magnetic number m, and a spin that can point up or down. In an atom the addresses are discrete energy levels. In a chunk of metal the addresses smear into a continuous band, but the rule is unchanged. Each address is filled by at most one electron. If you try to compress a piece of metal so hard that all the low-energy addresses fill up, the next electron you push in has to go into a high-energy address, and the price of admission is high. The total energy rises. The piece of metal pushes back.

In an ordinary chunk of copper this effect is invisible against the much larger pressure of thermal motion. In the core of a dead star, the densities are millions of times higher and the thermal motion, by comparison, is negligible. The exclusion-driven pressure becomes the only thing holding the core up.

The discovery of an actual specimen came long before anyone understood it. In 1844, Friedrich Bessel noticed that Sirius, the brightest star in the night sky, was wiggling. Its position drifted in a small wave on the sky with a period of about fifty years. Bessel concluded there must be an unseen companion, dark and massive, dragging Sirius around a common centre of mass. He died eighteen years before anyone saw the companion. In January 1862 the Massachusetts telescope-maker Alvan Graham Clark, testing a new eighteen-and-a-half-inch refractor on Sirius, saw a faint speck of light just off the primary, almost lost in the glare. The Sirius companion, soon called Sirius B, was real.

It made no sense. From the orbital motion the mass of Sirius B came out almost equal to that of our Sun. From its faintness, its luminosity was about a four-hundredth that of the Sun. By the usual relation between a star’s brightness and its surface temperature, a star that dim should have been cold and red. The spectroscopists found instead that Sirius B was hot and white. A hot, white, small, sun-mass star is not allowed by any ordinary stellar model. The star astronomers had found seemed to be the size of the Earth and yet to weigh as much as the Sun. Eddington called it “absurd.”

In 1915 Walter Adams, at Mount Wilson, did the experiment that pinned the absurdity down. He measured the wavelengths of the spectral lines of Sirius B and found them shifted toward the red. The shift was not the Doppler shift of orbital motion (he had already subtracted that). It was the gravitational redshift predicted by Einstein’s general relativity, published that same autumn. Photons climbing out of a deep gravitational well lose energy, and their wavelengths stretch. The size of the redshift implied a surface gravity hundreds of thousands of times stronger than the Sun’s, which implied that the entire mass of Sirius B really was packed into a sphere a few times the radius of the Earth. The density Eddington had refused to believe was right.

The theoretical answer arrived a decade after that, from a young Cambridge mathematician named Ralph Howard Fowler. Fowler had spent the war computing ballistic tables for the Royal Navy and had come back to Cambridge to apply the brand-new quantum statistics of Enrico Fermi and Paul Dirac to anything he could find. In December 1926, only months after Fermi and Dirac had separately written down the distribution that governs electrons, Fowler published a short paper titled “On Dense Matter.” In it he argued that the gas inside Sirius B could not possibly be an ordinary thermal gas. The electrons were degenerate, in the technical sense: every low-energy quantum state was occupied, all the way up to a maximum energy now called the Fermi energy. The pressure of such a gas does not depend on temperature at all. It depends only on density, and it can be enormous.

Fowler showed that this Fermi pressure was exactly the right size to support a sun-mass remnant at Earth-radius dimensions. The puzzle of Sirius B was solved. A white dwarf is a ball of carbon and oxygen ions floating in a sea of degenerate electrons, and what holds it up is not heat, not fusion, not anything classical, but Pauli’s refusal.

To get a feel for the size of the Fermi pressure, here is the back-of-an-envelope. Confine N electrons in a box of volume V. The exclusion principle says each electron needs at least a Compton-sized cube of phase space, so the highest-momentum electron has a momentum p_F set by N over V (the number density), and the corresponding kinetic energy is the Fermi energy E_F. Adding up the kinetic energies of all the electrons gives a total energy that scales as N times E_F, and the derivative of that total energy with respect to volume gives the pressure. In the non-relativistic regime the algebra comes out to

P ≈ (ℏ² / m_e) × (N/V)^(5/3),

where m_e is the electron mass. Squeeze the gas (raise N/V) and the pressure climbs as the five-thirds power of density. Notice the temperature does not appear. That is what “degenerate” means.

