Side-on 2D cross-section of the same reaction-force problem: a drill pushes into regolith, and by Newton's third law the regolith pushes back on the spacecraft. Along the anchor line that is a pure 1D radial problem, so the craft's height above the surface is integrated exactly as a 1D equation of motion.
Surface gravity: g = G·M / R², M = (4/3)π R³ ρ
Escape velocity: v_esc = √(2GM / R)
Craft (radial): F_net = F_drill − m·g − T_anchor(spring+damper)
Ejected regolith is different: once it leaves the surface its velocity generally has a sideways (tangential) component, so its trajectory is a genuine planar two-body orbit under the asteroid's central gravity, integrated as a real 2D vector equation each frame — not a radial fall.
Debris (2D): a = −GM·r̂/r², r̂ = (x,y)/|x,y|
Bound (falls back): specific energy ε = ½v² − GM/r < 0
Escapes for good: ε ≥ 0 (v ≥ v_esc at that radius)
- Radius / density — set the asteroid's mass, which sets g and vesc through the formulas above.
- Drill reaction force — the continuous reaction thrust felt by the craft while drilling, purely radial.
- Harpoon anchor — a spring-damper tether that supplies tension once it goes taut, capping the craft's drift; released, nothing but gravity opposes the drill.
- Ejected regolith — each grain gets a random launch angle, so some fall back on low ellipses, some swing wide before returning, and fast ones cross vesc and never come back — the counter for escaped grains ticks up permanently.
This is a real engineering constraint on missions such as OSIRIS-REx and Hayabusa2, which used brief contact-and-thrust maneuvers rather than sustained drilling precisely to avoid pushing the spacecraft away from a body with almost no gravity to hold it down.