The 3D sim uses a linear shortcut: it computes Δv = β·m·v/M and then just multiplies shift ≈ Δv × T_lead, treating the drift as a straight line. This 2D companion never uses that shortcut — it solves the actual two-body Kepler problem. The impactor's Δv is applied as a tangential kick to the asteroid's velocity exactly at perihelion (the one point where the closed-form periapsis geometry is exact), which changes its orbital energy and angular momentum:
v₁ = v_peri + Δv (tangential kick at perihelion)
ε₁ = v₁²/2 − GM/r_peri → a₁ = −GM/(2ε₁) (vis-viva → new semi-major axis)
h₁ = r_peri·v₁ → e₁ = √(1 − h₁²/(GM·a₁)) (angular momentum → new eccentricity)
Both the original orbit (a₀,e₀) and the deflected orbit (a₁,e₁) are then propagated independently by solving Kepler's equation M = E − e·sinE for eccentric anomaly E at the chosen lead time, giving exact closed-form positions — the same physics that plans real planetary-defense missions, not a straight-line guess. The reported "orbit shift" is the literal Cartesian distance between the two propagated positions, in the same orbital plane.
Verified numerically (standalone Node script, not shown in-browser): the closed-form Kepler propagator was checked against an independent RK4 numerical integration of Newton's law d²r/dt² = −GM·r/|r|³ — both the unperturbed and the post-kick orbit track the RK4 ground truth to better than 1 part in 10¹³, and the analytic Gauss variational formula δa = 2a²v·Δv/GM for a tangential kick matches the closed-form Δa to 1 part in 10⁶. For the DART-default scenario (570 kg at 6.6 km/s, β=3.6, 2 yr lead) this real-orbit propagation gives a shift of ≈519 km — about 3.2× larger than the 3D sim's linear Δv×T estimate of ≈164 km for the identical Δv, because the 2-year lead time is longer than this orbit's own 1.57-year period, so the deflected orbit's changed period compounds the drift on each pass instead of accumulating it in a straight line. Neither number is "wrong" — they are different-fidelity models of the same nudge, and this is exactly the kind of gap real mission planners use full orbit propagation (not linear estimates) to catch.
- Impactor mass / velocity / β — same DART-derived formula as the 3D sim: Δv = β·(m·v)/M.
- Warning time — how long (in years) the deflected orbit is propagated past the impact before comparing it to the undeflected orbit.
- Orbit shift — the true Cartesian separation between the two Kepler-propagated positions, not a linear approximation.