§ 03 · Sandbox · Celestial mechanics Filed 2026.05

Five quiet places, where gravity agrees with you.

The Sun and the Earth orbit a common centre, dragging space around them. In the rotating frame that travels with them, five points stand perfectly still — three precarious, two profoundly patient. Drop a test mass anywhere. Watch what space already knows.

Frame Rotating · co-orbital μ 0.0120 Bodies 0

Time 0.00 T⊕ State ▶ live

Click to drop · drag for initial velocity

Mass ratio · μ

0.012

Time speed

1.0×

Show

Sandbox

Reading the picture

§ 03 · Notes

Between the bodies, where Sun’s pull and Earth’s pull conspire to match orbital tempo. SOHO lives here. Unstable — drifts in months.

Behind the Earth, in the Sun’s shadow. JWST parks here so its sunshield faces one constant direction. Also unstable; held by tiny corrections.

Opposite the Earth, behind the Sun — forever invisible from home. Beloved of pulp science fiction; useless to engineers. Unstable.

L4 · L5

Sixty degrees ahead and behind, completing equilateral triangles with the two bodies. Stable when μ is small. Trojan asteroids gather here like ducks in a pond.

What you’re actually simulating

§ 03 · Method

Circular restricted three-body problem, normalized so the Sun–Earth distance is one and one orbital revolution is 2π. The test mass is small enough that it doesn’t pull on anything. Integrated with classical RK4. The five Lagrange points are computed numerically from the effective potential, including the centrifugal term — so as you change μ, they slide to where they actually belong.