GFDL:Lagrangian point

In celestial mechanics, the Lagrangian points, (also Lagrange point, L-point, or libration point) are the five stationary solutions of the circular restricted three-body problem. I.e. given two massive bodies in circular orbits around their common center of mass, there are five positions in space where a third body, of negligible mass, could be placed which would then maintain its position relative to the two massive bodies. As seen in a frame of reference which rotates with the same period as the two co-orbiting bodies, the gravitational fields of two massive bodies combined with the centrifugal force are in balance at the Lagrangian points, allowing the third body to be stationary relative to the first two bodies.

In the more general case of elliptical orbits, there are no longer stationary points in the same sense, but the Lagrangian points constructed at each point in time as in the circular case form stationary elliptical orbits which are similar to the orbits of the massive bodies. This is due to the fact that Newton's second law, <math>F = ma</math>, remains invariant if force and position are scaled by the same factor. That a body at a Lagrangian point orbits with the same period as the two massive bodies in the circular case implies that it has the same ratio of gravitational force to radial distance as they do. This fact is independent of the circularity of the orbits, and it implies that the elliptical orbits traced by the Lagrangian points are solutions of the equation of motion of the third body.

File:Lagrangepoint1.png

A diagram showing the five Lagrangian points in a two-body system

The five Lagrangian points are labelled and defined as follows:

L₁

On the line defined by the two large masses M₁ and M₂, and between them.

Example: An object which orbits the Sun more closely than the Earth would normally have a shorter orbital period than the Earth, but that ignores the effect of the Earth's own gravitational pull. If the object is directly between the Earth and the Sun, then the effect of the Earth's gravity is to weaken the force pulling the object towards the Sun, and therefore increase the orbital period of the object. The closer to Earth the object is, the greater this effect is. At the L₁ point, the orbital period of the object becomes exactly equal to the Earth's orbital period. The Solar and Heliospheric Observatory (SOHO) (NASA's site of SOHO project), for example, is stationed in a halo orbit around the Sun-Earth L₁ point.

L₂

On the line defined by the two large masses, and beyond the smaller of the two.

Example: A similar effect occurs on the other side of the Earth, further away from the Sun, where the orbital period of an object would normally be greater than that of the Earth. The extra pull of the Earth's gravity decreases the orbital period of the object, and at the L₂ point that orbital period becomes equal to the Earth's.

L2 is a commonly used spot for space-based observatories. Because an object around L2 will maintain the same orientation with respect to the Sun and Earth, shielding and calibration are much simpler.

The Wilkinson Microwave Anisotropy Probe is already in orbit around the Sun-Earth L₂. The proposed James Webb Space Telescope will be placed at the Sun-Earth L₂.

Distances from L₁ and L₂ to M₂

If M₂ is much smaller than M₁ then L₁ and L₂ are at approximately equal distances r from M₂, equal to the radius of the Hill sphere, given by:

<math>r \approx R \sqrt[3]{\frac{M_2}{3 M_1}}</math>

where R is the distance between the two bodies.

This distance can be described as being such that the orbital period corresponding to a circular orbit with this distance as radius around M₂ in the absence of M₁, is that of M₂ around M₁, divided by <math>\sqrt{3}\approx 1.73</math>.

Examples:

• Sun and Earth: 1,500,000 km from the Earth
• Earth and Moon: 61,500 km from the Moon

L₃

On the line defined by the two large masses, and beyond the larger of the two.

Example: A third Lagrangian point, L₃, exists on the opposite side of the Sun, a little further away from the Sun than the Earth is, where the combined pull of the Earth and Sun again causes the object to orbit with the same period as the Earth. When used with the Sun and the Earth as the two masses, the L₃ point was a popular place to put a "Counter-Earth" in pulp science fiction and comic books.

L₄ and L₅

At the third point of an equilateral triangle with the base of the line defined by the two masses, such that the point is ahead of, or behind, the smaller mass in its orbit around the larger mass.

L₄ and L₅ are sometimes called triangular Lagrange points or Trojan points.

Example: The Sun-Earth L₄ and L₅ points lie 60° ahead of and 60° behind the Earth in its orbit around the Sun.

Miscellaneous

The first three Lagrangian points are stable only in the plane perpendicular to the line between the two bodies. This can be seen most easily by considering the L₁ point. A test mass displaced perpendicularly from the central line would feel a force pulling it back towards the equilibrium point. This is because the lateral components of the two masses' gravity would add to produce this force, whereas the components along the axis between them would balance out. However, if an object located at the L₁ point drifted closer to one of the masses, the gravitational attraction it felt from that mass would be greater, and it would be pulled closer. (The pattern is very similar to that of tidal forces.)

The L₁ and L₂ points have some practical value since, although a satellite would wander away if left to itself, a relatively modest effort at station-keeping can prevent it from doing so. More subtly, they serve as "gateways" to control the chaotic trajectories of the Interplanetary Superhighway.

By contrast, L₄ and L₅ are stable equilibria (cf. attractor), provided the ratio of the masses M₁/M₂ is > 24.96. When a body at these points is perturbed, it moves away from the point, but the Coriolis force then acts, and bends the object's path into a stable, kidney bean-shaped orbit around the point (as seen in the rotating frame of reference). In the Sun-Jupiter system several thousand asteroids, collectively referred to as Trojan asteroids, are in such orbits. Other bodies can be found in the Sun-Saturn, Sun-Mars, Jupiter-Jovian satellite, and Saturn-Saturnian satellite systems. There are no known large bodies in the Sun-Earth system's Trojan points, but clouds of dust surrounding the L₄ and L₅ points were discovered in the 1950s. Clouds of dust, even fainter than the notoriously weak gegenschein, are also present in the L₄ and L₅ of the Earth-Moon system.

The Earth's companion object 3753 Cruithne is in a somewhat Trojan-like orbit around the Earth, but not in the same manner as a true Trojan. It has a regular solar orbit that is bumped at times by Earth. When the asteroid approaches Earth, the asteroid takes orbital energy from Earth and moves into a larger, higher energy orbit. When the asteroid (in a larger and slower orbit) is caught up by Earth, Earth takes the energy back and so the asteroid falls into a smaller, faster orbit and eventually catches Earth to begin the cycle anew. Epimetheus and Janus, satellites of Saturn, have a similar relationship, though they are of similar masses and so actually exchange orbits periodically. Another similar configuration is known as orbital resonance, in which orbiting bodies tend to have periods of a simple integer ratio, due to their interaction.

The Saturnian moon Tethys has two smaller moons in its L₄ and L₅ points, Telesto and Calypso. The Saturnian moon Dione has the moon Helene in its L₄ point.

The L5 Society is a precursor of the National Space Society, and promoted the possibility of establishing a colony and manufacturing facility in orbit around the L₄ and/or L₅ points in the Earth-Moon system.

See also

• Hill sphere

External links

• Explanation of Lagrange points by Prof. Neil J. Cornish
• Explanation of Lagrange points by Prof. John Baez