Introduction

The Lagrangian points are five points at which a smaller object can maintain a stable position with the Earth and the Sun – the gravitational pull of the earth and the Sun exactly equals the centripetal force required for the third object to move with them.

L4 and L5

L4 and L5 form the apex of two equilateral triangles that have the Sun and the Earth as their vertices. They are stable, whereas L1, L2, and L3, are unstable.

Trojans

Trojans are minor planets or moons that “share” another planet’s orbit by orbiting at L4 or L5. Most Trojans are known for Jupiter, although ones have been discovered for Mars, Neptune. The first known Earth Trojan was called 2010 TK7.

Reason for existence

All Lagrangian points are located where the gravitational force is zero. Look at a potential energy graph:

L1, L2, and L3 are located at “saddle points” – the potential reaches a maximum. So, it is very easy for them to escape; they are in unstable equilibrium.

However, L4 and L5 are at “hilltops”, at minimum potential. Whenever an object tries to wander off, it is stopped by the Coriolis force, which allows it to orbit in a stable equilibrium.

Sources