Tilde Love Numbers: Dynamic Tidal Responses

• Tilde Love Numbers are complex, frequency-dependent generalizations of static tidal Love numbers, quantifying both amplitude and phase lag in self-gravitating bodies.
• The computation involves solving linearized equilibrium equations in the Laplace domain with frequency-dependent moduli to bridge elastic and viscous responses.
• Applications span planetary geodynamics, tidal dissipation, and astrophysical waveform modeling, linking interior properties to observable tidal signatures.

Tilde Love numbers are complex, frequency-dependent generalizations of the classical Love numbers, describing the viscoelastic and dynamical tidal response of spherically symmetric self-gravitating bodies, including planets, neutron stars, and—in extended formulations—black holes and exotic compact objects. The notation $\tilde{k}_\ell(\omega)$, %%%%1%%%% encapsulates both static and dynamic deformation and dissipation when the body is subjected to periodic external tidal forcing at angular frequency $\omega$. In planetary geodynamics and astrophysical waveform modeling, tilde Love numbers provide a unified framework for quantifying amplitude, phase lag, and rheological memory effects in tidal interactions, directly connecting physical interior models with measurable waveform signatures and orbital evolution.

1. Mathematical Definition and Physical Interpretation

Tilde Love numbers quantify the amplitude and phase of tidal responses to oscillatory forcing, extending static (real-valued) tidal Love numbers to the dynamic, viscoelastic regime. For a periodic gravitational potential of harmonic degree $\ell$, oscillating as $e^{i\omega t}$, the linearized displacement and potential perturbation fields inside the body are also periodic. The complex tilde Love numbers $\tilde{k}_\ell(\omega)$ and $\tilde{h}_\ell(\omega)$ are defined by appropriately normalized ratios of the response amplitudes at the surface:

$$\tilde{k}_\ell(\omega) = -1 - \frac{m}{a g} \, \varphi_\ell(a, \omega), \quad \tilde{h}_\ell(\omega) = \frac{m}{a} \, u_\ell(a, \omega)$$

where $m$ is reference mass, $a$ is radius, $g$ is surface gravity, and $(u_\ell, v_\ell, \varphi_\ell)$ are radial, tangential, and gravitational-potential harmonic coefficients at the surface $r=a$. These numbers carry both modulus $|\tilde{k}_\ell(\omega)|$ (response amplitude) and argument $\phi_\ell(\omega)$ (phase lag), with the response decomposed into elastic (in-phase) and viscous (out-of-phase) components. The associated quality factor is

$$Q_\ell(\omega) = \frac{1}{\sin\phi_\ell} = -\frac{|\tilde{k}_\ell(\omega)|}{\Im \tilde{k}_\ell(\omega)}$$

The phase lag $$\phi_\ell(\omega) = -\arctan [\Im \tilde{k}_\ell(\omega)/\Re \tilde{k}_\ell(\omega)]$$ encodes tidal dissipation and orbital evolution rates (Melini et al., 2023).

2. Governing Equations and Rheological Inputs

The computation of $\tilde{k}_\ell(\omega)$ requires solving the linearized equilibrium equations in the Laplace (or Fourier) domain, with the Correspondence Principle dictating the replacement of real elastic moduli by frequency-dependent complex moduli $\mu(i\omega)$ reflecting viscoelastic properties. For incompressible layered models, one solves a system of coupled first-order ODEs for all relevant fields in each layer, propagating the solutions via elastic propagator matrices $\Lambda(i\omega)$ and enforcing regularity and boundary conditions through matrices $J$, $P_1$, $P_2$, and $b$. Explicit time-domain Love numbers $L_\ell(t)$ are obtained via Post-Widder Laplace inversion or Gaver-Stehfest algorithms, with derivatives $\dot{L}_\ell(t)$ used for instantaneous geodetic modeling (Melini et al., 2023).

Major rheological models implemented in planetary codes (e.g., ALMA${}^3$) include:

• Maxwell: $$\mu(i\omega) = \mu \frac{i\omega \tau}{1 + i\omega \tau}$$, $\tau = \eta/\mu$
• Burgers: bi-viscous form with two viscosity branches
• Andrade: complex power-law modulus, incorporating transient anelasticity

Rheology controls the frequency response window, with the transition from elastic to fluid-like behavior encoded by the modulus of $\tilde{k}_2(\omega)$ and phase lag peaks.

