Radiative Capture and Dynamical Binary Formation

An initially unbound pair becomes gravitationally bound when radiation and absorption during a close passage remove more center-of-mass energy than the positive orbital energy carried in from infinity. The capture criterion is invariant, but the commonly used weak-field cross section relies on a nearly parabolic Newtonian passage and fails near relativistic separatrices, direct collision, or strong finite-size effects.

Helpful background. The Einstein Equation fixes the spacetime, Weak Metric Perturbations and Gauge Freedom supplies radiation, Presymplectic Potentials and Currents supplies energy balance, Computational Relativity supplies strong-field benchmarks, Relativistic Two-Body Dynamics supplies encounter invariants, and Black-Hole Geometry and Mechanics supplies capture geometry.

The invariant capture inequality

Let the incoming center-of-mass energy be

$$E_{\mathrm{in}} =m_1+m_2+E_{\mathrm{orb},\infty}, \qquad E_{\mathrm{orb},\infty}\gt 0.$$

If the encounter emits energy $\Delta E_\infty$ to infinity and transfers $\Delta E_H$ from the orbit into horizons or internal body degrees of freedom, a leading nonrelativistic orbital-energy budget gives

$$E_{\mathrm{orb,out}} =E_{\mathrm{orb},\infty} -\Delta E_\infty-\Delta E_H.$$

Radiative capture occurs when

$$\Delta E_{\mathrm{tot}} \equiv\Delta E_\infty+\Delta E_H \gt E_{\mathrm{orb},\infty}.$$

Equality defines a marginally bound outgoing state. Capture by this criterion does not necessarily mean prompt merger: the pair can emerge from periapsis on a highly eccentric bound orbit.

The distinction between total and orbital energy is essential here. Absorbed energy remains in the bodies, whereas radiation to infinity leaves the system. In the incoming center-of-mass frame, the invariant mass of the remaining system is

$$W_{\mathrm{out}} =\sqrt{(E_{\mathrm{in}}-\Delta E_\infty)^2 -\lvert\mathbf P_{\mathrm{rad}}\rvert^2}.$$

Binding is assessed relative to the surviving bodies’ updated masses, $W_{\mathrm{out}}-m_{1,\mathrm{out}}-m_{2,\mathrm{out}}$, within a model that still resolves two bodies. Neglecting recoil and retaining the leading increase of their internal energies recovers the displayed orbital-energy formula. Adding absorption to the losses of total energy while also increasing those masses would count it twice. In a merger or disruption there need not be a two-body outgoing state to which this binding test applies.

Angular momentum must also be tracked,

$$J_{\mathrm{out}} =J_{\mathrm{in}}-\Delta J_{\mathrm{tot}},$$

because it determines the first bound periapsis, possible immediate plunge, and later circularization. Comparing energy alone can misclassify a trajectory that crosses the conservative plunge separatrix during the burst.

Weak-field nearly parabolic loss

For a Newtonian parabolic passage with periapsis $r_p$, the leading quadrupole energy loss is

$$\Delta E_{\mathrm{par}} =\frac{85\pi}{12\sqrt2} \frac{G^{7/2}\mu^2M^{5/2}}{r_p^{7/2}},$$

where $M=m_1+m_2$ and $\mu=m_1m_2/M$. The steep $r_p^{-7/2}$ scaling makes close passages disproportionately important.

For small asymptotic speed $v_\infty$, the incoming orbital energy is

$$E_{\mathrm{orb},\infty} \simeq\frac12\mu v_\infty^2.$$

Setting the two equal gives the largest periapsis predicted to bind at this order. With $C=85\pi/(6\sqrt2)$,

$$r_{p,\max} =GM\left(\frac{C\nu}{v_\infty^2}\right)^{2/7}, \qquad \nu=\frac{\mu}{M}.$$

Gravitational focusing relates impact parameter and periapsis:

$$b^2 \simeq r_p^2 \left( 1+\frac{2GM}{r_pv_\infty^2} \right) \simeq \frac{2GMr_p}{v_\infty^2}$$

in the focusing-dominated regime. Hence

$$b_{\max} \simeq\sqrt2\,GM(C\nu)^{1/7}v_\infty^{-9/7},$$

and the leading focusing cross section is $\sigma_{\mathrm{cap}}\simeq2\pi(GM)^2(C\nu)^{2/7}v_\infty^{-18/7}$. This is the cross section for meeting the energy-loss criterion. To count surviving binaries, exclude direct collisions, prompt plunges, and disrupted objects; those exclusions can remove the smallest-impact-parameter part.

