Compactness Relations and Hartle–Thorne Quadrupoles for Neutron-Star Spin Flip–Flops

Spin-induced quadrupole moments affect the secular spin dynamics of compact binaries. In nearly equal-mass binaries, large-amplitude quadrupole-induced spin flip–flops can occur when the body carrying the dominant spin has a dimensionless quadrupole factor near w=3. We investigate whether this value is attained by causal, radially stable neutron-star configurations described by four piecewise-polytropic equations of state (EoSs): APR4, MPA1, MS1, and SLy4. The quadrupole factor wHT is obtained by integrating the first-order frame-dragging and second-order quadrupolar Hartle–Thorne equations on nonrotating stellar backgrounds and is compared with the Urbanec compactness relation wU(C0). The direct condition wHT=3 is attained at background masses 1.727, 1.938, 2.283, and 1.739M⊙, respectively. At C0≃0.2205, where the Urbanec relation predicts wU=3, the directly calculated quadrupole factors exceed the fit by 8.0% on average, whereas their full EoS-dependent range is 4.3% of their mean. The direct crossing masses exceed the Urbanec-based estimates by 0.034–0.080M⊙. The signed residual varies nonmonotonically with compactness, distinguishing the common offset of the fit near w=3 from the smaller EoS-dependent variation at fixed compactness. Thus, all four EoSs admit causal, radially stable neutron-star configurations satisfying wHT=3. Whether these configurations undergo large-amplitude spin flip–flops depends additionally on the binary mass ratio, spin magnitudes and orientations, and secular spin dynamics.