For a variety of fully relativistic polytropic neutron star models we calculate the stars tidal Love number k2. Most realistic equations of state for neutron stars can be approximated as a polytrope with an effective index n~0.5-1.0. The equilibrium stellar model is obtained by numerical integration of the Tolman-Oppenheimer-Volkhov equations. We calculate the linear l=2 static perturbations to the Schwarzschild spacetime following the method of Thorne and Campolattaro. Combining the perturbed Einstein equations into a single second order differential equation for the perturbation to the metric coefficient g_tt, and matching the exterior solution to the asymptotic expansion of the metric in the stars local asymptotic rest frame gives the Love number. Our results agree well with the Newtonian results in the weak field limit. The fully relativistic values differ from the Newtonian values by up to ~24%. The Love number is potentially measurable in gravitational wave signals from inspiralling binary neutron stars.
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