Chandrasekhar Limit

What is modelled

A white dwarf is held up not by heat but by the pressure of degenerate electrons, which comes from the Pauli exclusion principle. The star is in hydrostatic balance, so a heavier star squeezes its electrons harder, they become relativistic, the pressure stops rising as fast with density and the star gets smaller and smaller. Above a certain mass no balance is possible.

Equations

• Degenerate electron pressure P(x) = (π me4c5 / 3h3) [ x(2x²−3)√(1+x²) + 3 asinh x ] with relativity parameter x = p_F/(m_ec).
• Mass density ρ = (8π/3)(mec/h)³ μe mu x³, μ_e = nucleon mass per electron (2 for carbon and oxygen).
• Structure dm/dr = 4π r² ρ, dP/dr = − G m ρ / r², integrated outward from the centre with RK4 until x reaches zero.
• Lane-Emden equation (1/ξ²) d/dξ (ξ² dθ/dξ) = − θn; the ultra-relativistic gas is the n = 3 polytrope and gives the limit MCh = (√(3π)/2) ξ²|θ'|1 (ℏc/G)3/2 / (μe mu)² about 1.456 solar masses for μ_e = 2 (Chandrasekhar's more careful treatment gives the same value).

Checks run in the page

• The Lane-Emden n = 3 constants ξ₁ = 6.8968 and 2.0182 from an RK4 integration.
• The limiting mass of the full structure at very high central density against the polytrope formula.
• The nonrelativistic slope d ln R / d ln M = −1/3 at low mass.
• The radius of a 1.02 solar mass star against the measured radius of Sirius B (about 5.8 thousand km).

Honest simplifications

Zero temperature, uniform composition, no Coulomb correction, no general relativity, no electron capture or pycnonuclear reactions, and no crust. The ignition density of 2 × 10⁹ g/cm³ used to trigger the explosion is a typical teaching value for carbon ignition in a Type Ia supernova, not a prediction for a given star. The explosion graphics are only a marker, not a simulation.

• Press Typical 0.6 and watch a normal white dwarf (about Earth-sized) slowly gain mass.
• Press Near the limit and watch the central density race upward until carbon ignition.
• Move the electrons-per-nucleon slider to 2.15 (iron) to see the limit drop.