Central density of a neutron star is unaffected by a binary companion at linear order in mu/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 start we prove analytically that the neutron starts central density is unaffected by the companion's presence to linear order in mu/R.