A sail bonded to (or towed alongside) the asteroid tilts its normal by a cone angle δ from the Sun line. Reflected sunlight pushes it with a force whose magnitude follows the standard flat solar-sail law:
F(δ) = (S₀ / d²) · A · (1 + r) · cos(δ) / c
radial component: F_r = F(δ)·cos(δ)
tangential component: F_t = F(δ)·sin(δ)
S₀ = 1361 W/m² (solar constant at 1 AU), d = Sun distance in AU, A = sail area, r = reflectivity (0 = black absorber, 1 = perfect mirror), c = speed of light. Only the tangential component — along or against the asteroid's orbital velocity — does useful work reshaping the orbit.
For a near-circular orbit, a small sustained tangential acceleration a_t secularly changes the semi-major axis (Gauss's planetary equation, circular-orbit limit):
da/dt = 2·a_t / n, n = √(μ☉/a³) (mean motion)
Each slice of thrust applied a time Δt before the encounter nudges the asteroid's along-track position at closest approach by roughly the planetary-defense "3× rule" used for kinetic impactors, generalised to continuous low thrust by integrating over the whole push:
miss shift ≈ ∫ 3 · a_t(t) · (L − t) dt
where L is the warning time and t runs from mission start to now. Posigrade (along-velocity) pushing raises the orbit and makes the asteroid arrive late; retrograde pushing lowers it and makes it arrive early — both grow the miss distance. This slow-push strategy needs years of warning, unlike a kinetic impactor's single instantaneous Δv, but needs no rendezvous propellant beyond attaching the sail.