White Dwarf & Neutron Star Compactness Explorer
Explore the inverted mass-radius relation of degenerate remnants, surface gravity log g, and relativistic compactness.
Supported by Pauli exclusion degeneracy pressure rather than thermal gas pressure, white dwarfs and neutron stars obey an inverted mass-radius relation (R ∝ M^(−1/3) in the non-relativistic regime): adding mass makes a white dwarf shrink and become denser until relativistic softening triggers collapse at the 1.44 M_☉ Chandrasekhar limit.
Interactive Physics Calculator
Governing Equation & Worked Benchmark
Relativistic White Dwarf Mass-Radius Relation & Gravitational Redshift
R / R_☉ ≈ 0.0126 (M / M_☉)^(−1/3) √[ 1 − (M / 1.44 M_☉)^(4/3) ] ; z_grav = (1 − 2GM / Rc²)^(−1/2) − 1Notice how the relativistic factor √[1 − (M/M_Ch)^(4/3)] drives the white dwarf radius rapidly toward zero as M approaches the Chandrasekhar limit M_Ch ≈ 1.44 M_☉.
| Benchmark Step | Physical Details & Result |
|---|---|
| Scenario | Compare a typical field white dwarf (M = 0.60 M_☉) with massive Sirius B (M = 1.018 M_☉) and an ultramassive 1.35 M_☉ white dwarf. |
| Calculation | At 0.60 M_☉: R ≈ 0.0124 R_☉ (8,630 km ≈ 1.35 R_⊕), ρ ≈ 1.3 × 10⁶ g/cm³, v_grav = c z_grav ≈ 30 km/s. At 1.018 M_☉ (Sirius B): R ≈ 0.0084 R_☉ (5,840 km ≈ 0.92 R_⊕), v_grav ≈ 76 km/s. At 1.35 M_☉: R shrinks to ~0.0044 R_☉ (3,060 km, smaller than Mars!). |
| Physical Takeaway | More massive white dwarfs are physically smaller, packing over a solar mass into a Moon-to-Mars-sized sphere just below the Type Ia supernova / neutron-star collapse threshold. |
Physical Mechanism & Interpretation
Because fermion phase space limits two electrons per h³ volume cell, squeezing N electrons into a smaller radius R raises their Fermi momentum p_F ∝ n_e^(1/3) ∝ M^(1/3)/R, generating non-relativistic degeneracy pressure P ∝ ρ^(5/3).
When Fermi velocity approaches the speed of light c, pressure softens to P ∝ ρ^(4/3), creating the Chandrasekhar ceiling (~1.4 M_☉ for electron degeneracy) and the Tolman-Oppenheimer-Volkoff ceiling (~2.2–2.3 M_☉ for neutron degeneracy + strong nuclear force).
Model Assumptions & Systematic Uncertainties
Thin non-degenerate hydrogen/helium envelopes inflate warm white dwarf radii by 3–8% above the zero-temperature carbon-oxygen Hamada-Salpeter curve.
Related Concepts & Methods
Scientific Sources & Provenance
- ESA Gaia Archive — Data Release 3 (DR3 (2022-06-13)) — Gaia Collaboration, Vallenari, A., et al. 2023, A&A, 674, A1 (Gaia Data Release 3). Archive / Reference ↗
- SIMBAD Astronomical Database (CDS) (2026-10 release) — Wenger, M., et al. 2000, A&AS, 143, 9 (The SIMBAD astronomical database). Archive / Reference ↗
- Gravitational-Wave Transient Catalog (GWTC-3) (O1–O3b Cumulative Catalog) — Abbott, R., et al. (LIGO-Virgo-KAGRA) 2023, PRX, 13, 041039 (GWTC-3: Compact Binary Coalescences). Archive / Reference ↗