Neutron Star Mass–Radius Relation (2D)

Equation of State

Central density ρ_c —

Adiabatic index Γ (stiffness) 2.00

EOS constant K 100

Star Structure

Mass

— M☉

Radius

— km

Compactness 2GM/Rc²

—

ρ_c / ρ_nuclear

—

Mass–radius curve for this Γ, K (dot = current star, ring = max-mass turnover)

How it works

A static, spherically symmetric star in general relativity obeys the Tolman–Oppenheimer–Volkoff equation instead of Newtonian hydrostatic balance:

dP/dr = -G(ρ+P/c²)(m+4πr³P/c²)
        ───────────────────────────
              r(r - 2Gm/c²)

dm/dr = 4πr² ρ

Matter is closed with a polytropic equation of state P = K ρ^Γ (a simplified stand-in for the true nuclear equation of state). Starting from a chosen central density ρ_c, the two equations are integrated outward with 4th-order Runge–Kutta until the pressure reaches zero — that radius and enclosed mass are the star's R and M. This 2D view integrates the identical TOV system as the 3D version and draws the star as a density cross-section instead of a rendered sphere.

ρ_c slider — denser core → more mass, but only up to a point.
Γ, K sliders — a stiffer equation of state (higher Γ) resists compression more, supporting a larger maximum mass at a larger radius.
Sweep ρ_c — animates the central density from low to high so you can watch the cross-section, mass and radius change live as the star climbs the M–R curve toward, and past, its turnover.
The M–R curve is traced by repeating the integration for many central densities. It always turns over: past the peak, dM/dρ_c < 0 and the configuration is dynamically unstable and would collapse — this peak is the TOV maximum-mass limit, the neutron-star analogue of the Chandrasekhar limit. Real neutron stars are believed to top out around 2–2.3 M☉, which is exactly why heavier merger remnants promptly collapse to black holes.
Compactness 2GM/(Rc²) compares the star's radius to its Schwarzschild radius; a neutron star typically reaches 30–50% of that ratio, far closer to a black hole than any ordinary star.

Density shells (bright = dense core) Schwarzschild radius (2GM/c²)

Past the maximum-mass point: this configuration is unstable and would collapse.