Electron Degeneracy Pressure

Where the pressure comes from

The pressure that supports a white dwarf has almost nothing to do with the heat in it. A typical white dwarf cools for billions of years from 100 000 K toward absolute zero, but its radius barely changes. That is because its support is quantum-mechanical: a degeneracy pressure that survives at zero temperature.

The starting point is Pauli's exclusion principle. Two identical fermions cannot occupy the same single-particle quantum state — and "state" includes spin. In a confined volume V the allowed single-particle momentum states are discrete and equally spaced in momentum space, with two spin states per spatial state. Pack N electrons into V at low temperature and they fill the lowest-momentum states first, all the way up to a Fermi sphere of radius p_F in momentum space. Beyond that boundary, all states are empty; below, all states are full.

Because the lowest-momentum boxes are already taken, additional electrons must sit in higher-momentum states. Their kinetic energy is not zero, and it is not a function of temperature — it is dictated purely by the requirement that each state hold no more than one electron. When you try to compress the gas, the available volume in real space shrinks, but the number of electrons stays the same. The Fermi momentum p_F must therefore rise — there must be more states between zero momentum and p_F to fit everyone in. Higher p_F means higher kinetic energy means higher pressure. The pressure is a direct consequence of exclusion.

The equation of state

Let n_e be the number density of electrons. With two spin states per cell of phase-space volume h³, the count of momentum states with magnitude less than p_F gives

n_e = (8π / 3 h³) p_F³, so p_F = ℏ (3π² n_e)^(1/3).

For non-relativistic electrons, each occupied state contributes kinetic energy p²/2m_e. Integrating and dividing by 2/3 (the standard kinetic-pressure relation) yields the cold non-relativistic Fermi pressure

P_NR = (1/5) (3π²)^(2/3) (ℏ²/m_e) n_e^(5/3).

Replacing the electron density with the mass density ρ and the mean molecular weight per electron μ_e gives

P_NR ≈ 1.00 × 10¹³ (ρ / μ_e)^(5/3) Pa (with ρ in kg/m³).

This is the "P ∝ ρ^(5/3)" relation that supports a typical white dwarf. The slope 5/3 means a small compression produces a steeply rising pressure — a stiff equation of state — which is what makes the white dwarf a stable mechanical object.

For ultra-relativistic electrons (p_F ≫ m_e c), each state contributes kinetic energy p c instead of p²/2m. The integrals run differently and the result is

P_UR = (1/4) (3π²)^(1/3) (ℏ c) n_e^(4/3) ≈ 1.24 × 10¹⁵ (ρ / μ_e)^(4/3) Pa.

Slope 4/3 instead of 5/3 — a softer equation of state. The crossover happens around p_F ~ m_e c, which corresponds to ρ / μ_e ~ 10⁹ kg/m³ — exactly the density regime of a heavy white dwarf.

Thermal versus degenerate

An ideal classical gas has pressure P = n k T. Its pressure vanishes as T → 0. A fully degenerate Fermi gas has pressure P = (1/5)(3π²)^(2/3)(ℏ²/m_e) n^(5/3). Its pressure does not vanish as T → 0; the only way to reduce it is to spread the electrons out.

The boundary is set by E_F vs kT. Define the Fermi temperature T_F = E_F / k. When T ≪ T_F the gas is "deeply degenerate" and quantum pressure dominates. When T ≫ T_F the gas is classical. In a white dwarf interior:

• n_e ~ 3 × 10³⁶ m⁻³ at ρ = 10⁹ kg/m³, μ_e = 2.
• p_F = ℏ (3π² n_e)^(1/3) ~ 0.7 m_e c — already mildly relativistic.
• E_F ~ 0.5 MeV → T_F ~ 6 × 10⁹ K.
• Actual interior T ~ 10⁷ K → T / T_F ~ 0.002. Deeply degenerate.

That is why temperature drops out of the white dwarf mass–radius relation almost entirely. Sirius B at T_eff = 25 000 K and a cool 4 000 K white dwarf have nearly the same mass–radius point because the equation of state is the same in both.

Anatomy at a glance

Regime	p_F / m_e c	Equation of state	Polytropic index	Stable star?
Classical (kT ≫ E_F)	—	P = n k T	n = ∞ (isothermal)	Yes, with internal heat
Non-relativistic degenerate	≲ 0.1	P ∝ ρ^(5/3)	n = 3/2 (γ = 5/3)	Yes — typical white dwarf
Mildly relativistic	~ 0.5–1	Smooth crossover	n ~ 2	Yes; radius shrinks with mass
Ultra-relativistic	≫ 1	P ∝ ρ^(4/3)	n = 3 (γ = 4/3)	Only at the critical mass
Beyond Chandrasekhar	—	Insufficient support	—	No — collapse to NS or Type Ia SN
Inverse β-decay onset	~ 1 (high ρ)	Neutronisation	—	Neutron-rich, drops to NS regime

Worked example — pressure inside Sirius B

Sirius B is a famous nearby white dwarf with M ≈ 1.02 M_sun and R ≈ 0.0084 R_sun ≈ 5 850 km. A characteristic central density follows from M / (4/3 π R³):

ρ_c ≈ 3 (1.02 × 1.989 × 10³⁰ kg) / (4π × (5.85 × 10⁶ m)³) ≈ 2.4 × 10⁹ kg/m³.

