Gravitational Slingshot & Gravity Assist

Planetary flyby · Heliocentric energy gain · Oberth effect · Trajectory bending

Heliocentric frame — Sun at center · Planet orbiting · Spacecraft trajectory

Planet frame — hyperbolic trajectory

Spacecraft speed over time

Mission Design

Speed before flyby:
Speed after flyby:
ΔV gain:
Deflection angle:
Hyperbolic excess:
Flyby energy gain:
ΔE = 2Vₚ·v∞·sin(δ/2)
where δ = deflection angle
δ = 2·arcsin(1/e) where e = eccentricity

Oberth effect: Burn at periapsis for maximum ΔE (v² is highest there).