Like L 1, L 2 is about 1.5 million kilometers or 0.01 au from Earth (away from the sun). The extra pull of Earth's gravity decreases the object's orbital period, and at the L 2 point, that orbital period becomes equal to Earth's. On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than Earth's. Here, the gravitational forces of the two large masses balance the centrifugal effect on a body at L 2. The L 2 point lies on the line through the two large masses beyond the smaller of the two. L 1 is about 1.5 million kilometers from Earth in the direction of Sun, or 0.01 au. At the L 1 point, the object's orbital period becomes exactly equal to Earth's orbital period.
The closer to Earth the object is, the greater this effect is. If the object is directly between Earth and the Sun, then Earth's gravity counteracts some of the Sun's pull on the object, increasing the object's orbital period. An object that orbits the Sun more closely than Earth would typically have a shorter orbital period than Earth, but that ignores the effect of Earth's gravitational pull. It is the point where the gravitational attraction of M 2 and that of M 1 combine to produce an equilibrium. The L 1 point lies on the line defined between the two large masses M 1 and M 2. The five Lagrange points are labelled and defined as follows: See also: List of objects at Lagrange points From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits. In the first chapter he considered the general three-body problem. In 1772, Lagrange published an "Essay on the three-body problem". The three collinear Lagrange points (L 1, L 2, L 3) were discovered by Leonhard Euler around 1750, a decade before Joseph-Louis Lagrange discovered the remaining two. Artificial satellites, for example the James Webb Space Telescope, have been placed at L 1 and L 2 with respect to the Sun and Earth, and with respect to the Earth and the Moon. The Lagrange points have been proposed for uses in space exploration. Several planets have trojan asteroids near their L 4 and L 5 points with respect to the Sun Jupiter has more than one million of these trojans. When the mass ratio of the two bodies is large enough, the L 4 and L 5 points are stable points meaning that objects can orbit them, and that they have a tendency to pull objects into them. L 1, L 2, and L 3 are on the line through the centers of the two large bodies, while L 4 and L 5 each act as the third vertex of an equilateral triangle formed with the centers of the two large bodies. There are five Lagrange points for the Sun–Earth system, and five different Lagrange points for the Earth–Moon system. This can make Lagrange points an excellent location for satellites, as few orbit corrections are needed to maintain the desired orbit.įor any combination of two orbital bodies there are five Lagrange points, L 1 to L 5, all in the orbital plane of the two large bodies. At the Lagrange points, the gravitational forces of the two large bodies and the centrifugal force balance each other. Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. Mathematically, this involves the solution of the restricted three-body problem. In celestial mechanics, the Lagrange points ( / l ə ˈ ɡ r ɑː n dʒ/ also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies.
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