Gravity Assists & Lagrange Points: Free Speed From Orbital Mechanics

Gravity itself is the whole story — again

Every trick in orbital mechanics is still just F = G·m₁m₂/r² playing out over time. An orbit is a continuous sideways free-fall: a spacecraft moving fast enough that the ground curves away beneath it exactly as fast as it falls, so it never lands. What changes between a stable circular orbit, a slingshot manoeuvre and a Lagrange point is nothing about the underlying law — only the geometry and relative motion of the bodies involved.

The gravity-assist slingshot: free speed from a moving planet

A gravity assist lets a spacecraft change speed and direction using a planet's gravity instead of its own fuel. The trick lies in reference frames: in the planet's own frame, the spacecraft enters and leaves the encounter at exactly the same speed — only its direction has been bent. But the planet itself is racing around the Sun, and that motion doesn't cancel out. If the flyby geometry is arranged so the spacecraft passes behind the planet's direction of travel, the planet's orbital velocity adds onto the spacecraft's speed in the Sun's frame — a true, permanent gain, paid for by an immeasurably tiny loss in the planet's own enormous orbital momentum.

This is exactly why interplanetary missions plan multi-planet flyby chains rather than flying straight to their destination — Voyager 2's tour of Jupiter, Saturn, Uranus and Neptune used successive gravity assists to reach speeds no onboard engine could ever have provided with the fuel it could actually carry.

The 5 Lagrange points: parking spots that hold themselves

In any two-body system — Sun and Earth, say — there are exactly five positions where a small third object's own orbital period naturally matches the two larger bodies' period, letting it stay in a fixed position relative to both without constant course correction. These are the Lagrange points, L1 through L5:

L1, L2, L3 — on the line through both bodies, unstable saddle points
L4, L5     — 60° ahead / behind the smaller body, stable troughs
             (for large enough mass ratio between the two bodies)

L1, L2 and L3 sit directly on the line joining the two bodies and are unstable — a spacecraft parked there, like the James Webb Space Telescope at Sun-Earth L2, needs periodic small burns to stay put. L4 and L5, by contrast, form genuinely stable gravitational troughs when the mass ratio between the two bodies is large enough (as it is for the Sun and any planet), which is why Jupiter's Trojan asteroids — thousands of them — collect at its L4 and L5 points and stay there for the age of the solar system with no station-keeping at all.

Hohmann transfers: the cheapest way between two orbits

Moving a spacecraft from one circular orbit to another — Earth orbit to a higher geostationary orbit, for instance — doesn't require constant thrust. A Hohmann transfer does it with exactly two burns: the first kicks the spacecraft off its starting circle onto an elliptical path tangent to both orbits, and the second, timed for when the ellipse reaches the target altitude, circularises it there. Of every possible two-burn transfer between two circular orbits, the Hohmann transfer uses the least total propellant — the tradeoff is time, since the spacecraft coasts along the slow, wide arc of the transfer ellipse rather than taking a more direct, higher-energy route.

Frequently asked questions

Try it live

Everything above runs in your browser — open Orbital Mechanics, fire a Δv burn during a close flyby, and watch how much orbital energy a well-timed slingshot can add or remove. Nothing is installed, nothing is uploaded.