page_title: Lagrange point
section_title: L4 and L5
hierarchical_section_title: Lagrange point / Mathematical details / L4 and L5
caption_attribution_description: English: Net radial acceleration of a point orbiting along the Earth-Moon line. A positive value means the object will be forced to the right. Lagrangian points L3,L1,L2 occur where the line crosses the x-axis, but due to the positive slope on crossover, none of them are stable.
context_page_description: In celestial mechanics, the Lagrange points are orbital points near two large co-orbiting bodies. At the Lagrange points the gravitational forces of the two large bodies cancel out in such a way that a small object placed in orbit there is in equilibrium relative to the center of mass of the large bodies.
  There are five such points, labeled L₁ to L₅, all in the orbital plane of the two large bodies. L₁, L₂, and L₃ are on the line through the centers of the two large bodies, while L₄ and L₅ each act as the third vertex of an equilateral triangle formed with the centers of the two large bodies. L₁, L₂, L₃ are unstable equilibria, whereas L₄ and L₅ are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies.
  For each given combination of co-orbiting planetary bodies there are five Lagrange points L₁ to L₅ for the Sun–Earth system, and in a similar way there are five different Lagrange points for the Earth–Moon system. Several planets have trojan satellites near their L₄ and L₅ points with respect to the Sun. Jupiter has more than a million of these trojans.
context_section_description: The reason these points are in balance is that, at L₄ and L₅, the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the barycenter of the system; additionally, the geometry of the triangle ensures that the resultant acceleration is to the distance from the barycenter in the same ratio as for the two massive bodies. The barycenter being both the center of mass and center of rotation of the three-body system, this resultant force is exactly that required to keep the smaller body at the Lagrange point in orbital equilibrium with the other two larger bodies of system. (Indeed, the third body need not have negligible mass.) The general triangular configuration was discovered by Lagrange in work on the three-body problem.
caption: Net radial acceleration of a point orbiting along the Earth–Moon line.
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