Fowler’s 1926 analysis was non-relativistic. It used the kinetic energy formula p²/2m, which is fine as long as the electrons are moving slowly compared to the speed of light. In a sun-mass white dwarf the electrons at the Fermi surface are moving at about a quarter of the speed of light. That is fast, but not yet fast enough to break the non-relativistic story badly. Push the white dwarf to higher masses, though, and the central density climbs, and the Fermi momentum climbs with it. At some density the most energetic electrons are moving very close to the speed of light. The kinetic energy stops growing as p² and starts growing only as p. The pressure stops climbing as ρ^(5/3) and softens to ρ^(4/3). And then a peculiar thing happens. A pressure that grows only as the four-thirds power of density is exactly the borderline case at which gravity wins.

In the summer of 1930, a nineteen-year-old physics student named Subrahmanyan Chandrasekhar boarded the steamer Pilsna in Bombay, bound for Cambridge to begin his PhD with R.H. Fowler. The voyage took eighteen days. Chandrasekhar carried a sheaf of papers, a copy of Eddington’s “The Internal Constitution of the Stars,” and a problem that nobody at home had been able to talk him out of working on. He had read Fowler’s paper in Madras and noticed the non-relativistic assumption. He wanted to redo the calculation with the right relativistic kinetic energy.

He did the integral on the boat. Between Bombay and Suez he had the non-relativistic case nailed down. Between Suez and Port Said he had the ultra-relativistic limit. By the time he reached Southampton he had what is now called the Chandrasekhar mass, the upper limit above which electron degeneracy pressure cannot hold up a star against its own gravity. The number came out to about 1.4 times the mass of the Sun. Above that, no equilibrium exists.

The argument is short enough to sketch. In hydrostatic equilibrium the inward gravitational pull on a shell is balanced by the outward pressure gradient. For a polytropic equation of state P proportional to ρ^γ, a self-consistent solution requires γ greater than 4/3 for stability. The non-relativistic Fermi gas has γ equal to 5/3. Stable. The ultra-relativistic Fermi gas has γ equal to 4/3. Marginal. At the mass where the electrons go fully relativistic the polytrope has no stable radius. There is no white dwarf solution. The star must keep contracting.

The Chandrasekhar limit () is the maximum mass of a stable white dwarf star. These stars resist gravitational collapse primarily through electron degeneracy pressure, compared to main sequence stars, which resist collapse through thermal pressure. The Chandrasekhar limit is the mass above which electron degeneracy pressure in the star's core is insufficient to balance the star's own gravitational self-attraction. The value of the Chandrasekhar limit depends upon the ratio of the number of…

What Chandrasekhar carried into Cambridge in 1930 was, in retrospect, one of the cleanest results in astrophysics. He had taken a young man’s confidence in a new formalism, a sea voyage’s worth of time, and the Fermi-Dirac statistics that Fowler had introduced to him, and he had derived a hard upper bound on the mass of a stable degenerate star. He spent the next four years polishing the calculation, computing the full mass-radius relation, and writing it up for the Monthly Notices of the Royal Astronomical Society. In January 1935 he presented the work in person at a meeting of the Society in Burlington House.

Eddington spoke after him. Eddington was the most respected stellar theorist in Britain, the man who had measured the bending of starlight in 1919 and made Einstein famous, and he did not believe a word of Chandrasekhar’s result. He stood up and said, in front of the audience, that there must be a law of nature that prevented stars from behaving in this absurd way, that the relativistic degeneracy formula must be wrong, and that the conclusion was a “reductio ad absurdum” of relativistic Fermi-Dirac statistics. He gave no derivation; he was making a personal appeal to taste. Chandrasekhar, twenty-four years old and a long way from home, sat through the talk in silence. He went home and cried, and wrote to his father in Madras.

