The Tolman–Oppenheimer–Volkoff (TOV) equations describe hydrostatic equilibrium of a spherically symmetric self-gravitating fluid in general relativity using the Schwarzschild metric. In the mass–radius plane, realistic neutron-star models depend sensitively on the cold dense-matter equation of state above nuclear saturation density. This toy page fixes a simple polytropic closure P = K ρ^Γ with Γ = 2 (polytropic index n = 1) or Γ = 5/3 (n = 3/2), integrates outward in radius with RK4, and normalizes K separately for each choice so that scanning central density yields a maximum mass near ~2 M☉ — a pedagogical anchor, not a nuclear-physics fit. Rotation, magnetic fields, finite temperature, crust physics, and causal EOS constraints are omitted.
Who it's for: Advanced undergraduate GR / astrophysics after the Chandrasekhar white-dwarf page; prelude to tabulated neutron-star EOS discussions.
Key terms
TOV equation
neutron star
mass–radius relation
polytrope
Schwarzschild metric
compact object
How it works
This is a minimalTolman–Oppenheimer–Volkoff integrator in Schwarzschild geometry for a coldpolytropeP = K ρ^Γ with Γ = 2 (n = 1) or Γ = 5/3 (n = 3/2). K is fixed per choice so that scanning central density ρ_c gives a maximum mass near ~2 M☉ — a teaching normalization, not a fit to nuclear physics. dm/dr = 4π r² ρ and the standard pressure gradient with (ρ + p/c²) and (m + 4π r³ p/c²) are included; no crust, no rotation, no realistic EOS tabulated from QCD. The plotted branch truncates at the mass peak along the ρ_c sequence to emphasize the stable segment before collapse.
Frequently asked questions
Why does the cyan curve stop at a peak?
Along increasing central density the stable sequence typically reaches a maximum mass; beyond that, configurations are secularly unstable toward collapse. The plot truncates at the largest mass found on the scanned ρ_c grid to emphasize the stable segment.
Is ~2 M☉ from nuclear data?
No — K is hand-tuned for each polytrope so the scan peaks near 2 M☉. Observed high-mass pulsars (~2 M☉) constrain real EOS models, which are much stiffer and structured than a single polytrope.