Dynamical Tides and Resonances

An adiabatic tide assumes that a star’s quadrupole follows the companion’s field instantaneously. That approximation fails when an orbital driving harmonic approaches an internal mode frequency or when the forcing changes on the mode’s response time. A resonance can transfer energy and angular momentum from the orbit into an f-, g-, inertial-, or interface mode and thereby change the inspiral phase. The size is controlled by mode overlap, damping, rotation, and how quickly the binary sweeps through resonance.

Helpful background. Stress–Energy and Conservation supplies mode energy; the Einstein equations supply stellar perturbations; Relativistic Two-Body Dynamics supplies orbital evolution; Relativistic Matter and Compact Objects supplies the mode spectrum; Gravitational Radiation and Waveforms supplies the inspiral flux; and Computational Relativity supplies direct resonance checks.

A forced stellar mode

Expand a linear Lagrangian displacement in normalized stellar modes,

$$\boldsymbol{\xi}(\boldsymbol{x},t) =\sum_a q_a(t)\boldsymbol{\xi}_a(\boldsymbol{x}).$$

For a complex harmonic basis, include conjugate partners so that the physical displacement is real.

For a nonrotating Newtonian star, a useful explicit normalization is $\int\rho\,\boldsymbol\xi_a^*\cdot\boldsymbol\xi_b\,d^3x=MR^2\delta_{ab}$, so $\boldsymbol\xi_a$ has dimensions of length and $q_a$ is dimensionless. For one weakly damped mode the projected fluid equation is

$$\ddot q_a+2\gamma_a\dot q_a+\omega_a^2q_a =F_a(t),$$ $$F_a(t)=W_{\ell m}Q_{a\ell m}\frac{GM'R^{\ell-2}}{D^{\ell+1}} e^{-im\Phi(t)} ,$$

Here $W_{\ell m}$ is the dimensionless geometric coefficient in the harmonic expansion of the companion potential, $M'$ is companion mass, $D$ separation, and $\dot\Phi=\Omega_{\rm orb}$. With unit-normalized spherical harmonics, the dimensionless overlap is

$$Q_{a\ell m}=\frac{1}{MR^\ell} \int\rho\,\boldsymbol\xi_a^*\mathbin{\cdot} \boldsymbol\nabla(r^\ell Y_{\ell m})\,d^3x.$$

The extra factor $R^{\ell-2}$ in the forcing is required by this normalization: $F_a$ has units of inverse time squared for every $\ell$. Projecting $-\boldsymbol\nabla\Phi_{\rm tide}$ on the eigenfunction yields the overlap integral, so a symmetry-orthogonal mode is not excited. Lai 1994, §2, Eqs. (2.8)–(2.11) uses an equivalent unit-normalized mode basis; amplitudes and overlaps must be translated together. A relativistic or rotating mode problem requires its own inner product and coupling coefficients rather than blindly retaining this Newtonian normalization.

The inputs are a stellar equilibrium and mode spectrum, eigenfunction normalization, damping, binary masses and spins, tidal multipoles, and an orbital-frequency evolution. Frequencies have dimension $L^{-1}$; a mode overlap cannot carry hidden dimensions that depend on an arbitrary eigenfunction rescaling.

Static response and resonance

Far below resonance and for slow forcing,

$$q_a\simeq\frac{F_a}{\omega_a^2}.$$

Summing the relevant modes recovers the equilibrium tidal deformability. This is the principal low-frequency check: a dynamical response function must tend to the static Love response without double counting an additional equilibrium quadrupole.

For a monochromatic drive $F_a=F_0e^{-i\sigma t}$, the steady response is

$$q_a(t)=\frac{F_0e^{-i\sigma t}} {\omega_a^2-\sigma^2-2i\gamma_a\sigma}.$$

The sign of the damping term follows by differentiating the phase twice. For $\gamma_a\gt0$ the response poles lie below the real frequency axis, as a stable retarded response must. This formula explains both the static limit and the large amplitude near resonance; a chirping orbit does not remain in this steady state indefinitely.

For the positive-frequency, $m\gt0$ driving component of a circular orbit around a nonrotating star, resonance occurs when

$$m\Omega_{\rm orb}\simeq\omega_a .$$

Rotation splits the spectrum. With perturbations $e^{-i\omega t+im\phi}$ and stellar spin $\Omega_s$, $\omega_{\rm rot}=\omega-m\Omega_s$; the forcing and mode frequency must be compared in the same frame. A sign error can turn a retrograde rotating-frame mode into the wrong inertial resonance. Flanagan and Racine analyze the gravitomagnetic excitation of rotational modes and its phase effect in Flanagan and Racine 2007, §§II.A–II.B and V.B–V.E.

