Lagrange Points

An interactive website about Lagrange points in a Sun-Earth system. A Sun-Earth co-rotating reference frame is used, in which the Sun and Earth are stationary. You can move a test particle and inspect acceleration vectors from the Sun (g_sun), Earth (g_earth), the centrifugal term, and their vector sum, while tracking where L1-L5 appear for different mass ratios.

Website

https://lagrange-points.thesmirnov.com

What You Can Do

• Move the test particle around the scene and inspect local vector balance.
• Lock/unlock the particle to freeze a point and compare vectors precisely.
• Change the Earth mass using the slider.
• Use quick presets for Earth and Jupiter-like mass ratios.
• Toggle grid, Lagrange points, force vectors, and orbit overlays.

Notes

• This is an educational, normalized model intended for intuition and visual exploration.
• Labels Sun and Earth are used as intuitive naming to make the scene immediately readable; physically this is just a two-body system where one body is more massive than the other.
• CM means center of mass.
• The default Earth mass is set to 25% of the Sun mass on purpose to keep the visualization readable. With realistic Sun-Earth ratios, the mass contrast is so extreme that many effects become hard to notice at a glance.

Controls

Mouse / touch:

• Move pointer to move the particle (when unlocked).
• Click/tap canvas to lock or unlock the particle.

Keyboard hotkeys:

• 1 Toggle g_sun
• 2 Toggle g_earth
• 3 Toggle centrifugal
• 4 Toggle vector sum
• L Toggle Lagrange points
• G Toggle grid
• Space Lock/unlock particle
• O Toggle Sun-Earth orbit
• C Toggle CM-Earth orbit
• R Reset state

Why This Exists

One night around 2 a.m., I was already half asleep when my brain suddenly resurrected a random podcast fragment from about two weeks earlier: Lagrange points. Naturally, that was the perfect moment to become curious.

I started digging in. L1, L2, and L3 clicked pretty quickly, but L4 and L5 still felt slippery. The equilateral-triangle part looked almost like cheating, and the fact that these points stay in place regardless of the two-body mass ratio felt like pure mathematical black magic.

So instead of sleeping like a reasonable person, I built this visualization to see the vector balance with my own eyes. It finally made everything click. Then I went to bed.

Next day I was tired. Absolute mystery.