Moments of inertia for neutron and strange stars: limits derived for the Crab pulsar

Recent estimates of the properties of the Crab nebula are used to derive constraints on the moment of inertia, mass and radius of the pulsar. To this purpose, we employ an approximate formula combining these three parameters. Our “empirical formula” I ≃ a(x)MR, where x = (M/M⊙)(km/R), is based on numerical results obtained for thirty theoretical equations of state of dense matter. The functions a(x) for neutron stars and strange stars are qualitatively different. For neutron stars aNS(x) = x/(0.1 + 2x) for x ≤ 0.1 (valid for M > 0.2 M⊙) and aNS(x) = 2 9 (1 + 5x) for x > 0.1. For strange stars aSS(x) = 2 5 (1 + x) (not valid for strange stars with crust and M < 0.1 M⊙). We obtain also an approximate expression for the maximum moment of inertia Imax,45 ≃ (−0.37 + 7.12 · xmax)(Mmax/M⊙)(RMmax/10 km), where I45 = I/10 g · cm, valid for both neutron stars and strange stars. Applying our formulae to the evaluated values of ICrab, we derive constraints on the mass and radius of the pulsar. A very conservative evaluation of the expanding nebula mass, Mneb = 2 M⊙, yields MCrab > 1.2 M⊙ and RCrab = 10− 14 km. Setting the most recent evaluation (“central value”) Mneb = 4.6 M⊙ rules out most of the existing equations of state, leaving only the stiffest ones: MCrab > 1.9 M⊙, RCrab = 14− 15 km.