The picture that emerges from Fowler and Chandrasekhar is this. A white dwarf is a corpse. Its fusion has stopped; its luminosity comes entirely from the slow leak of leftover heat through a thin atmosphere. The interior is a Coulomb crystal of carbon and oxygen ions floating in a degenerate electron sea. The Fermi pressure of that sea is what holds the star up. The star can have any mass from a few hundredths of a solar mass (a relic of a very small dying star) up to but not exceeding the Chandrasekhar limit. The higher the mass, the smaller the radius, because the gravitational binding wins more decisively over the rest-mass energy of the electrons and packs them tighter. At the limit, the radius formally falls to zero.

Surface temperatures of observed white dwarfs run from about 100,000 K for the youngest, hottest examples down to about 4,000 K for the oldest cooled remnants. The Sun, for comparison, has a surface temperature of about 5,800 K. A young white dwarf is hot and blue-white; an old one is dim and red. Because the interior is held up by degeneracy and not heat, cooling does not contract the star. It just dims it. A white dwarf radiates its leftover heat over billions of years, sliding slowly down the cooling curve, going from white to yellow to orange to red to (in principle, given enough time) a final cold “black dwarf” state that no example of has yet been observed. The universe is not old enough; the coolest known white dwarfs are still around 4,000 K, well above any final temperature.

What if a white dwarf accretes matter from a binary companion, gradually building up mass and edging toward the Chandrasekhar limit? The answer is one of the most violent events in astrophysics. As the central density climbs toward the limit, carbon ignites under degenerate conditions, which is to say without a thermostat. A normal fusion process raises the temperature, which lowers the density (the gas expands), which slows the fusion. Under degeneracy the pressure does not depend on temperature, so raising the temperature does not expand the gas, and the fusion runs away into a thermonuclear explosion that consumes the entire star. This is a Type Ia supernova, and because every white dwarf at the Chandrasekhar limit has nearly the same mass, every Type Ia supernova has nearly the same peak luminosity. That uniformity is what made them the “standard candles” used in the 1998 discovery that the expansion of the universe is accelerating.

Although white dwarfs are known with estimated masses as low as and as high as , the mass distribution is strongly peaked at , and the majority lie between . The estimated radii of observed white dwarfs are typically 0.8–2% the radius of the Sun; this is comparable to the Earth's radius of approximately 0.9% solar radius. A white dwarf, then, packs mass comparable to the Sun's into a volume that is typically one millionth of the Sun's; the…

So a white dwarf is, in the end, an extraordinary illustration of what quantum mechanics looks like at astronomical scales. A rule written down by Pauli in 1924, to count electrons in atoms, holds up a star of solar mass against its own gravity. A correction worked out by a teenager on a steamer, accounting for the relativistic kinematics of those same electrons, sets an exact upper bound on how heavy that star can be. Cross that bound and the star cannot stay a white dwarf. It must do something more extreme.

What it does is the subject of the next chapter.

↘ Where to read more

• The clearest derivation of the mass-radius relation, all the way to the limit, is Chandrasekhar’s own “An Introduction to the Study of Stellar Structure” (1939), chapters 10 and 11. It is unfussy and surprisingly readable.
• For the human story, Kameshwar Wali’s “Chandra: A Biography of S. Chandrasekhar” (1990) is the standard reference. The chapter on the 1935 confrontation with Eddington is painful and worth reading in full.
• The Walter Adams 1925 paper on the gravitational redshift of Sirius B (“The Relativity Displacement of the Spectral Lines in the Companion of Sirius,” Proc. Nat. Acad. Sci. 11, 382) is short and uses no equations a careful undergraduate cannot follow.
• For the modern view, Shapiro and Teukolsky’s “Black Holes, White Dwarfs, and Neutron Stars” (1983) treats the full equation of state and is the canonical graduate text.
• If you want a feel for what a Fermi gas actually is, Ashcroft and Mermin’s “Solid State Physics” chapter 2 is the cleanest introduction in print.

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