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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.

Figure 1. The Tolman-Oppenheimer-Volkoff equation integrated for four equations of state: an ideal free-neutron Fermi gas (the 0.71 solar-mass Oppenheimer-Volkoff limit), a stiff and a soft polytrope, and MIT-bag quark matter. Sweeping the central density traces the mass-radius diagram; only equations of state whose maximum mass exceeds the observed two-solar-mass pulsars survive. Method: RK4 integration of the relativistic hydrostatic equations; Canvas2D, deterministic.
equation of state
log central density3.5e18

WHAT TO TRY

  • Step along an equation-of-state track: every point is a star in hydrostatic balance, but past the mass peak adding matter shrinks the radius and the star is unstable to collapse.
  • Switch the equation of state: a stiffer one (steeper P versus density) supports more mass. Only curves whose peak clears the dashed two-solar-mass pulsar line survive observation.
  • Watch the interior panel: pressure and enclosed mass fall from the dense core to the surface, the structure the TOV equation balances against general-relativistic gravity.