🌑 Orbital Mechanics — Gravity Assist & Lagrange Points
Explore two key concepts in spacecraft navigation: the gravitational slingshot (gravity assist) where a spacecraft steals orbital energy from a planet, and the 5 Lagrange equilibrium points of a two-body system where gravitational forces balance.
Preset orbit
Orbital elements
State (vis-viva & Kepler)
a = -GM · r / |r|^3
integrated with velocity-Verlet.
Kepler 3: T = 2π√(a^3/GM)
Vis-viva: v = √(GM(2/r - 1/a))
Angular momentum L = r × v is conserved; sweeps equal areas in equal time (Kepler 2).
Drag to orbit the camera.
How Gravity Assists Work
A gravity assist (gravitational slingshot) uses a planet's gravity and orbital motion to accelerate or decelerate a spacecraft — essentially "stealing" kinetic energy from the planet's enormous mass. In the planet's reference frame, the spacecraft's speed is unchanged but its direction rotates. In the Sun's frame, the change in direction produces a net speed gain of up to 2·v_planet. NASA used this technique to send Voyager 1 & 2 on trajectories that would otherwise be impossible with available propellant. Lagrange points L1–L3 are unstable saddle points; L4 and L5 (equilateral triangle with the two masses) are stable if the mass ratio m₂/m₁ < 0.0385 (Routh criterion).