This is the real linearized planetary-defense model used to plan missions like NASA's DART: a spacecraft of mass m hits the asteroid head-on at closing speed v. Because the impact throws ejecta backward, the actual momentum delivered is larger than the impactor's own momentum by the momentum-enhancement factor β (DART's post-impact analysis of Dimorphos found β ≈ 2.2–4.9, depending on the model):
Δv = β · m · v / M
That Δv is tiny — millimeters per second — but it is applied lead-time before the asteroid would have reached Earth's orbit. Over that whole remaining travel time the small velocity change accumulates into a real, linearized miss distance:
miss distance ≈ Δv × lead_time
This is why early detection matters more than a bigger rocket: doubling the lead time doubles the miss distance for the *same* Δv, while doubling β or the impactor mass only doubles Δv itself. A deflection mission launched with years of warning can nudge an asteroid clear of Earth with a small spacecraft; the same mission with only weeks of warning cannot.
- Orbit view — the asteroid's original Earth-crossing path (red) vs. its deflected path (green) after the kinetic impact, visually exaggerated for legibility.
- Miss-distance curve — for the current impactor and β, how the resulting miss distance grows with lead time, log-scaled from 10 days to 10 years.
- DART preset — loads the real 2022 mission numbers: 580 kg impactor at 6.6 km/s into ~5.0×10⁹ kg Dimorphos, β≈3.6, and shows what a multi-year lead time on an Earth-crossing target of that size would achieve.