Physics · Mechanics · common

Solar System: Two-Body Orbits

everything orbits the shared centre of mass

Two gravitating bodies circle their shared barycentre: radii split by mass, period set by total mass. Plus a stable n-body integrator for games.

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Interactive visual

The cyan and orange bodies circle the yellow barycentre ring along their faint orbits; sphere size grows with mass at equal density. Raise the mass ratio and the orange body moves inward while the cyan one swings wide, and the whole pair speeds up.

What it is

Gravity is mutual: the star pulls the planet and the planet pulls the star with an equal and opposite force. With no outside forces the centre of mass (barycentre) never accelerates, so both bodies orbit that point, not each other's centres. The heavier body traces the smaller circle: m₁·r₁ = m₂·r₂.

For a circular pair the separation d and total mass fix the angular rate through Kepler's third law: ω² = G(m₁+m₂)/d³. Make one body heavier and the whole system spins faster; make them equal and they chase each other around a point exactly halfway between.

In a game with more than two bodies there is no closed form, so you integrate. Sum the pairwise forces, then advance with a symplectic integrator (leapfrog/velocity Verlet) at a fixed timestep. Add a softening length so close passes do not fling bodies to infinity. For background planets that never interact, Kepler rails are cheaper and drift-free.

Key formulas

Newton's law of gravitation: $F_{ij} = G \frac{m _{i} m _{j}}{∣ r _{ij} ∣ ^{3}} r_{ij}$
Barycentre: $R_{c m} = \frac{\sum m _{i} r _{i}}{\sum m _{i}}$
Radii around the barycentre: $r_{1} = \frac{m _{2}}{m _{1} + m _{2}} d, r_{2} = \frac{m _{1}}{m _{1} + m _{2}} d$
Angular rate of a circular pair: $ω = \frac{G ( m _{1} + m _{2} )}{d ^{3}}$
Circular orbit speed (light satellite): $v_{c} = GM / r$

Rules of thumb

Pick game units so G = 1 and masses/distances are near 1–100; real SI values lose float precision fast.
Softened gravity: use |r|² + ε² in the denominator with ε ≈ 1–5% of a typical separation.
Leapfrog/velocity Verlet keeps orbits closed for thousands of revolutions; explicit Euler spirals out within a few.
Start a stable circular orbit by giving the satellite v = √(GM/r) perpendicular to the radius, and give the star the opposite momentum so the barycentre stays put.
Sun–Jupiter's barycentre lies just outside the Sun's surface: about 1.07 solar radii from its centre.

Implementation

Velocity-Verlet n-body step with softening (Three.js vectors):

import * as THREE from "three"

type Body = { m: number; p: THREE.Vector3; v: THREE.Vector3; a: THREE.Vector3 }
const d = new THREE.Vector3()

function accelerations(bodies: Body[], G = 1, eps2 = 0.01) {
  for (const b of bodies) b.a.set(0, 0, 0)
  for (let i = 0; i < bodies.length; i++)
    for (let j = i + 1; j < bodies.length; j++) {
      const bi = bodies[i], bj = bodies[j]
      d.subVectors(bj.p, bi.p)
      const r2 = d.lengthSq() + eps2
      const s = G / (r2 * Math.sqrt(r2))
      bi.a.addScaledVector(d, s * bj.m)
      bj.a.addScaledVector(d, -s * bi.m)
    }
}

/** Call accelerations(bodies) once before the first step. */
export function step(bodies: Body[], dt: number) {
  for (const b of bodies) {
    b.v.addScaledVector(b.a, dt / 2) // half kick
    b.p.addScaledVector(b.v, dt) // drift
  }
  accelerations(bodies)
  for (const b of bodies) b.v.addScaledVector(b.a, dt / 2)
}

Common mistakes

Pinning the star at the origin while the planet orbits it: momentum is not conserved and heavy moons look wrong.
Updating each body's position before computing the others' forces, so the result depends on array order.
Integrating with explicit Euler or a variable frame time: orbits gain energy and spiral outwards.
No softening: a near collision gives a huge acceleration and slingshots bodies out of the system.

Prompt

Implement a TypeScript n-body gravity simulator for a Three.js scene. Inputs: an array of bodies {mass, position: THREE.Vector3, velocity: THREE.Vector3}, gravitational constant G, softening ε and a fixed timestep dt. Use velocity Verlet (half-kick, drift, recompute accelerations, half-kick), compute each pair once and apply equal and opposite forces, and never allocate inside step(). Provide totalEnergy() and barycentre(). Edge cases: zero or one body, coincident bodies (softening). Tests: a two-body circular orbit with v = √(G(m₁+m₂)/d) keeps its separation within 0.5% over 100 orbits, and the barycentre stays fixed to 1e-9.