Tidal Love numbers of a slowly spinning neutron star

TL;DR

This work extends slow-rotation perturbation theory to neutron stars to compute tidal Love numbers to linear order in spin, uncovering spin-tidal couplings that generate a new class of rotational Love numbers and couplings between electric and magnetic distortions. The authors formulate the perturbation equations for axisymmetric, stationary tidal fields, solve the exterior vacuum problem analytically for multipoles up to $\ell=4$, and match interior and exterior solutions to extract the rotational Love numbers. Numerical results across realistic equations of state show that spin can modify the mass quadrupole and higher multipoles by substantial fractions (e.g., $\sim$10–13% near merger for $χ\approx0.05$), and that the previously observed approximate universality of static Love numbers degrades with increasing spin, particularly in the electric-led sector. These spin-tidal effects are relevant for gravitational-wave modeling of spinning neutron-star binaries in the late inspiral and near-merger phases, motivating future work on nonaxisymmetric spin, higher-order corrections, and integration into effective one-body/PN frameworks.

Abstract

By extending our recent framework to describe the tidal deformations of a spinning compact object, we compute for the first time the tidal Love numbers of a spinning neutron star to linear order in the angular momentum. The spin of the object introduces couplings between electric and magnetic distortions and new classes of spin-induced ("rotational") tidal Love numbers emerge. We focus on stationary tidal fields, which induce axisymmetric perturbations. We present the perturbation equations for both electric-led and magnetic-led rotational Love numbers for generic multipoles and explicitly solve them for various tabulated equations of state and for a tidal field with an electric (even parity) and magnetic (odd parity) component with $\ell=2,3,4$. For a binary system close to the merger, various components of the tidal field become relevant. In this case we find that an octupolar magnetic tidal field can significantly modify the mass quadrupole moment of a neutron star. Preliminary estimates, assuming a spin parameter $χ\approx0.05$, show modifications $\gtrsim10\%$ relative to the static case, at an orbital distance of five stellar radii. Furthermore, the rotational Love numbers as functions of the moment of inertia are much more sensitive to the equation of state than in the static case, where approximate universal relations at the percent level exist. For a neutron-star binary approaching the merger, we estimate that the approximate universality of the induced mass quadrupole moment deteriorates from $1\%$ in the static case to roughly $6\%$ when $χ\approx0.05$. Our results suggest that spin-tidal couplings can introduce important corrections to the gravitational waveforms of spinning neutron-star binaries approaching the merger.

Figures (11)

• Figure 1: (color online). Scheme of the spin-tidal coupling in the slow-rotation approximation to second order in the spin for the electric-led system (top diagram) and for the magnetic-led system (bottom diagram). ${\cal P}_\ell$ and ${\cal A}_\ell$ generically denote polar (i.e., even parity, or electric) and axial (i.e., odd parity, or magnetic) perturbations with harmonic index $\ell$. The quantity enclosed in a box denotes the component of the external tidal field. Perturbations with different parity and harmonic index are coupled to the external perturbations by arrows, each arrow (taken in either direction) denoting a coupling of linear order in the spin. For example, in the top diagram ${\cal P}_\ell$ sources ${\cal P}_{\ell+2}$ deformations to second order in the spin, since two arrows are needed to go from ${\cal P}_\ell$ to ${\cal P}_{\ell+2}$. Purple horizontal arrows denote Zeeman-like couplings which are nonzero only in the nonaxisymmetric case, $m\neq0$ (cf. Paper I and the review Pani:2013pma for details.)
• Figure 2: (color online). Ratio between the (dimensionless) rotational tidal Love number $\delta\tilde{\lambda}_E^{(23)}$ and the standard electric quadrupolar tidal Love number $\tilde{\lambda}_E^{(2)}$ for the various EoSs considered in this work. This ratio is crucial for the estimate \ref{['dM1']}.
• Figure 3: (color online). Left panel: mass-radius relation for the slowly-spinning, perfect-fluid stellar models analyzed in this work. We consider various tabulated EoS covering a wide range of NS deformability (cf. Table \ref{['tab:EoS']}). Right panel: moment of inertia as a function of the stellar mass.
• Figure 4: (color online). Dimensionless electric (left panels) and magnetic (right panels) Love numbers for $\ell=2,3,4$ (from top to bottom) for various nonrotating NS models with different EoS and as a function of the stellar compactness $C=M/R$. Note that, according to our definitions, the electric Love numbers $\lambda_E^{(\ell)}$ and the magnetic Love number $\lambda_M^{(3)}$ are negative. Nonetheless, modulo a negative factor, our definitions are fully equivalent to the usual ones (cf. e.g. Ref. Binnington:2009bb). Note that $\tilde{\lambda}_E^{(2)}$, $\tilde{\lambda}_M^{(3)}$, $\tilde{\lambda}_E^{(4)}$ are associated with equatorial-symmetric tidal perturbations, whereas $\tilde{\lambda}_M^{(2)}$, $\tilde{\lambda}_E^{(3)}$, $\tilde{\lambda}_M^{(4)}$ break this symmetry.
• Figure 5: (color online). Dimensionless rotational electric (left panels) and magnetic (right panels) tidal Love numbers for a spinning NS as a function of the compactness $M/R$ and for the various tabulated EoS adopted in this work. In our notation, $\delta\tilde{\lambda}_E^{(23)}$ denotes the correction to $\tilde{\lambda}_E^{(2)}$ arising through the coupling to the magnetic octupolar ($\ell=3$) tidal field to first order in the spin. Note that $\delta\tilde{\lambda}_M^{(12)}$ would represent a tidally induced spin shift that can be reabsorbed in the definition of $\chi$ and therefore $\delta\tilde{\lambda}_M^{(12)}=0$.