Bound, Scattering, Capture, and Plunge Regimes

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Bound orbit, scattering, capture, and plunge are statements about global motion, not the sign of an instantaneous radial velocity. In a conservative stationary problem they can be classified by asymptotic energy, angular momentum, and the turning points of an effective radial potential. Radiation makes the boundaries dynamical: an initially unbound pair can lose enough energy to become bound, while a slowly inspiralling bound system can cross a separatrix and plunge.

Required background. The Einstein Equation fixes the gravitational dynamics, and Weak Metric Perturbations and Gauge Freedom explains why a coordinate radius is not by itself a regime classifier.

Helpful background. Presymplectic Potentials and Currents supplies conserved generators, while Black-Hole Perturbation Theory gives the fixed-background radial-potential picture used as a controlled limit.

Radial motion and invariant data

When the conservative dynamics admits a separated radial equation, as for equatorial Schwarzschild or Kerr geodesics, the radial motion can be written schematically as

\left(\frac{dr}{d\lambda}\right)^2=\mathcal R(r;E,J,\ldots),

where $\lambda$ is a convenient parameter and $\mathcal R$ depends on conserved energy $E$ , angular momentum $J$ , masses, spins, and possibly a Carter-like constant. Stationarity and axisymmetry alone do not guarantee separability or a Carter constant. Under a smooth invertible radial-coordinate transformation with nonzero derivative, the coordinate location of a root changes but its existence and multiplicity do not. Singular transformations at an endpoint require separate treatment.

For Newtonian relative motion,

\frac{\mu\dot r^2}{2}+V_{\mathrm{eff}}(r)=E_{\mathrm{NR}}, \qquad V_{\mathrm{eff}}(r)=\frac{J^2}{2\mu r^2}-\frac{GM\mu}{r}.

Negative nonrelativistic energy gives a bound ellipse, zero energy a parabola, and positive energy a hyperbola. General relativity preserves the usefulness of asymptotic energy but adds unstable circular orbits, strong-field separatrices, radiation, and horizon boundary conditions. The Newtonian sign rule is therefore a limit, not a universal definition.

For relativistic scattering in asymptotically flat spacetime, specify incoming four-momenta and the total angular momentum at past infinity. A useful invariant relative energy is

\gamma=-u_1\!\cdot u_2, \qquad s=-(p_1+p_2)^2 =m_1^2+m_2^2+2m_1m_2\gamma.

The outcome is determined by the complete solution at fixed $(s,J)$ , not by an impact parameter alone; $b$ is related to $J$ through the asymptotic momentum and depends on the chosen center convention.

Bound motion and separatrices

A conservative bound orbit oscillates between two radial turning points while remaining in a compact radial interval. Stable circular motion occurs at a double root,

\mathcal R(r_c)=0, \qquad \partial_r\mathcal R(r_c)=0,

with the stability sign determined by the second derivative in the chosen radial equation. An unstable circular orbit also gives a double root but separates qualitatively different motions.

In Schwarzschild geodesic motion, using inverse dimensionless radius $u=GM/r$ and dimensionless angular momentum $j=L/(\mu GM)$ , the radial equation can be expressed through

V_{\mathrm{eff}}(u)=(1-2u)(1+j^2u^2).

The innermost stable circular orbit lies at $r=6GM$ ; unstable circular orbits form the strong-field boundary relevant to zoom–whirl motion. For eccentric bound orbits the separatrix is a curve in energy–angular-momentum space, not a single radius. These geodesic statements become the $\nu\to0$ check on two-body models.

A dissipative inspiral is only approximately bound. On orbital times it tracks a sequence of conservative tori; on the radiation-reaction time its energy and angular momentum drift. “Bound” then means that no escape channel is reached during the modeled interval and that an adiabatic description remains valid. Near the separatrix the radial period grows, the timescale hierarchy deteriorates, and a transition-to-plunge treatment is needed.

Scattering and conservative capture

A scattering solution arrives from past infinity and returns to future infinity. Its invariant conservative signature is an outgoing asymptotic state and a finite scattering angle

\chi(E,J)=\Delta\phi-\pi.

