Asteroid Impact Risk: B-Plane Keyhole Analysis

Orbit Determination

Observation arc 200 days

Approach speed v_∞ 8.0 km/s

Keyhole Geometry

Keyhole offset 8.0 R⊕

Keyhole radius 1.0 km

Live Readouts

LOV uncertainty σ

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Capture radius

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This-pass impact chance (Monte Carlo, N clones)

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Keyhole passage → future resonant-return risk

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How it works

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.

Earth Gravitational capture disc Resonant-return keyhole Orbit-solution clones

B-plane view. Axes independently normalised — not to true relative scale. Drag to orbit, scroll to zoom.

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