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.

Neutron star TOV: toy M–R (polytrope)

Schwarzschild TOV for polytropes n = 1 or 3/2; K normalized so the ρ_c scan peaks near ~2 M☉ (schematic only).

About this model

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 minimal Tolman–Oppenheimer–Volkoff integrator in Schwarzschild geometry for a cold polytrope P = 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.

Continue with similar topics in this category — or all 51 in Astronomy & The Sky.

View category →

NewUniversity / research

Black Hole Shadow (Schematic)

Launch Simulator

Silhouette and stylized ring; Rₛ scales with mass — not full GR ray tracing.

NewUniversity / research

Chandrasekhar limit (Lane–Emden toy)

Launch Simulator

White-dwarf M–R track for n = 3/2 cold degenerate electrons; UR n = 3 marks the ~1.44 M☉ scale (μ_e dependent).

NewSchool

Stellar Life Cycle

Launch Simulator

Cloud → MS → giant/SN → WD / NS / BH vs initial mass (schematic).

NewUniversity / research

Sphere of Influence (Hill)

Launch Simulator

r_H ≈ a (m/3M)^(1/3): schematic secondary orbit and Hill radius vs masses and a.

NewUniversity / research

CMB Power Spectrum (Acoustic Peaks)

Launch Simulator

Cosmic Microwave Background temperature D_ℓ vs ℓ with Sakharov peaks: tune Ω_b h², Ω_c h², n_s, A_s, τ, h and watch the parity flip between odd / even peaks, the Silk damping tail, and the Sachs–Wolfe plateau move. Pedagogical parametric ΛCDM model.

NewUniversity / research

Big Bang Nucleosynthesis (BBN)

Launch Simulator

Light-element abundance curves H, ⁴He, D, ³He, ⁷Li vs cosmic time / temperature. Weak freeze-out, neutron decay gap, deuterium bottleneck → Y_p ≈ 0.245. Slide η₁₀ and N_eff over the classic BBN curves; observed values overlaid.

Neutron star TOV: toy M–R (polytrope)

About this model

How it works

Frequently asked questions

Related simulators

Black Hole Shadow (Schematic)

Chandrasekhar limit (Lane–Emden toy)

Stellar Life Cycle

Sphere of Influence (Hill)

CMB Power Spectrum (Acoustic Peaks)

Big Bang Nucleosynthesis (BBN)

Neutron star TOV: toy M–R (polytrope)

About this model

How it works

Frequently asked questions