Capture means that no outgoing asymptotic branch exists for the same conservative data. In a black-hole background, an infalling trajectory crosses the future horizon; in an effective two-body model, the absence of an outer turning point indicates entry into the plunge region, but the subsequent merger is beyond a conservative radial potential.

At the conservative threshold, two radial roots merge:

\mathcal R(r_\ast;E,J_\ast)=0, \qquad \partial_r\mathcal R(r_\ast;E,J_\ast)=0.

The critical angular momentum $J_\ast(E)$ is invariant when defined from asymptotic charges, while $r_\ast$ is representation dependent. Close to the threshold, trajectories spend a long time near the unstable orbit and the scattering angle develops logarithmic growth. The structure of relativistic scattering and its use in PM/EOB dynamics is reviewed in Damour 2020, §§II–III and VI.

This conservative classification must be separated from absorption or radiation. A trajectory can cross the conservative capture threshold even in a time-symmetric model. Conversely, an orbit that would scatter conservatively can radiate enough energy and angular momentum during periapsis passage to become bound.

Radiative capture and plunge

Let incoming asymptotic energy above the two-body rest energy be

E_\infty=\sqrt{s}-M\gt 0.

If a close encounter radiates energy $\Delta E_{\mathrm{rad}}\gt 0$ , the leading capture condition is

E_\infty-\Delta E_{\mathrm{rad}}\lt 0,

with angular-momentum loss included consistently when determining the final orbit. This inequality is necessary for radiative binding but not sufficient for a long-lived binary: the post-encounter state can immediately plunge or collide, and finite-size disruption may invalidate a point-particle model.

For a bound inspiral, plunge begins when the evolving orbit can no longer follow stable conservative motion. It is not defined by a universal coordinate speed or by crossing the Schwarzschild ISCO radius in every problem. Spin, eccentricity, mass ratio, and the chosen effective description shift the separatrix. A horizon crossing is a later geometric event, while waveform merger and peak amplitude are still different markers.

The hierarchy is therefore:

Regime	Asymptotic or turning-point criterion	Essential extra data
Bound	confined radial motion in conservative limit	stability and radiation timescale
Scattering	incoming and outgoing asymptotic states	$E$ , $J$ , spins, radiated momentum
Conservative capture	incoming state, no outgoing outer branch	horizon or absorbing inner condition
Radiative capture	outgoing energy becomes bound after losses	$\Delta E$ , $\Delta J$ , finite-size outcome
Plunge	stable-orbit tracking fails past a separatrix	transition model and inner boundary

The comparison axes are the existence of asymptotic branches, the number and stability of radial turning points, and the role of dissipative losses. Bound and scattering motion retain a future orbital branch, whereas conservative capture and plunge do not; radiative capture differs from conservative capture because the classification changes only after energy and angular momentum have been emitted.

Checks and limitations

Three checks prevent common misclassification. First, the weak-field limit must recover the conic-section energy rule. Second, the test-mass limit must recover the background-geodesic separatrix and capture threshold. Third, time reversal must exchange incoming and outgoing conservative scattering, while a retarded radiative solution need not be time symmetric.

An effective radial potential is not always available. Generic precessing, resonant, or chaotic motion can require a higher-dimensional phase-space description. In numerical relativity, finite-radius trajectories and coordinate puncture positions are not asymptotic observables; capture and scattering should be inferred from horizons, asymptotic radiation, and extrapolated incoming/outgoing charges.

Exercises

For Newtonian relative motion, find the radius and value of the minimum of $V_{\mathrm{eff}}$ at fixed nonzero $J$ . What does it represent?

Solution

Setting $\partial_rV_{\mathrm{eff}}=0$ gives

r_c=\frac{J^2}{GM\mu^2}.

Substitution yields

V_{\mathrm{eff}}(r_c) =-\frac{G^2M^2\mu^3}{2J^2}\lt 0.

The second derivative is positive, so this is a stable circular orbit. Energies between this minimum and zero give bound eccentric motion; positive energy gives unbound motion.

Why is the coordinate radius $r_\ast$ of a relativistic capture threshold less useful for comparing two calculations than $J_\ast(E)$ ?