Tidal Love numbers of neutron stars

TL;DR

This paper addresses how the internal structure of neutron stars (NS) affects their tidal response in gravitational waves. It computes the relativistic tidal Love number $k_2$ for fully relativistic polytropic NS models by solving the Tolman-Oppenheimer-Volkoff equations and static $l=2$ perturbations, connecting the interior solution to the exterior metric via $y=R H'(R)/H(R)$ to obtain $k_2$. The results show relativistic $k_2$ values can differ from Newtonian predictions by up to about $24\%$, with a stronger dependence on the polytropic index $n$ than on compactness, and they provide a practical fitting formula and implications for GW-based NS EoS constraints, including upper bounds on the tidal deformability $\lambda$ for inspiraling binaries. These findings demonstrate how GW observations can constrain NS radii and the high-density EoS through tidal effects in the early inspiral phase.

Abstract

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.

Figures (3)

• Figure 1: The relativistic Love numbers $k_2$.
• Figure 2: The difference in percent between the relativistic dimensionless Love numbers $k_2$ and the Newtonian values $k_2^N$.
• Figure 3: The range of Love numbers for the estimated NS parameters from X-ray observations. Top to bottom sheets: EXO0748-676, $\omega$Cen, M 13, NGC 2808. For an inspiral of two $1.4M_{\odot}$ NSs at a distance of 50 Mpc, LIGO II detectors will be able to constrain $\lambda$ to $\lambda \leqslant 20.1 \times 10^{36} {\rm g}{\,}{\rm cm}^2{\rm s}^2$ with $90\%$ confidence tidal.