Near a narrow resonance, write the detuning $\delta(t)=\omega_a-m\Omega_{\rm orb}(t)$. Linearizing about the crossing gives $\delta\simeq-m\dot\Omega_{\rm orb}(t-t_{\rm res})$. The coherent interaction time scales as

$$t_{\rm res}\sim(m\dot\Omega_{\rm orb})^{-1/2}.$$

Slow sweep and large overlap increase the transferred energy; strong damping or a rapid sweep suppresses it. A direct integration should agree with the narrow-resonance asymptotic result when $\gamma_a t_{\rm res}\ll1$ and the orbital parameters change little across the layer.

Energy transfer and orbital phase

For a conservative free mode with real inertial frequency $\omega_a$ and $m\ne0$, canonical energy and angular momentum obey the pattern-speed relation

$$E_{c,a}=\frac{\omega_a}{m}J_{c,a}.$$

The rotating-frame canonical energy is $E_{c,a}^{\rm rot}=E_{c,a}-\Omega_s J_{c,a}$, as in the CFS instability. In a nonrotating star, exciting a positive-energy mode removes orbital energy and advances the inspiral: the orbit reaches a given frequency sooner.

For a rotating star, the total stellar energy change also involves its spin; one must not equate a positive rotating-frame mode energy with energy lost by the orbit. Excitation of a negative-canonical-energy branch can extract spin energy and return energy to the orbit, delaying the inspiral. The relevant time shift scales as the total stellar energy change divided by the orbital radiation-loss rate near resonance. Obtaining the phase shift also requires matching the orbital phase through the resonance. These distinctions are worked out in Flanagan and Racine 2007, Appendix B, Eqs. (B7)–(B11).

The overlap can be as decisive as the frequency. A mode at the right frequency but with a symmetry-forbidden or tiny tidal overlap produces negligible phase change. Low-frequency g- and interface modes may resonate earlier but can have small coupling; f-modes couple strongly but are encountered late, where adiabatic evolution and clean separation from merger become less accurate. Rotation permits inertial and r-mode resonances and changes selection rules.

Hinderer and collaborators show how frequency-dependent quadrupolar response can be organized relative to the adiabatic limit in Hinderer et al. 2016, Eqs. (1)–(2). The subsequent placement of that response into full binary dynamics belongs in Adiabatic Tidal Dynamics; detector likelihood and parameter estimation are separate from establishing the stellar response.

Limits of the oscillator description

The linear model assumes that the background star and mode basis vary slowly outside a controlled resonance layer. It can fail when the mode reaches nonlinear amplitude, couples to daughter modes, breaks waves, heats the star, or triggers mass transfer; strong magnetic fields and differential rotation can also replace isolated eigenfrequencies with continua or mixed modes. Close to contact, the tidal field is not a small perturbation and a direct relativistic evolution is needed.

Independent checks include invariance under eigenfunction rescaling, recovery of the static Love response, energy–angular-momentum balance, agreement between asymptotic and direct integration through a narrow resonance, and disappearance of the effect as $Q_{a\ell m}\to0$. A claimed measurable phase shift must also state the waveform baseline and avoid counting the same equilibrium tide in both static and dynamical pieces.

Check the resonance frequency

A nonrotating star has an $\ell=m=2$ mode with $f_a=\omega_a/(2\pi)=1.8\,{\rm kHz}$. At what orbital and dominant quadrupolar gravitational-wave frequencies does resonance occur?

Solution

For $m=2$,

$$2\Omega_{\rm orb}=\omega_a .$$

Thus $f_{\rm orb}=\Omega_{\rm orb}/(2\pi)=f_a/2=0.9\,{\rm kHz}$. The dominant quadrupolar gravitational-wave frequency is $f_{\rm GW}=2f_{\rm orb}=1.8\,{\rm kHz}$, equal to the mode frequency. Rotation would shift the inertial-frame mode frequency, and the actual relevance of this late crossing depends on overlap, damping, contact frequency, and the validity of the linear inspiral model.

References

• Flanagan, Éanna É., and Étienne Racine. “Gravitomagnetic Resonant Excitation of Rossby Modes in Coalescing Neutron Star Binaries.” Physical Review D 75 (2007): 044001. DOI.
• Hinderer, Tanja, et al. “Effects of Neutron-Star Dynamic Tides on Gravitational Waveforms within the Effective-One-Body Approach.” Physical Review Letters 116 (2016): 181101. DOI.
• Lai, Dong. “Resonant Oscillations and Tidal Heating in Coalescing Binary Neutron Stars.” Monthly Notices of the Royal Astronomical Society 270 (1994): 611–629. DOI.