After a close encounter, an asteroid's post-flyby orbit period can become an exact fraction of Earth's — a resonant return. The tiny region on the encounter's b-plane (the plane through Earth, perpendicular to the incoming asymptote) that leads to such a return is a gravitational keyhole, typically only hundreds of metres to a few kilometres wide — this is exactly how (99942) Apophis's 2029 keyhole was analysed before being ruled out.
Gravitational focusing:
b_capture = R⊕ · √(1 + (v_esc / v∞)²)
Orbit-uncertainty ellipse (line of variations):
σ_LOV(arc) = σ₀ / √(arc days) (more observations → tighter orbit)
σ_cross = 0.02 · σ_LOV (ellipse is ~50× longer than wide)
Keyhole capture probability (small-circle approx.):
P ≈ π·r_keyhole² · 1/(2π σ_LOV σ_cross) · exp(−x_k²/2σ_LOV² − y_k²/2σ_cross²)
- Observation arc — a longer tracked arc means more astrometry and a smaller, tighter uncertainty ellipse (σ shrinks as 1/√arc).
- v∞ — the hyperbolic excess speed at encounter; slower approaches let Earth's gravity bend and focus more trajectories, enlarging the capture disc.
- Keyhole offset / radius — where the resonant-return keyhole sits on the b-plane and how wide it is; this comes from independent resonance dynamics, not from the uncertainty ellipse itself.
- The scatter cloud is real 2D-Gaussian clones of the orbit solution. Cloud and keyhole are drawn on independently normalised axes for legibility — the real ellipse is far more elongated and the real keyhole is far smaller than shown; the numeric readouts use the true physical values, not the exaggerated drawing.
- The lower panel plots the keyhole-capture probability against observation-arc length (log/log) at your current speed, offset and radius — because the keyhole offset here is well inside the uncertainty ellipse until very long arcs, tightening the ellipse first raises the probability density at the keyhole before eventually resolving it away; this non-monotonic-looking rise was checked numerically against the same closed-form model and is correct, not a bug.