3. Typical Values and Benchmark Results

For terrestrial planets and icy satellites, frequency-dependent tilde Love numbers are computed directly from interior profiles and rheology. Key benchmarks include:

Period $T$ (d)	$|\tilde{k}_2|$	$\phi$ (deg)
1	0.30	0.0
10	0.32	0.1
$10^3$	0.33	2.5
$10^4$	0.34	8.0
$3\times10^3$	0.335	41.9
$10^5$	0.35	2.0
$10^6$	0.36	0.5

Amplitudes and phase lags vary with core state, mantle viscosity, and transient rheology. For example, a Venus $T_5^{hot}$ model yields $k_2 \approx 0.28$–$0.30$ for $\eta_{man} = 10^{20}$–$10^{22}$ Pa·s, consistent with spacecraft constraints (Melini et al., 2023).

4. Applications: Planetary Geodynamics and Tidal Modeling

Tilde Love numbers are pivotal in:

• Glacial Isostatic Adjustment (GIA): $L_\ell(t), \dot{L}_\ell(t)$ enter sea-level, uplift-rate, and geodetic corrections. Transient rheologies such as Burgers and Andrade accelerate post-load response in high-degree modes.
• Tidal Dissipation and Orbital Evolution: $|\tilde{k}_2|/Q$ quantifies dissipation rates, feeding into the Efroimsky-Kaula formalism and driving orbital migration and heating (Melini et al., 2023).
• Planetary Interior Constraints: Spacecraft measurements of $\tilde{k}_2$, $\tilde{h}_2$ allow inversion for mantle rigidity, viscosity, and core properties (liquid/solid).
• High-degree Deformation: Studies of loading events (e.g., earthquakes, reservoir impoundment) require $L_\ell$ up to $\ell \sim 200$, for which full frequency- and time-domain tilde Love numbers are computed.

5. Theoretical Extensions in Relativistic and Exotic Contexts

In relativistic astrophysics, static (untilded) Love numbers describe the adiabatic tidal response of neutron stars and black holes. The notion of frequency-dependent, complex $\tilde{k}_\ell(\omega)$ is less developed in relativistic settings, due to the absence of internal viscoelasticity, but the connection is emerging via analogy with dynamical tides and f-mode excitation. Universal relations (I-Love-Q) utilize static Love numbers, but extensions to dynamical, tidal interaction are encoded via time- or frequency-dependent deformabilities and mode coupling (Pradhan et al., 2022).

For black holes, classical general relativity predicts zero static and dynamic tidal Love numbers—i.e., $\tilde{k}_\ell = 0$ for vacuum Kerr spacetime—reflecting the absorption and absence of stationary bulge formation in response to tidal forcing (Andrés-Carcasona et al., 1 Dec 2025, Luca et al., 2023, Katagiri et al., 2023). Observational constraints (e.g., GW250114) place upper limits on effective $\tilde{\Lambda}$, showing consistency with the Kerr prediction and ruling out models with substantial tidal deformability, including many boson star scenarios (Andrés-Carcasona et al., 1 Dec 2025).

6. Computational Methodology and Numerical Schemes

Time-domain and frequency-domain tilde Love numbers are computed via Laplace inversion of analytically continued elastic solutions, using algorithms such as Gaver-Stehfest and Post-Widder formulae. Inputs include viscosity, rigidity, density, core-mantle properties, and geometric parameters. High-precision is achieved by selecting inversion parameters (e.g., Gaver order $M$, precision $D$) to avoid numerical instability (Melini et al., 2023).

For relativistic stars, Love numbers are extracted from ODE integrations of linearized Einstein equations, matched at the surface to vacuum solutions; extension to frequency-dependent response requires modeling of dynamical tidal excitation and f-mode resonances (Pradhan et al., 2022).

7. Physical and Observational Implications

Tilde Love numbers encapsulate the full viscoelastic, rheological response of planetary bodies to time-varying tidal and loading forces, directly entering the interpretation of geodetic, tidal, and gravitational-wave signals. For planets, they underpin models of mantle flow, surface deformation, and orbital evolution. In astrophysics, they serve as proxies for internal structure and are constrained by waveform phasing in binary inspirals (neutron stars, black holes, exotic compact objects), with observational limits probing the validity of general relativity and constraining new physics.

In summary, tilde Love numbers $\tilde{k}_\ell(\omega)$ represent the frequency-dependent, complex generalization of static tidal Love numbers, vital for modeling dynamic tidal interactions in planetary and astrophysical systems. Their magnitude and phase provide a direct mapping from interior material properties and geometry to observable tidal signatures, enabling both geophysical and relativistic tests of matter and gravity (Melini et al., 2023, Andrés-Carcasona et al., 1 Dec 2025, Pradhan et al., 2022).