The tidal deformation of a neutron star in a binary inspiral driven by the emission of gravitational waves affects the orbital dynamics and produces a measurable modulation of the waves. Late in the inspiral, the external timescale associated with changes in the tidal field becomes comparable to internal timescales associated with hydrodynamical processes within the star, and a regime of dynamical tides takes over from a prior regime of static tides. A recent analysis by Yu et al. [Mon. Not. R. Astron. Soc. 519, 4325 (2022)] reveals that nonlinear aspects of the tidal interaction are important during the regime of dynamical tides. Their theoretical framework is grounded in Newtonian gravity and fluid mechanics and relies on a representation of the tidal deformation in terms of the star’s normal modes of vibration; the nonlinearities are incorporated through a coupling of the modes and a nonlinear driving by the external tidal field. We confirm their observation in a general relativistic treatment of the tidal deformation of a neutron star, without relying on a mode representation of this deformation. The starting point of our description is a simultaneous time-derivative and nonlinear expansion of the tidal deformation, expressed in terms of three encapsulating constants, the static $k_2$, dynamic ${\ddot{k}}_2$, and nonlinear $p_2$ tidal constants. We describe the neutron star’s deformation in terms of a well-defined quadrupole moment tensor ${\mathcal{Q}}_{ab}$, which is related to the tidal quadrupole moment ${\mathcal{E}}_{ab}$ through a frequency-domain response function ${\tilde{k}}_2(\omega )$. In a pragmatic extension of our simultaneous time-derivative and nonlinear expansion, we express this in a form proportional to $(1-{\omega }^2/{\omega }_{*}^2{)}^{-1}$, the characteristic response of a harmonic oscillator subjected to a driving force of frequency $\omega$, with a natural-frequency parameter ${\omega }_{*}$ constructed from the tidal constants. We compute these for polytropic stellar models and show that the nonlinear constant $p_2$ lowers the frequency parameter by as much as 15% relative to an estimation based on a purely linear treatment of the tidal deformation. This means that the approach to resonance is triggered at an earlier frequency during an inspiral, which causes a significant enhancement of the star’s tidal response.

Close

PrevNext

View figure in article

• General relativity
• Gravitation
• Gravitational waves