Nonradial oscillations of stratified neutron stars with solid crusts: Mode characterization and tidal resonances in coalescing binaries

Dynamical tides of neutron stars in the late stages of binary inspirals provide a viable probe into dense matter through gravitational waves, and potentially trigger electromagnetic precursors. We model the tidal response as a set of driven harmonic oscillators, where the natural frequencies are given by the quasinormal modes of a nonrotating neutron star. These modes are calculated in general relativity by applying linear perturbation theory to stellar models that include a solid crust and compositional stratification. For the mode spectrum, we find that the canonical interface mode associated with the crust-core boundary vanishes in stratified neutron stars and is replaced by compositional gravity modes with mixed gravity-interfacial character, driven primarily by strong buoyancy in the outer core. We also find that fluid modes such as the core gravity mode and the fundamental mode can penetrate the crust, and we establish a criterion for such penetration. Regarding the tidal interaction, we find that transfer of binding energy to oscillations is dominated by the fundamental mode despite its frequency being too high to resonate with the tidal forcing. In general, we find that lower-frequency modes induce gravitational-wave phase shifts smaller than $\sim 10^{-3},\rm rad$ for the equation of state we consider. We discover that nonresonant fundamental and crustal shear modes can trigger crust breaking already near the first gravity-mode resonance, while gravity-mode resonance concentrates strain at the base of the crust and may marginally crack it. These results suggest that both resonant and nonresonant excitations can overstress the crust and may channel energy into the magnetosphere prior to merger, potentially powering electromagnetic precursors. Our work represents an important step toward realistic modeling of dynamical tides of neutron stars in multimessenger observations.

💡 Research Summary

This paper presents a comprehensive study of dynamical tides in non‑rotating neutron stars (NSs) that possess both a solid crust and compositional stratification. The authors construct equilibrium stellar models using the unified relativistic density‑functional EOS TW99, which self‑consistently provides the pressure, energy density, and chemical potentials for cold npeµ matter in β‑equilibrium. Two adiabatic indices are defined: Γ₀, the index of the background matter, and Γ₁, the index relevant for perturbations when the composition is frozen on the oscillation timescale. The difference Γ₁ − Γ₀ generates buoyancy (the Brunt‑Väisälä frequency) and therefore determines the spectrum of composition‑g‑modes. The index varies strongly near the neutron‑drip density (∼3 × 10¹¹ g cm⁻³) and at the crust‑core transition (∼1.0 × 10¹⁴ g cm⁻³), indicating strong buoyancy in those layers.

The solid crust is modeled as a body‑centered cubic lattice. Using the standard expression for the shear modulus μ, the authors compute the relativistic shear speed vₛ = √(μ/(ε + p)). vₛ is of order 10⁻³–10⁻² c throughout the crust, decreasing sharply once free neutrons appear above the drip point. This elastic property gives rise to shear (s) modes and to an interfacial (i) mode that would exist at a sharp density jump in a fluid star.

The perturbation problem is solved in full general relativity without invoking the Cowling approximation. Polar (even‑parity) perturbations of the static spherically symmetric metric are expanded, and the coupled system of Einstein‑fluid‑elastic equations is integrated numerically. Boundary conditions enforce regularity at the centre, continuity of metric and fluid variables across the crust‑core interface, and vanishing Lagrangian pressure perturbation at the stellar surface.

The resulting quasinormal mode (QNM) spectrum is classified into four families:

1. Fundamental (f) mode – a high‑frequency (∼2 kHz) pressure‑restored acoustic mode, present in all fluid stars.
2. Shear (s) modes – low‑frequency (∼0.5–1 kHz) elastic modes confined to the solid crust.
3. Gravity (g) modes – low‑frequency (∼100–300 Hz) buoyancy‑restored modes arising from the Γ₁ − Γ₀ stratification, predominantly located in the outer core.
4. Mixed gravity‑interfacial (g‑i) modes – in stratified stars the traditional crust‑core interfacial mode disappears; instead a hybrid mode appears with characteristics of both a buoyancy‑driven g‑mode and an interface mode, strongly influenced by the strong buoyancy in the outer core.

A key result is the identification of a “crust‑penetration criterion”: fluid‑dominated modes (f and core g) can propagate through the solid crust if their radial displacement does not decay appreciably across the elastic layer. Numerical evaluation shows that for a canonical 1.4 M⊙ star the f‑mode and the core g‑mode satisfy this condition, whereas pure crustal s‑modes remain confined.

The tidal response of the star to the companion’s gravitational field is modeled as a set of driven harmonic oscillators, each characterized by its eigenfrequency ωₙ and tidal coupling coefficient Qₙ. The authors compute Qₙ for all identified modes and evaluate the energy transferred from the orbit to each mode during inspiral. Although resonant excitation (when the orbital frequency matches ωₙ) dramatically enhances energy transfer, the analysis reveals that the f‑mode dominates the total transferred energy even without exact resonance, because its coupling is orders of magnitude larger than that of low‑frequency modes.

Gravitational‑wave (GW) phase shifts ΔΦ induced by the dynamical tides are calculated. All modes produce ΔΦ ≲ 10⁻³ rad for the adopted EOS, well below the current detector sensitivity but potentially observable with next‑generation facilities. The authors note that nonlinear effects could amplify the impact of g‑mode resonances, but such effects are not included in the present linear analysis.

A novel aspect of the work is the investigation of crust breaking. The authors adopt a breaking strain ε_break ≈ 10⁻², motivated by molecular‑dynamics studies of neutron‑star crusts. They compute the induced strain for each mode as a function of orbital separation. Both non‑resonant excitation of the f‑mode and resonant excitation of s‑ and g‑i‑modes can exceed ε_break already near the first g‑mode resonance (∼100 Hz). In particular, the g‑mode resonance concentrates strain at the base of the crust, making it especially prone to localized cracking. Crust failure would release elastic energy into the magnetosphere, offering a plausible mechanism for electromagnetic precursors (e.g., short‑GRB precursor flares) observed seconds before merger.

In summary, the paper delivers a state‑of‑the‑art relativistic treatment of neutron‑star oscillations that simultaneously incorporates elastic crust physics and realistic compositional stratification. It demonstrates that (i) the traditional crust‑core interfacial mode is replaced by mixed gravity‑interfacial modes in stratified stars, (ii) high‑frequency f‑mode dominates tidal energy transfer even without resonance, (iii) all dynamical‑tide‑induced GW phase shifts are modest for the considered EOS, and (iv) both resonant and non‑resonant excitations can overstress the crust, potentially powering observable electromagnetic precursors. The work thus bridges neutron‑star asteroseismology, GW data analysis, and multimessenger astrophysics, and sets the stage for future extensions that include rotation, magnetic fields, and nonlinear mode coupling.