Neutron Stars: the TOV Equation and the Mass-Radius Diagram

A neutron-star structure playground built on the relativistic hydrostatic-equilibrium (TOV: Tolman-Oppenheimer-Volkoff) equation. The TOV system is integrated by RK4 from the centre to the surface for four equations of state: an ideal degenerate free-neutron Fermi gas (which reproduces the historic Oppenheimer-Volkoff maximum mass of $0.71 M_{\odot}$ at about 9 km), a stiff and a soft polytrope anchored at nuclear density, and self-bound MIT-bag quark matter. Sweeping the central density traces the mass-radius diagram, whose turning point is the maximum mass; only equations of state whose maximum exceeds the observed two-solar-mass pulsars (J0740, J0348) survive. Panel A is the mass-radius diagram for all four equations of state with the $2 M_{\odot}$ line; Panel B is the selected star interior (pressure, energy density, enclosed mass) with a density-shaded cross-section; Panel C is the equation of state itself on a log-log pressure-density plane. The free-neutron equation of state reproduces the Oppenheimer-Volkoff maximum of $0.71 M_{\odot}$, stiffer equations of state reach higher masses, and the mass-radius turning point marks the onset of instability.