  For a variety of fully relativistic polytropic neutron star models we
calculate the star's 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 star's 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.
