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 1772, famed Italian mathematician, Joseph Louis Lagrange was working on the infamous “3 body problem” when he discovered an interesting quirk in the results. Originally, he had set out to discover a way to easily calculate the gravitational interaction between arbitrary numbers of bodies in a system, because Newtonian mechanics conclude that such a system results in the bodies orbiting chaotically until there is a collision, or a body is thrown out of the system so that equilibrium can be achieved. The logic behind this conclusion is that a system with one body is trivial, as it is merely static relative to itself; a system with two bodies is very simple to solve for, as the bodies orbit around their common center of gravity. However, once more than two bodies are introduced, the mathematics go crazy. A situation arises where you would have to calculate every gravitational interaction between every object at every point along its trajectory. Lagrange, however, wanted to make this simpler. He did so with a simple conclusion: “the trajectory of an object is determined by finding a path that minimizes the action over time.” This is found by subtracting the potential energy from the kinetic energy. With this way of thinking, Lagrange re-formulated the classical Newtonian mechanics to give rise to Lagrangian mechanics. With his new system of calculations, Lagrange’s work led him to hypothesize how a third body of negligible mass would orbit around two larger bodies which were already orbiting one another. This third body at specific points in its orbit would become stationary to one of its host bodies (planets), these points were named “Lagrangian points” in Lagrange's honor.

In the more general case of elliptical orbits, there are no longer stationary points in the same sense: it becomes more of a Lagrangian “area” where the third body makes small odd-shaped orbits about the invisible Lagrangian point; these orbits are commonly referred to as halo orbits. 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>p = mv</math> (<math>p</math> the impulse, <math>m</math> the mass and <math>v</math> the velocity), 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 labeled and defined as follows:

L₁

The L₁ point lies 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 Sun-Earth L₁ is ideal for making observations of the Sun. Objects here are never shadowed by the Earth or the Moon. Solar and Heliospheric Observatory (SOHO) (NASA's site of SOHO project) is stationed in a halo orbit around L₁. The Earth-Moon L₁ allows easy access to lunar and earth orbits with minimal Delta-v and would be ideal for a half-way manned space station for transporting cargo and personnel to the Moon and back.

L₂

The L₂ point lies on the line defined by the two large masses, and beyond the smaller of the two.

Example: On the other side of the Earth, further away from the Sun, 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.

Sun-Earth L₂ is a good 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 Earth-Moon L₂. Earth-Moon L2 would be a good location for a communication satellite covering the Moon's far side.

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

The L₃ point lies 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

The L₄ and L₅ points lie 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.

Stability

The first three Lagrangian points are technically 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.)

Although the L₁, L₂, and L₃ points are nominally unstable, it turns out that it is possible to find stable periodic orbits around these points, at least in the restricted three-body problem. These perfectly periodic orbits, referred to as "halo" orbits, do not exist in a full n-body dynamical system such as the solar system. However, quasi-periodic (i.e. bounded but not precisely repeating) Lissajous orbits do exist in the n-body system. These quasi-periodic orbits are what all libration point missions to date have used. Although they are not perfectly stable, a relatively modest effort at station-keeping can allow a spacecraft to stay in a desired Lissajous orbit for an extended period of time. It also turns out that, at least in the case of Sun-Earth L₁ missions, it is actually preferable to place the spacecraft in a large amplitude (100,000 - 200,000 km) Lissajous orbit instead of having it sit at the libration point, since this keeps the spacecraft off of the direct Sun-earth line and thereby reduces the impacts of solar interference on the Earth-spacecraft communications links. Another interesting and useful property of the collinear libration points and their associated Lissajous orbits is that 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. This is the case for the Sun/Earth and Earth/Moon systems, though by a smaller margin in the latter. 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).

Examples

In the Sun-Jupiter system several thousand asteroids, collectively referred to as Trojan asteroids, are in orbits around the Sun-Jupiter L₄ and L₅ points. 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, called Kordylewski clouds, even fainter than the notoriously weak gegenschein, are also present in the L₄ and L₅ of the Earth-Moon system.

The Saturnian moon Tethys has two smaller moons in its L₄ and L₅ points, Telesto and Calypso. The Saturnian moon Dione also has two Lagrangian co-orbitals, Helene at its L₄ point and Polydeuces at L₅. The moons wander azimuthally about the Lagrangian points, with Polydeuces describing the largest deviations, moving up to 32 degrees away from the Saturn-Dione L₅ point. Tethys and Dione are hundreds of times more massive than their "escorts" (see the moons' articles for exact diameter figures, masses are not known in several cases), and Saturn is far more massive still, which makes the overall system stable.

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 Space colonisation).

Other co-orbitals

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. Rather, it occupies one of two regular solar orbits, periodically alternating between the two due to close encounters with Earth. When the asteroid approaches Earth, it takes orbital energy from Earth and moves into a larger, higher energy orbit. Some time later, the Earth catches up with the asteroid (which is in a larger and slower orbit), at which time Earth takes the energy back and so the asteroid falls into a smaller, faster orbit and eventually catches Earth to begin the cycle anew. This has no noticeable impact on the length of the year, since Earth masses over 20 billion times more than 3753 Cruithne.

Epimetheus and Janus, satellites of Saturn, have a similar relationship, though they are of similar masses and so actually exchange orbits with each other periodically. (Janus is roughly 4 times more massive, but still light enough for its orbit to be altered.) 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.

See also

• List of objects at the Sun-Earth Lagrangian points
• Interplanetary Superhighway
• Hill sphere
• Kordylewski clouds
• Lunar space elevator
• Big Splash

External links

• Explanation of Lagrange points by Prof. Neil J. Cornish
• Explanation of Lagrange points by Prof. John Baez
• Home On Lagrange (filksong)