When a low-to-intermediate-mass star ($M le 8 M_odot$) exhausts its thermonuclear core fuel, it sheds its outer envelope to expose a glowing, ultra-dense ember: a White Dwarf. With mass comparable to the Sun packed into a volume comparable to Earth, white dwarfs are sustained not by thermal gas pressure, but by the purely quantum mechanical resistance of degenerate electrons governed by the Pauli Exclusion Principle. In 1930, nineteen-year-old Subrahmanyan Chandrasekhar proved that this quantum balance has an absolute relativistic upper ceiling: the Chandrasekhar Mass Limit.
1. Fermi-Dirac Statistics & Degeneracy Pressure
Electrons are spin-$frac{1}{2}$ fermions obeying the Pauli Exclusion Principle: no two identical electrons may occupy the same quantum state. As matter is compressed, electrons are forced to occupy successively higher energy levels up to the Fermi Energy $E_F$. In completely degenerate matter ($T to 0$ approximation), the maximum momentum is the Fermi momentum:
p_F = hbar left( 3pi^2 n_e right)^{1/3} = hbar left( frac{3pi^2 rho Y_e}{m_u} right)^{1/3}
Where $n_e$ is the electron number density, $rho$ is mass density, $Y_e = Z/A$ is the mean electron-to-nucleon ratio ($approx 0.5$ for carbon-oxygen white dwarfs), and $m_u$ is the atomic mass unit.
2. Non-Relativistic vs Ultra-Relativistic Equations of State
The degeneracy pressure exerted by electrons depends critically on whether their Fermi velocities approach the speed of light ($v sim c$):
- Non-Relativistic Regime ($p_F ll m_e c$): Kinetic energy scales as $E_k = p^2 / (2m_e)$. Integrating over momentum states yields:
P_{text{non-rel}} = frac{(3pi^2)^{2/3} hbar^2}{5 m_e} n_e^{5/3} propto rho^{5/3}
This adiabatic index $Gamma = 5/3$ yields a stable polytrope ($n = 1.5$) where star radius scales inversely with mass: $R propto M^{-1/3}$. Adding mass compresses the star into a smaller, higher-density volume.
- Ultra-Relativistic Regime ($p_F gg m_e c$): As density increases, electron velocities approach $c$, and energy scales linearly with momentum ($E_k approx p c$):
P_{text{rel}} = frac{(3pi^2)^{1/3} hbar c}{4} n_e^{4/3} propto rho^{4/3}
The adiabatic index softens from $Gamma = 5/3$ to $Gamma = 4/3$ ($n = 3$ polytrope).
3. The Instability Criterion & The Chandrasekhar Threshold
In hydrostatic equilibrium, gravitational pressure scales as:
P_{text{grav}} sim frac{G M^2}{R^4} sim G M^{2/3} rho^{4/3}
Notice that gravitational pressure and ultra-relativistic degeneracy pressure share the exact same density dependence: $P propto rho^{4/3}$. Therefore, as a white dwarf becomes more massive and relativistic, its ability to support itself against gravity becomes independent of radius. Setting $P_{text{rel}} = P_{text{grav}}$ defines a unique mass where hydrostatic equilibrium can barely be sustained:
M_{text{Ch}} = frac{omega_3^0 sqrt{3pi}}{2} left( frac{hbar c}{G} right)^{3/2} left( frac{Y_e}{m_u} right)^2 approx 1.44 M_odot
Where $omega_3^0 approx 2.018$ is the Lane-Emden boundary parameter for an $n = 3$ polytrope. If a white dwarf’s mass exceeds $1.44 M_odot$, electron degeneracy pressure fails catastrophically, triggering runaway collapse into a neutron star or thermonuclear detonation as a Type Ia Supernova.