Plug into the non-relativistic formula (μ_e = 2):

P_NR ≈ 1.0 × 10¹³ × (2.4 × 10⁹ / 2)^(5/3) Pa ≈ 1.0 × 10¹³ × (1.2 × 10⁹)^(5/3) Pa ≈ 1.7 × 10²² Pa.

That is roughly 1.7 × 10¹⁷ atmospheres of pressure at the centre of Sirius B — supplied not by thermal motion but by Pauli exclusion. The Fermi momentum is p_F ≈ ℏ (3π² × 7.2 × 10³⁵)^(1/3) ≈ 4 × 10⁻²² kg·m/s, giving p_F / m_e c ≈ 1.5. Sirius B sits in the mildly relativistic regime: the full Chandrasekhar interpolation would give a slightly softer pressure, but the non-relativistic estimate is already close.

Why a maximum mass exists

The intuitive argument is mechanical scaling. Gravity per particle scales as M / R. Non-relativistic Fermi pressure scales as ρ^(5/3) ~ M^(5/3) / R⁵. Setting them equal gives R ∝ M^(−1/3) — more massive white dwarfs are smaller. So far, so stable.

Now switch to ultra-relativistic. Pressure scales as ρ^(4/3) ~ M^(4/3) / R⁴, while gravity still scales as M² / R⁴. The R dependence cancels. Equilibrium reduces to a single algebraic condition on M alone:

M = M_Ch ≈ 1.4 (μ_e / 2)⁻² M_sun.

For any M < M_Ch the relativistic stiffness still exceeds what is needed; for any M > M_Ch no radius supplies enough pressure and the configuration collapses. Real white dwarfs near the limit reach densities of 10⁹ kg/m³ or higher, where general relativistic corrections and inverse β-decay (electron capture onto nuclei) further destabilise the structure. The endpoints are well known: thermonuclear runaway in C/O white dwarfs gives a Type Ia supernova; in O/Ne/Mg ones, electron capture triggers accretion-induced collapse to a neutron star.

A brief history

R. H. Fowler in 1926 applied the brand-new Fermi-Dirac statistics — published earlier that year — to a stellar interior. He showed that a degenerate electron gas could provide enormous pressure even at the low temperatures of a contracting stellar remnant. White dwarfs immediately had a physical theory. The young Subrahmanyan Chandrasekhar, on a voyage from Madras to Cambridge in 1930, extended the calculation to include special relativity, found that the equation of state softens from γ = 5/3 to γ = 4/3 as p_F approaches m_e c, and computed the critical mass at which the n = 3 polytrope ceases to support itself. Arthur Eddington at the Royal Astronomical Society meeting on 11 January 1935 publicly attacked the result, declaring there must be "a law of Nature to prevent a star from behaving in this absurd way." Chandrasekhar held his ground; observation eventually confirmed him; the 1983 Nobel Prize recognised the work.

Where else degeneracy pressure matters

• Metals at room temperature. Conduction electrons in copper have T_F ≈ 80 000 K, so they are nearly fully degenerate at 300 K. This is why metals' heat capacity is far below the classical 3R/2 prediction — only a thin shell near the Fermi surface contributes.
• Brown dwarfs. Substellar objects with M ≲ 0.08 M_sun. Their cores are partially degenerate; that is part of why all brown dwarfs above ~0.013 M_sun have similar radii.
• Earth's core. Inner-core iron is hot and dense enough that a measurable fraction of its pressure is electron-degenerate.
• Neutron-star outer crust. Down to ρ ~ 10¹¹ kg/m³ the crust is supported by electrons; below that, neutron drip and neutron degeneracy take over.
• Helium flash in red giants. A red-giant core that becomes electron-degenerate before helium ignition behaves explosively when fusion does start — pressure decoupled from temperature means runaway. The same idea underlies the Type Ia explosion of a near-Chandrasekhar white dwarf.

Common pitfalls

• Calling it electrostatic repulsion. It is not — neutral helium and unionised hydrogen do not provide degeneracy pressure. The cause is Pauli exclusion, a statistics effect, not a Coulomb effect.
• Imagining electrons "pushing each other away." Each electron is a delocalised wavefunction; there is no classical "pushing." Exclusion is a constraint on which states the gas can occupy, not on local forces.
• Confusing degeneracy with thermal pressure. Degenerate pressure is independent of temperature. Mixed-pressure regimes do exist (e.g. partially degenerate, like the surfaces of white dwarfs), but the deep interiors are essentially cold.
• Assuming P ∝ ρ^(5/3) always. It softens to P ∝ ρ^(4/3) once electrons turn relativistic — and that softening is what makes the Chandrasekhar mass finite.
• Conflating Chandrasekhar with the TOV limit. Chandrasekhar applies to electron-supported objects. The Tolman-Oppenheimer-Volkoff mass (~ 2 M_sun) is the analogue for neutron-supported objects; the physics differs (nucleon-nucleon forces, GR effects, exotic-matter equations of state).
• Thinking degeneracy "fails" gradually. Below M_Ch the equilibrium is stable. Above it there is no stable radius — the result is dynamical collapse on the free-fall time of seconds, not a quiet adjustment.