Orbital Mechanics: Kepler, Verlet & Lagrange Points

Kepler's laws, and where they stop being enough

For a single planet orbiting a single, much heavier star, Kepler's three laws describe the motion exactly: orbits are ellipses with the star at one focus; a line from star to planet sweeps equal areas in equal times (so a planet moves faster near perihelion, slower near aphelion); and the square of the orbital period is proportional to the cube of the semi-major axis. This two-body problem has an exact closed-form solution — you can compute the planet's position at any future time directly from six orbital elements, no simulation required.

The moment a third body enters — a second planet, a moon, a spacecraft — the system generally has no closed-form solution. This is the famous three-body problem: Poincaré showed in the 1890s that its trajectories can be chaotic and are not expressible in general as a finite combination of elementary functions. The practical response is numerical integration: advance every body's position and velocity by a small time step, using the gravitational force each body currently exerts on every other, and repeat.

Why the integrator has to be Velocity Verlet

This simulation advances every scenario with Velocity Verlet, not a fancier higher-order method, because orbital mechanics cares about a property most integrators ignore: long-run energy behaviour. Verlet is symplectic — it conserves a shadow energy close to the true energy indefinitely, so a bound orbit stays bound and a periodic orbit stays periodic over millions of steps. A higher-order but non-symplectic method like RK4 is more accurate per step, but leaks energy slowly and steadily, so a planet integrated with RK4 for long enough visibly spirals in or out even though each individual step looks nearly exact.

x += v * h + 0.5 * aOld * h * h;
const aNew = gravityFrom(allOtherBodies, x);   // sum of GM_i / r_i^2 terms
v += 0.5 * (aOld + aNew) * h;
aOld = aNew;

Lagrange points and the Trojan asteroids

In the restricted three-body problem (one body's mass is negligible next to the other two, e.g. an asteroid orbiting the Sun–Jupiter system), there are five special positions, the Lagrange points L1–L5, where the combined gravity of the two large bodies and the orbital centrifugal effect balance so a third, massless body can sit in a fixed position relative to the other two, co-rotating with them. L1–L3 lie on the line through both large bodies and are unstable saddle points — useful for spacecraft (like solar observatories at Sun–Earth L1) that need frequent small correction burns, not for anything left alone.

L4 and L5 are different: they sit 60° ahead of and behind the smaller large body along its orbit, forming an equilateral triangle with the two large masses, and for a large enough mass ratio between the two primaries they are dynamically stable. Jupiter's Trojan asteroids are trapped at its L4 and L5 points in exactly this way — tens of thousands of them, some over 100 km across, orbiting the Sun in lockstep with Jupiter without ever needing to be individually integrated as if they were random debris.

Gravity assists: stealing momentum from a planet

A gravity assist (or flyby) changes a spacecraft's speed relative to the Sun without burning any propellant, by exchanging momentum with a planet. In the planet's own reference frame the encounter is elastic — the spacecraft leaves with the same speed it arrived at, only its direction changed by the hyperbolic deflection angle. Transformed back into the Sun's frame, though, that direction change adds (or subtracts) the planet's orbital velocity vector to the spacecraft's, which can produce a substantial net speed change relative to the Sun — the mechanism behind essentially every outer-solar-system mission's actual flight path.

Frequently asked questions

Try it live

Everything above runs in your browser — open Orbital Mechanics and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.