Central Density of a Neutron Star is Unaffected by a Binary Companion at Linear Order inμ/R

Recent numerical work by Wilson et al. on binary neutron star coalescence shows a striking instability as the stars come close together: Each star's central density increases by an amount proportional to $1/R$, where $R$ is the orbital radius. This overwhelms tidal stabilization effects [which scale as ${1/R}^6$] and causes the stars to collapse before they merge. By considering the perturbation limit, where a point particle of mass $\mu$ orbits a neutron star, we prove analytically that the neutron star's central density is unaffected by the companion's presence to linear order in $\mu /R$.