What Are Lagrange Points?

In the restricted three-body problem (two massive bodies + a massless test particle), there are five special positions where the gravitational forces and the centrifugal force in the rotating reference frame exactly balance. These are the Lagrange points L1–L5.

• L1 — between the two bodies; used for solar observation (SOHO, DSCOVR).
• L2 — behind the smaller body; host to James Webb Space Telescope, Gaia, Planck.
• L3 — directly opposite the smaller body (unstable).
• L4 & L5 — 60° ahead and behind in the orbit; stable if M₁/M₂ > 24.96. Home to Trojan asteroids (2000+ in Sun–Jupiter L4/L5).

Effective Potential

The colour map shows the effective potential in the co-rotating frame: $\Phi_{eff} = -\frac{GM_1}{r_1} - \frac{GM_2}{r_2} - \frac{1}{2}\omega^2\rho^2$. Lagrange points are saddle points (L1–L3) or local maxima (L4–L5) of this surface.

Real-World Applications

Dozens of spacecraft occupy L1 and L2 of the Sun–Earth system. The Trojan asteroids co-orbit with Jupiter. NASA is studying L4/L5 of Earth–Moon for future space stations. The concept was first described by Joseph-Louis Lagrange in 1772.