Both methods change the asteroid's along-track speed before it reaches the encounter point. For a small tangential Δv applied a warning time Δt before encounter, the resulting displacement at arrival is the standard first-order planetary-defense budgeting relation:
Δs ≈ 3 · Δv · Δt
Kinetic impactor (DART-style, instantaneous): a spacecraft of mass m hits the asteroid at closing speed v; ejecta recoil multiplies the delivered momentum by an enhancement factor β (DART measured β≈3.6 on Dimorphos):
Δv = β·m·v / M_asteroid
Gravity tractor (slow-pull, continuous): a spacecraft hovers at distance d and uses its own gravity to tug the asteroid for most of the warning time (here modelled as 90% of it, reserving margin for approach/verification):
a = G·m / d², Δv = a · (0.9·Δt)
Asteroid mass is estimated from diameter assuming a typical rocky density of 2,600 kg/m³. Miss distance is compared against two real reference thresholds: Earth's radius plus atmosphere (~6,871 km, below which the pass counts as an impact) and lunar distance (~384,400 km, a commonly used "clearly safe" bar in mission planning). This linear Δv×Δt relation is a first-order budgeting tool, not a full orbit propagation — it is the same order-of-magnitude formula planetary-defense teams use to compare candidate missions before running a full numerical trajectory solve.