Compact-Star Binaries and Tidal Dynamics

Finite-size compact stars enter binary dynamics through their constraint- consistent equilibrium, static and dynamical tidal response, merger support, and matter outflows. This chapter follows those material inputs from a quasiequilibrium inspiral snapshot through postmerger spectra. It does not repeat point-particle orbital dynamics or detector-likelihood machinery; instead it states what stellar calculation supplies to those subjects and where each approximation ceases to apply.

Helpful background. Stress–Energy and Conservation supplies fluid and response budgets; the Einstein equations supply constraints and perturbations; Relativistic Two-Body Dynamics supplies orbital evolution; Relativistic Matter and Compact Objects supplies stellar structure; Gravitational Radiation and Waveforms supplies radiative observables; and Computational Relativity supplies merger simulations.

Route through finite-size binary physics

The six pages follow a controlled loss of approximation.

1. Relativistic Binary Equilibrium and Initial Data uses approximate helical symmetry, irrotational or spinning circulation, and Einstein’s constraints to build a late-inspiral snapshot.
2. Tidal Deformability and Love Response matches a regular static stellar perturbation to the exterior growing and decaying fields.
3. Dynamical Tides and Resonances adds mode frequency, damping, overlap, rotation, and controlled resonance passage.
4. Merger and Remnant Classification distinguishes prompt collapse, hypermassive and supramassive support, and long-lived stars using explicit mass limits and timescales.
5. Ejecta, Disks, and Kilonova Inputs separates dynamical ejecta, bound disk, fallback, and later neutrino, viscous, or magnetic outflows.
6. Postmerger Spectra and Dense-Matter Signatures connects high-frequency peaks and collapse time to versioned strain and simulation data with calibration and model uncertainty.

Quasiequilibrium fails as radiation reaction and tidal deformation accelerate; static response fails near stellar resonances; perturbative tides fail near contact; and short merger calculations cannot determine long disk or magnetic timescales. The chapter makes each transition explicit.

Static response and its boundaries

For the convention used here,

$$Q_{ij}=-\lambda{\cal E}_{ij},\qquad \lambda=\frac{2}{3G}k_2R^5,\qquad \Lambda=\frac{\lambda}{G^4M^5} =\frac{2}{3}k_2\left(\frac{GM}{R}\right)^{-5}.$$

The interior calculation starts from a TOV background and a regular quadrupolar perturbation, then matches at the stellar surface. Hinderer derives this material-star response and the corrected matching formula in Hinderer 2008, equations (10)–(23), pp. 1217–1220, with 2009 erratum. The response depends on EOS derivatives and needs jump conditions at density discontinuities.

This convention is not silently transferred to a black hole. Vacuum response and gauge or subtraction questions are developed in Relativistic Love Numbers and Gauge Invariance. The insertion of a stellar $\Lambda$ into orbital dynamics is developed in Adiabatic Tidal Dynamics.

At finite frequency, a mode amplitude obeys a forced oscillator equation,

$$\ddot q_a+2\gamma_a\dot q_a+\omega_a^2q_a=F_a(t).$$

Far below resonance it must recover the static response. Near $m\Omega_{\rm orb}\simeq\omega_a$, energy transfer depends on overlap, damping, rotation, and sweep rate. Eigenfunction normalization must cancel from a physical energy or phase shift.

Merger outcomes are support statements

Remnant labels compare consistently defined baryon masses with equilibrium limits. A stable remnant can settle below the cold nonrotating limit; a supramassive remnant needs uniform rotation; and a hypermassive remnant exceeds the uniform-rotation limit and needs differential rotation, thermal pressure, or other transient support. Prompt collapse requires a declared postmerger time and a resolved horizon, not only lapse or density behavior. Baiotti and Rezzolla synthesize the binary and remnant sequence in Baiotti and Rezzolla 2017, §§II and V.B, Fig. 1.

Threshold fits are calibrated relations, not universal laws. Their EOS, mass-ratio, spin, thermal, neutrino, magnetic, and resolution domains must be stated. Energy, angular momentum, and baryon budgets should separate central remnant, disk, unbound matter, fallback, radiation, and boundary losses.

The same separation prevents double counting kilonova inputs. Dynamical tidal and shock ejecta are distinct from bound disk material that becomes unbound later. Foucart compares merger-neutrino methods and their composition effects in Foucart 2023, §§4.1–4.3 and 5.2. Unbound criteria, extraction radius, weak interactions, nuclear network, geometry, and wavelength-dependent opacity remain part of the result.

Validation and evidence standards

The conventions are $(-,+,+,+)$, $u^\mu u_\mu=-1$, explicit $G$, and $c=\hbar=k_B=1$. Dimension checks use $GM\Omega$, $GM/R$, and $\Lambda$. Initial data should recover isolated TOV stars at large separation and satisfy constraint and virial checks. Static tides should recover a Newtonian low-compactness benchmark. Dynamical tides should conserve mode-plus-orbit energy and angular momentum. Merger calculations should converge budgets, horizon time, disk, and ejecta quantities.

Computational results need pinned EOS and microphysics releases, code environment, grid and tolerance, benchmark, convergence, and failure diagnostics. Frontier ejecta claims need a dated comparison of channels and tests that could distinguish them. Observational postmerger claims need event-level strain, calibration, selections, likelihood and priors, simulation training version, systematics, and a reproduction boundary. A nondetection is conditional on those choices and is not absence of emission.

Review the chapter

A star has $k_2=0.090$ and compactness $C=GM/R=0.18$. Compute its dimensionless quadrupolar deformability in this chapter’s convention.

Solution

Using $\Lambda=(2/3)k_2C^{-5}$,

$$\Lambda=\frac{2}{3}(0.090)(0.18)^{-5} \simeq3.18\times10^2.$$

Thus $\Lambda\simeq318$. The answer is dimensionless and belongs to the material-star convention stated above. It is an input to later orbital dynamics, not by itself a waveform phase or detector constraint.

References

• Baiotti, Luca, and Luciano Rezzolla. “Binary Neutron Star Mergers: A Review of Einstein’s Richest Laboratory.” Reports on Progress in Physics 80 (2017): 096901. arXiv.
• Foucart, François. “Neutrino Transport in General Relativistic Neutron Star Merger Simulations.” Living Reviews in Computational Astrophysics 9 (2023): 1. DOI.
• Hinderer, Tanja. “Tidal Love Numbers of Neutron Stars.” Astrophysical Journal 677 (2008): 1216–1220; erratum Astrophysical Journal 697 (2009): 964. DOI.