These scalings underlie weak-field dynamical-capture estimates. The energy spectrum and parabolic limit of the Peters–Mathews treatment are derived in Berry and Gair 2010, Eqs. (25)–(30), p. 3, Open PDF.

For a scale estimate, let $\nu=1/4$ and $v_\infty=10^{-3}$ (about $300\,\mathrm{km\,s^{-1}}$). The formula gives $r_{p,\max}\simeq93.4GM$ and $b_{\max}\simeq1.37\times10^4GM$. At that periapsis the parabolic speed is about $0.146c$ and the focusing factor $2GM/(r_pv_\infty^2)$ is about $2.14\times10^4$. These values check the assumed hierarchy; they do not establish a precise relativistic error bar for the quadrupole prediction.

Checks and regime boundaries

The formula must satisfy three checks:

1. $\Delta E_{\mathrm{par}}$ has dimensions of energy;
2. it decreases monotonically with periapsis;
3. the capture boundary reduces to $\Delta E_{\mathrm{tot}}=E_{\mathrm{orb},\infty}$.

It assumes $r_p\gg GM$, orbital speeds in the emission region compatible with the quadrupole approximation, negligible spin and finite size, and an incoming orbit close enough to parabolic that its hyperbolic correction is small. It fails as $r_p$ approaches an unstable relativistic orbit, where zoom–whirl residence enhances radiation and the trajectory changes during emission.

For black holes, horizon absorption can be included with its sign and spin dependence. For material bodies, tidal excitation, shocks, mass loss, or disruption can remove orbital energy but may not create a stable compact binary. The outcome definition must distinguish bound survivors, collision, and debris-producing capture.

A strong-field calculation should fix incoming ADM energy and angular momentum, measure radiated energy and angular momentum after junk radiation, and test resolution and extraction extrapolations. If an analytic result is calibrated to the same points, those points do not constitute validation.

After capture

The first post-capture orbit is often extremely eccentric. Between bursts, a secular orbit-averaged treatment may be adequate if the orbital period is short relative to environmental perturbations and the next periapsis remains within the model’s regime. Near one-pass merger, a burst-by-burst or fully nonlinear evolution is required.

The capture inequality is local to the encounter and says nothing by itself about formation rates. A population prediction additionally needs the distribution of relative velocities, impact parameters, masses, spins, and third-body perturbations. Those astrophysical inputs are outside this dynamical derivation.

Exercises

Derive the velocity exponent of the weak-field capture cross section.

Solution

Equating $r_p^{-7/2}$ energy loss to $v_\infty^2$ gives $r_{p,\max}\propto v_\infty^{-4/7}$. Strong focusing gives $b_{\max}^2\propto r_{p,\max}v_\infty^{-2}$, hence $b_{\max}^2\propto v_\infty^{-18/7}$. Since $\sigma=\pi b_{\max}^2$, the cross section has the same exponent.

Why can a pair satisfy the energy capture inequality yet merge promptly?

Solution

Radiation also removes angular momentum. If the outgoing $(E,J)$ lies below the plunge separatrix, the system has no bound outer turning point and proceeds directly toward merger. Negative outgoing orbital energy alone only proves binding, not the existence of a long-lived eccentric orbit.

References

• Berry, Christopher P. L., and Jonathan R. Gair. “Gravitational Wave Energy Spectrum of a Parabolic Encounter.” Physical Review D 82 (2010): 107501. DOI.

Further reading

• O’Leary, Ryan M., Bence Kocsis, and Abraham Loeb. “Gravitational Waves from Scattering of Stellar-Mass Black Holes in Galactic Nuclei.” Monthly Notices of the Royal Astronomical Society 395 (2009): 2127–2146. DOI.