The tidal response of a relativistic star

Abstract

We develop a fully relativistic approach for determining the frequency-dependent tidal response of a compact star. The strategy involves matching the solution for the linearised fluid dynamics in the star's interior to the spacetime perturbations in the near-zone surrounding the body, along with an identification of the tidal driving and the star's response. Notably, this identification is exact in Newtonian gravity and we provide strong evidence that it remains robust also in the relativistic case. The argument does not involve a sum over the star's quasinormal modes and hence circumvents one of the obstacles that have held up the development of models for relativistic tides. Numerical results are provided, at the proof-of-principle level, for a realistic matter equation of state from the BSk family, including composition stratification leading to the presence of low-frequency gravity modes. We also sketch the connection with the field-theory inspired approach to the problem, in which the tidal response is expressed in terms of asymptotic scattering amplitudes.

Figures (3)

• Figure 1: A schematic illustration of the tidal problem. A star with mass $M$ and radius $R$ is tidally deformed by a binary companion with mass $M'$. The orbital separation is $D$. Our focus is on the star's response to the tidal interaction, a problem that is naturally explored in a body centered coordinate system associated with the star 2013PhRvD..88b4046V2024NatAs...8.1277R and the weak-field near zone in the vicinity of the star. In addition, the orbital problem involves the near zone that envelops the two bodies and the emitted gravitational radiation, a problem that involves the wave zone.
• Figure 2: A comparison of the effective Love number obtained from (i) the standard mode sum \ref{['modesum1']} (orange+dashed) and (ii) the proposed matching approach \ref{['eq:Love-alt']} (blue+solid). The results, which correspond to a compressible model with $\Gamma=2$ and the $\Gamma_1=2.05$ case from 2021MNRAS.504.1273P, are in perfect agreement. The singularities associated with the first two gravity modes are particularly prominent in the tidal response. It is also worth noting that the effective Love number approaches the static result, $k_2\approx 0.2599$, in the low-frequency limit.
• Figure 3: Top panel: The effective Love number $k_2(\omega)$ obtained from the matching relation \ref{['version1']} for a $1.4M_\odot$ neutron star and the BSk22 equation of state (solid blue curve). The static result, $k_2\approx0.0949$ obtained following the standard calculation from 2008ApJ...677.1216H, is indicated (horizontal dashed line) as are the first two core g-modes and a crustal g-mode (vertical dashed lines). Also shown (as a dashed orange curve) is the result obtained from the inferred mode sum. Bottom panel: The relative (percentage) error, $\Delta$, in the inferred mode sum compared to the result for $k_2$ obtained from matching at the star's surface. With a typical error below the 1% level, it is evident that the analysis leads to robust results for the $\mathcal{A}_n$ amplitudes. The result also shows that, for all practical purposes one can safely replace the matching calculation with a sum over the modes. This agrees with the intuition gleaned from the Newtonian case and neatly circumvents the non-Hermitian nature of the relativistic problem.