Astronomy
Gravity assist
One flyby in two reference frames, and the chains missions build from them: Voyager 2 and Cassini on the catalogue's orbits, each flyby's turn, periapsis and the manoeuvres it leaves.
The model and its limits
Unpowered two-body hyperbola; the planet moves at constant velocity in the local Sun frame. Solar tides, orbital curvature, moons, atmosphere and ring impacts are omitted. This is not a mission planner. The solar escape condition uses asymptotic speed and the planet's solar distance, not the temporary close-approach speed. Recoil is a first-order diagnostic and does not move the planet. The probe is enlarged; planet size and path use one length scale. Surfaces, spin and rings are illustrative. Velocity arrows in 3D show directions only; the separate vector diagram has one km/s scale and projects onto the ecliptic. Its circle is the projection boundary for |v∞|; tilted exits can lie inside it.
Equations used in the module
Equations e = 1 + r_p v∞²/GM; δ = 2 asin(1/e) V = Vp + v∞; |Δv| = 2 v∞ sin(δ/2) Δε = Vp · (v∞out − v∞in) Δspeed = |Vout| − |Vin| ≠ |Δv| chain: v∞ = V_leg − V_planet; e = 1/sin(δ/2); r_p = GM(e − 1)/v∞²
