Degenerate Stars and the Mass Limit

White dwarfs are stellar remnants left behind after stars up to roughly eight times the Sun’s mass exhaust their nuclear fuel. Without sustained fusion to counteract gravity, the star collapses until electron degeneracy pressure halts the implosion. This quantum mechanical pressure arises from the Pauli exclusion principle, which forbids electrons from occupying the same quantum state. As electrons are squeezed into a smaller volume, they must occupy higher momentum states, creating a pressure that resists further compression even at near-zero temperature. Subrahmanyan Chandrasekhar, working in the early 1930s, combined relativistic quantum mechanics with stellar structure equations to determine the maximum mass that electron degeneracy alone can support. Above this threshold, now called the Chandrasekhar limit, gravity overwhelms electron degeneracy and the star must continue collapsing into a neutron star or black hole.

Chandrasekhar modeled a white dwarf as a sphere of electrons and nuclei in hydrostatic equilibrium. In the nonrelativistic regime, the pressure scales with density to the five-thirds power, providing ample support. However, at very high densities the electrons become relativistic, and their pressure scales only as the four-thirds power of density. This softer equation of state cannot withstand gravity beyond a finite mass. Solving the Lane–Emden equation for a relativistic Fermi gas yields a characteristic mass:

_Ch = 5.83 μ_e2 M_☉

Here _e is the mean molecular weight per electron, essentially the average number of nucleons per free electron. The constant 5.83 arises from fundamental constants such as the Planck constant, the speed of light, and the gravitational constant after the equations are non-dimensionalized and expressed in solar masses _☉. For a typical carbon–oxygen white dwarf with _e=2, the limit becomes roughly 1.44 solar masses. If accretion from a companion star pushes the mass toward this value, any additional matter can trigger catastrophic collapse.

Astrophysical Significance

The Chandrasekhar limit underpins our understanding of several cosmic phenomena. Many Type Ia supernovae occur when a white dwarf in a binary system accretes material from its companion and approaches this mass limit. Once exceeded, the star undergoes runaway thermonuclear fusion and explodes, leaving no remnant behind. Because the explosion happens near a fixed mass, Type Ia supernovae have nearly uniform peak luminosities, enabling astronomers to use them as standard candles for measuring cosmic distances. Observations of these supernovae led to the discovery of the accelerating expansion of the universe.

The limit also constrains models of stellar evolution. Stars born with more than about eight solar masses will never become stable white dwarfs because their cores exceed the Chandrasekhar mass before fusion ceases. Instead they collapse into neutron stars, supported by neutron degeneracy pressure, or continue to black holes if the mass is large enough. Consequently the mass distribution of white dwarfs we observe today, peaking around 0.6 solar masses, reflects both the initial mass function of stars and the physics captured by Chandrasekhar’s theory.

Dependence on Composition

The mean molecular weight per electron _e depends on composition. For fully ionized helium, carbon, or oxygen each nucleus contributes two protons and typically two neutrons, yielding one free electron per two nucleons and _e=2. Heavier elements like magnesium or iron contain more nucleons per electron due to a higher neutron fraction, so _e rises and the mass limit falls. Hydrogen, which provides one electron per proton, would have _e=1, implying a very large limit, but hydrogen does not form white dwarfs because fusion begins before degeneracy sets in. Real white dwarfs thus have _e between about 2 and 2.2.

Given this dependence, the calculator asks for _e and the star’s actual mass. It computes the Chandrasekhar limit and evaluates stability. If the actual mass exceeds the limit, the star cannot remain a white dwarf in hydrostatic equilibrium. Accretion from a companion or mergers of two white dwarfs are mechanisms that can drive the mass beyond the limit, precipitating collapse or supernova.

Sample Limits by Composition

The table below lists representative compositions and their corresponding limits. These values assume complete ionization and ignore corrections from finite temperature, rotation, and magnetic fields, which can modify the precise limit by several percent.

Composition	μe	Limit (M☉)
Helium	2.00	1.46
Carbon/Oxygen	2.00	1.46
Neon/Magnesium	2.15	1.26
Iron	2.15	1.26

More exotic possibilities, such as white dwarfs with a core of oxygen-neon-magnesium, remain topics of active research. Observations of unusually massive white dwarfs help test theories of dense matter and can illuminate whether rotation or crystallization significantly alters the limit.

Limitations and Extensions

Chandrasekhar’s original calculation assumes a cold, non-rotating, non-magnetic white dwarf composed of fully ionized matter in equilibrium. Real stars may rotate rapidly, reducing effective gravity at the equator and allowing slightly higher masses. Strong magnetic fields change the electron energy levels, potentially modifying degeneracy pressure. General relativistic corrections become important near the limit, decreasing the mass by a few percent. Despite these refinements, the simple formula captures the essential physics and remains a cornerstone of stellar astrophysics.

Understanding the limit also informs models of accretion in binary systems and the end states of stellar evolution. When a white dwarf siphons matter from a companion, astronomers monitor the accretion rate and composition to predict whether the system might become a Type Ia supernova. Space-based observatories seek out such supernova progenitors to improve distance measurements across the universe.

In summary, the Chandrasekhar limit encapsulates the delicate balance between quantum mechanical pressure and gravity. This calculator allows students and enthusiasts to explore how composition affects the maximum mass and to test hypothetical scenarios. By entering a mean molecular weight per electron and an actual mass, one can immediately see whether a modeled white dwarf would remain stable or collapse to a more exotic state. The legacy of Chandrasekhar’s work resonates far beyond theoretical astrophysics, influencing cosmology, nucleosynthesis, and our understanding of how stars die.