Tidal Disruption and the Capture Boundary

An extended compact body produces observable debris only if tidal disruption begins outside the strong-field capture boundary and proceeds rapidly enough before plunge. The leading scale comparison is simple, but the boundary is not one universal radius: black-hole spin and inclination, mass ratio, orbital energy, stellar compactness, and material response all matter. Numerical claims must distinguish the onset of mass shedding from complete disruption and from matter remaining outside the final horizon.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon and capture scales; Black-Hole Perturbation Theory supplies extreme-mass-ratio limits; Formulations of General Relativity supplies coupled evolution; Relativistic Two-Body Dynamics supplies orbital invariants; Computational Relativity supplies convergence methods.

The invariant scale comparison

Let an extended body have mass $M_*$, areal radius $R_*$, and compactness

$$\mathcal C_*=\frac{GM_*}{c^2R_*}.$$

For a black hole of mass $M_{\rm BH}$ and mass ratio $q=M_{\rm BH}/M_*$, balance the companion’s tidal acceleration across the star with its self-gravity:

$$\frac{GM_{\rm BH}R_*}{r_t^3} \sim\frac{GM_*}{R_*^2}.$$

This gives

$$r_t\sim R_*q^{1/3}, \qquad \frac{r_t}{GM_{\rm BH}/c^2} \sim\frac{1}{\mathcal C_*}q^{-2/3}.$$

Disruption outside capture is favored by smaller $q$, smaller compactness, and prograde black-hole spin, which can move the relevant last-stable or capture scale inward. The coefficient in $r_t$ depends on stellar structure and relativistic dynamics; the formula is a regime diagnostic, not a precision threshold.

For a quasi-circular black-hole–neutron-star system, compare the mass-shedding frequency with the innermost stable orbit and plunge. Kyutoku, Shibata, and Taniguchi develop this comparison and its dependence on mass ratio, equation of state, and spin in Kyutoku, Shibata, and Taniguchi 2021, §§2.4.1–2.4.3 and 3.3.1–3.3.4. Hyperbolic or parabolic encounters instead require pericenter and orbital-energy dependent capture thresholds.

What “disruption” must mean

Several events can occur in order:

1. a cusp forms and mass shedding begins;
2. the star loses hydrostatic coherence;
3. streams avoid immediate capture;
4. matter remains outside the apparent horizon after merger;
5. a bound disk or unbound ejecta survives at a declared time.

These are not equivalent. A simulation that shows surface distortion but leaves negligible external rest mass has not established an astrophysically significant disruption. Conversely, some stripping can occur on a scattering orbit without merger.

The equation of state, temperature, magnetic fields, and neutrinos determine the matter model through compact-object physics. Here we compare the resulting self-gravity scale with the capture geometry.

Continuum calculation and benchmark

The continuum problem is Einstein plus a declared matter system, commonly relativistic hydrodynamics with a specified equation of state. Record constraint-satisfying initial data, masses, spins, stellar compactness, orbital energy and angular momentum, atmosphere treatment, gauge, grid or spectral geometry, refinement, horizon finder, and outer boundary.

Benchmark the isolated star against its equilibrium solution and mode frequencies. Benchmark a small companion or low-tide case against geodesic or perturbative capture, and compare quasi-circular threshold trends with published converged sequences. Evolve at least three effective resolutions or an equivalent tolerance sequence.

Converged outputs should include:

$$M_{\rm out}(t), \qquad M_{\rm unbound}(t), \qquad M_{\rm disk}(t), \qquad A_H(t),\ J_H(t),$$

plus constraint norms and baryon-mass, energy, and angular-momentum residuals. State the unboundness criterion and time; Bernoulli and geodesic criteria need not agree in hot, pressure-supported flow.

The review’s numerical-method and result sections document formulation choices and systematic trends Kyutoku, Shibata, and Taniguchi 2021, §3.1 and §§3.3–3.4. No simulation was rerun for this page.

Failure modes and uncertainty

A principal failure mode is artificial disruption by a low-density numerical atmosphere or under-resolved stellar surface. Others include initial-data eccentricity, gauge-dependent coordinate separation, loss of baryon mass at refinement boundaries, horizon-finder delay, insufficient duration to classify a disk, and equation-of-state error mistaken for discretization error.

Report numerical uncertainty separately from model uncertainty. Resolution may converge $M_{\rm out}$ for one equation of state while the physical spread across admissible equations of state remains larger. A later clean rerun must reproduce equilibrium benchmarks, regenerate convergence and budget tables, and retain machine-readable horizon and matter diagnostics.

Check the scaling

For $\mathcal C_*=0.17$, compare $r_t/(GM_{\rm BH}/c^2)$ at $q=2$ and $q=8$ using the leading estimate.

Solution

The estimate is $\mathcal C_*^{-1}q^{-2/3}$. At $q=2$ it is about

$$5.88\times2^{-2/3}\simeq3.70,$$

while at $q=8$ it is $5.88/4\simeq1.47$. The larger mass ratio moves disruption deeper in black-hole length units and makes capture before disruption more likely. A precise conclusion still requires spin, inclination, orbital energy, and the material model.

References

• Kyutoku, Koutarou, Masaru Shibata, and Keisuke Taniguchi, “Coalescence of Black Hole–Neutron Star Binaries”, Living Reviews in Relativity 24, 5, 2021. Open PDF.

Further reading

• Foucart, François, “Black-Hole–Neutron-Star Mergers: Disk Mass Predictions”, Physical Review D 86, 124007, 2012. Open PDF.
• Shibata, Masaru, and Keisuke Taniguchi, “Coalescence of Black Hole–Neutron Star Binaries”, Living Reviews in Relativity 14, 6, 2011. Open PDF.