Relativistic excitation of compact stars

TL;DR

This work develops a fully relativistic scattering framework to study tidal excitation of nonrotating compact stars by external gravitational driving, linking the scattering coefficient's poles to stellar quasinormal modes and the residues to a star excitation factor. It derives both transient and steady tidal responses: transient mode excitation during resonances is tied to the mode residue and drives energy transfer, while steady tides relate the scattering phase to the total tidal energy and are connected to self-force results and the EOB formalism. Compared with Newtonian predictions, relativistic corrections significantly reduce transient energies and alter tidal energetics, producing potentially observable phase shifts of order ~0.5 radians in late inspiral for binary neutron stars, which has important implications for accurate GW waveform modeling. The framework enables a gauge-invariant, principled incorporation of dynamical and equilibrium tides into PN/EOB pipelines, improving predictions for matter effects in neutron-star mergers and guiding future third-generation detector analyses.

Abstract

In this work, we study the excitation of a compact star under the influence of external gravitational driving in the relativistic regime. Using a model setup in which a wave with constant frequency is injected from past null infinity and scattered by the star to future null infinity, we show that the scattering coefficient encodes rich information of the star. For example, the analytical structure of the scattering coefficient implies that the decay rate of a mode generally plays the role of ``star excitation factor'', similar to the ``black hole excitation factor'' previously defined for describing black hole mode excitations. With this star excitation factor we derive the transient mode excitation as a binary system crosses a generic mode resonance of a companion star during the inspiral stage. This application is useful because previous description of resonant mode excitation of stars still relies on the mode and driving force decomposition based on the Newtonian formalism. In addition, we show that the scattering phase is intimately related to the total energy of spacetime and matter under the driving of a steady input wave from infinity. We also derive the relevant tidal energy of a star under steady driving and compare that with the dynamic tide formula. We estimate that the difference may lead to $\mathcal{O}(0.5)$ radian phase modulation in the late stage of the binary neutron star inspiral waveform.

Figures (10)

• Figure 1: Schematic illustration of (a) Wave-scattering picture: gravitational waves are injected from past null infinity and scattered back to future null infinity by the star. (b) Duality between the wave-scattering picture and point-mass scenario.
• Figure 2: $|{\rm W}_n|$ and $\gamma_n$ as functions of stellar compactness, $\mathcal{C}:=M/R$, for f-mode. The results are calculated from a set of stellar models using the polytropic equation of state with $n=1$ and fixed stellar mass $M=1.4\,M_\odot$. The inset shows the relative difference between $|{\rm W}_n|$ and $\gamma_n$.
• Figure 3: The ratio between the amplitude-based [Eq. \ref{['eq:E22']}] and Newtonian definitions ($[\mathcal{E}_{ij}\mathcal{E}^{ij}]_{\rm Newt}=\frac{9}{4}(\frac{m_2}{m_1+m_2})^2\Omega^4$) of external tidal field as function of the GW frequency $\omega=m\Omega$ for $(\ell,m)=(2,2)$ mode. $\omega_f$ is the f-mode frequency.
• Figure 4: Top: Transient mode energy of f-mode (blue) and g$_1$-mode (orange) resonance as a function of the stellar compactness $\mathcal{C}$. Both results calculated from the relativistic formula [Eq. \ref{['eq:mode energy']}, solid] and the Newtonian formula [Eq. \ref{['eq:Newt energy']}, dashed] are plotted. The stellar models are constructed from polytropic equation of state with $n=1$. Bottom: Comparison of the transient energy between the two formula.
• Figure 5: Same as Fig. \ref{['fig:transient']}, but plotted as a function of stellar mass $M$. In this figure, the relativistic stars are modeled using a polytropic equation of state with $n=1$ and $\kappa = 86\,M_\odot^{2}$. The corresponding Newtonian stars are constructed to have the same masses and radii as the relativistic models.