The Formation and Evolution of White Dwarfs
Author: 丛雨
Reviewer: 円岛
Sirius is the brightest star in Earth’s night sky and is visible every winter. In 1844, the German astronomer and mathematician Friedrich Bessel calculated that Sirius must actually be a binary system with a companion about as massive as the Sun. The companion was so faint that it remained unseen for years. It was finally photographed in 1862, shining at only one-thousandth the brightness of Sirius. Its spectrum was obtained later, and astronomers began to recognize a new class of object: the white dwarf.
The Late Evolution of Sun-like Stars
To understand how white dwarfs form, we must begin with the post-main-sequence evolution of low- and intermediate-mass stars. Stars below 2.3 solar masses are generally classed as low-mass stars, while those between 2.3 and 8 solar masses are intermediate-mass stars. The boundary reflects how such stars evolve and how their lives end, as we will see below.
As hydrogen fusion proceeds in a star’s core, the fuel is gradually exhausted and an increasingly massive core composed mainly of helium develops. Gravity makes this core contract, raising its temperature, pressure, and density as the star slowly enters the red-giant phase. From there, its evolution follows one of two paths. In an intermediate-mass star, the core becomes hot enough to ignite helium and begins the triple-alpha process, fusing helium into carbon. The core of a low-mass star is initially too cool for helium fusion, so it cannot use radiation pressure to resist gravity. Instead, its electrons become degenerate, and electron degeneracy pressure opposes further contraction.
What is degeneracy pressure? In a system of fermions, no two particles can occupy the same microscopic quantum state, a rule known as the Pauli exclusion principle. In a degenerate electron gas, for example, each energy level can hold at most two electrons, one in each spin state; additional electrons must occupy other states. The resulting resistance among fermions is degeneracy pressure. Because the lower energy levels in a degenerate gas fill quickly, most particles have far more energy than they would in an ordinary gas. That high energy corresponds to high momentum, so pressure generated through the exchange of particle momentum can greatly exceed the pressure of a normal gas. It can therefore resist stronger gravity and support a much denser core. Theory shows that electron degeneracy pressure depends on density. At the same density and temperature, particles with less mass become degenerate more readily, so electrons are the first particles in a stellar core to enter a degenerate state.
As a degenerate core contracts further, its temperature continues to rise. In low-mass stars between 0.5 and 2.3 solar masses, the core eventually reaches the temperature required for helium fusion and ignites explosively in a “helium flash.” Degeneracy is then lifted. Once the core’s helium is exhausted, these stars, like intermediate-mass stars, develop a core made mainly of carbon and oxygen. The star has now reached the final phase of its life: the asymptotic giant branch (AGB). The carbon–oxygen core also becomes electron-degenerate, giving it the basic form of a white dwarf. Enormous luminosity and powerful stellar winds strip away the envelope, leaving an isolated carbon–oxygen white dwarf while the expelled material forms a planetary nebula. Stars below 0.5 solar masses never develop a core hot enough to fuse helium; their final remnants are helium white dwarfs.
The Structure and Properties of White Dwarfs
Nuclear reactions have ceased inside a white dwarf. It is the ash left after a star’s active life has ended. White dwarfs lie in the lower-left region of the Hertzsprung–Russell diagram: they are faint, hot, and dense. Sirius’s companion, for example, is about as massive as the Sun but only about as large as Earth. Its surface temperature is roughly 27,000 K, yet its luminosity is only 1/360 that of the Sun. Although a white dwarf’s interior may be as hot as 10^8 K, electron degeneracy pressure still dominates over thermal pressure. Degenerate electrons also conduct heat efficiently, leaving the interior nearly isothermal. Around the core, which occupies most of the volume, lies a thin envelope of nondegenerate ideal gas. This cooler envelope transports energy through convection and radiation far less efficiently than the isothermal core conducts it. By limiting the loss of internal energy, the envelope helps explain why white dwarfs cool so slowly.
The white-dwarf mass–radius relation tells us that as a white dwarf gains mass, its radius shrinks. The resulting increase in internal density and electron degeneracy pressure is needed to counter stronger gravity, and the electron gas gradually changes from nonrelativistic to relativistic degeneracy. A white dwarf cannot shrink without limit, however, so its mass has an upper bound: the Chandrasekhar limit. This is the maximum mass that a degenerate electron gas can support. At the limit, the star would be compressed toward a singularity. The numerical value depends only on the ratio of atomic mass to atomic number. That ratio is 2 for both helium and carbon–oxygen white dwarfs, so both have the same upper mass, the familiar value of about 1.45 solar masses.
Accretion and Eruptions
If a white dwarf belongs to a close binary system, its companion may expand late in life until it fills its Roche lobe. Matter from the companion then flows through the first Lagrange point and is accreted by the white dwarf. What happens next depends strongly on the accretion rate. If the rate is too low, the accreted hydrogen undergoes runaway fusion on the white dwarf’s surface. A nova ejects the newly accumulated material, and the cycle may repeat to form a recurrent nova. If the rate is too high, the hydrogen-rich envelope expands rapidly and the white dwarf instead becomes an asymptotic giant once again.
Stable accretion occurs only across a narrow range of rates. When a white dwarf’s mass approaches the Chandrasekhar limit, a thermonuclear explosion follows. It releases more energy than the star needs to maintain hydrostatic equilibrium, so the result is catastrophic: the white dwarf is torn apart in a Type Ia supernova. Because the progenitors of Type Ia supernovae have roughly the same mass—the Chandrasekhar limit—their explosions have similar luminosities and absolute magnitudes. This makes them one of astronomy’s standard candles for measuring distance.
White-dwarf explosions usually leave no remnant, but there are exceptions. If both stars in a binary system are white dwarfs, gravitational-wave emission drains angular momentum from the system until the two draw close enough to merge. Numerical simulations suggest that the less massive, less dense white dwarf is completely disrupted and accreted by the other. If accretion proceeds rapidly, electron capture by the products of carbon ignition reduces the pressure, causing the star to collapse and ultimately form a neutron star.
References
Introduction to Astrophysics, Xiang Shouping, University of Science and Technology of China Press
Introduction to Stellar Structure and Evolution, Li Yan, Peking University Press
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