# THE LAGRANGE POINTS

The **Lagrangian points** (pron.: /ləˈɡrɑːndʒiən/; also **Lagrange points**, **L-points**, or **libration points**) are the five positions in anorbital configuration where a small object affected only by gravity can theoretically be part of a constant-shape pattern with two larger objects (such as a satellite with respect to the Earth and Moon). The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to orbit with them.

Lagrangian points are the constant-pattern solutions of the restricted three-body problem. For example, given two massive bodies in orbits around their common center of mass, there are five positions in space where a third body, of comparatively negligible mass, could be placed so as to maintain its position relative to the two massive bodies. As seen in a rotating reference frame matching the angular velocity of the two co-orbiting bodies, the gravitational fields of two massive bodies combined with the satellite's acceleration are in balance at the Lagrangian points, allowing the third body to be relatively stationary with respect to the first two bodies.

THE 5 LAGRANGE POINTS OF THE EARTH - SUN SYSTEM

Lagrange points are named in honor of Italian-French mathematician Joseph-Louis Lagrange. There are five special points where a small mass can orbit in a constant pattern with two larger masses. The Lagrange Points are positions where the gravitational pull of two large masses precisely equals the centripetal force required for a small object to move with them. This mathematical problem, known as the "General Three-Body Problem" was considered by Lagrange in his prize winning paper (*Essai sur le Problème des Trois Corps*, 1772).

Of the five Lagrange points, three are unstable and two are stable. The unstable Lagrange points - labeled L1, L2 and L3 - lie along the line connecting the two large masses. The stable Lagrange points - labeled L4 and L5 - form the apex of two equilateral triangles that have the large masses at their vertices. L4 leads the orbit of earth and L5 follows.

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. It is the most intuitively understood of the Lagrangian points: the one where the gravitational attraction of M2 partially cancels M1 gravitational attraction.

**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 Earth's gravity weakens the force pulling the object towards the Sun, and therefore increases 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. L1 is about 1.5 million kilometers from the Earth.

The location of L1 is the solution to the following equation balancing gravitation and centrifugal force:

where *r* is the distance of the L1 point from the smaller object, *R* is the distance between the two main objects, and M1 and M2 are the masses of the large and small object, respectively. (The quantity in parenthesis on the right is the distance of L1 from the centre of mass.) Solving this for *r* involves solving a quintic, but if the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then L1 and L2 are at approximately equal distances *r* from the smaller object, equal to the radius of the Hill sphere, given by:

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 :

The Sun–Earth L1 is suited for making observations of the Sun–Earth system. Objects here are never shadowed by the Earth or the Moon. The first mission of this type was the International Sun Earth Explorer 3 (ISEE3) mission used as an interplanetary early warning storm monitor for solar disturbances. The feasibility of this orbit was the result of a PhD thesis by the astrodynamicist Robert W. Farquhar.[6] Subsequently the Solar and Heliospheric Observatory (SOHO) was stationed in a Halo orbit at L1, and the Advanced Composition Explorer (ACE) in a Lissajous orbit, also at the L1 point. WIND is also at L1.

The Earth–Moon L1 allows comparatively easy access to lunar and earth orbits with minimal change in velocity and has this as an advantage to position a half-way manned space station intended to help transport cargo and personnel to the Moon and back.

In a binary star system, the Roche lobe has its apex located at L1; if a star overflows its Roche lobe then it will lose matter to its companion star.

### L2

The **L2** point lies on the line defined by the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal effect on a body at L2.

**Example:**On the side of the Earth 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.

The Sun–Earth L2 is a good spot for space-based observatories. Because an object around L2 will maintain the same relative position with respect to the Sun and Earth, shielding and calibration are much simpler. It is, however, slightly beyond the reach of Earth's umbra, so solar radiation is not completely blocked. The space observatories Herschel and Planck are already in orbit around the Sun–Earth L2, as was Chang'e 2 (until April 2012) and the Wilkinson Microwave Anisotropy Probe (until October 2010). The Gaia probe and James Webb Space Telescope are to be placed at the Sun–Earth L2. Earth–Moon L2 would be a good location for a communications satellitecovering the Moon's far side. Earth–Moon L2 would be "an ideal location" for a propellant depot as part of the proposed depot-based space transportation architecture.

The location of L2 is the solution to the following equation balancing gravitation and centrifugal force:

with parameters defined as for the L1 case. Again, if the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then L2 is at approximately the radius of the Hill sphere, given by:

#### Examples

- Sun and Earth: 1,500,000 km (930,000 mi) from the Earth
- Earth and Moon: 60,000 km (37,000 mi) from the Moon

### L3

The **L3** point lies on the line defined by the two large masses, beyond the larger of the two.

**Example:**L3 in the Sun–Earth system exists on the opposite side of the Sun, a little outside the Earth's orbit but slightly closer to the Sun than the Earth is. (This apparent contradiction is because the Sun is also affected by the Earth's gravity, and so orbits around the two bodies' barycenter, which is, however, well inside the body of the Sun.) At the L3 point, the combined pull of the Earth and Sun again causes the object to orbit with the same period as the Earth.

The location of L3 is the solution to the following equation balancing gravitation and centrifugal force:

with parameters defined as for the L1 and L2 cases except that *r* is now how far L3 is below the point at distance *R* from the more massive object. If the mass of the smaller object (M2) is much smaller than the mass of the larger object (M1) then:

The Sun–Earth L3 point was a popular place to put a "Counter-Earth" in pulp science fiction and comic books. Once space-based observation became possible via satellites and probes, it was shown to hold no such object. The Sun–Earth L3 is unstable and could not contain an object, large or small, for very long. This is because the gravitational forces of the other planets are stronger than that of the Earth (Venus, for example, comes within 0.3 AU of this L3 every 20 months). In addition, because Earth's orbit is elliptical and because the barycenter of the Sun–Jupiter system is unbalanced relative to Earth (that is, the Sun orbits the Sun–Jupiter center of mass, which is outside of the Sun itself), such a Counter-Earth would frequently be visible from Earth.

A spacecraft orbiting near Sun–Earth L3 would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a 7-day early warning could be issued by the NOAA Space Weather Prediction Center. Moreover, a satellite near Sun–Earth L3 would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for manned mission to near-Earth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth L3 were studied and several designs were considered.

One example of asteroids which visit an L3 point is the Hilda family whose orbit brings them to the Sun–Jupiter L3 point.

### L4 and L5

The **L4** and **L5** points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies behind (L5) or ahead of (L4) the smaller mass with regard to its orbit around the larger mass.

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 system, this resultant force is exactly that required to keep a body at the Lagrange point in orbital equilibrium with the rest of the system. (Indeed, the third body need not have negligible mass). The general triangular configuration was discovered by Lagrange in work on the 3-body problem.

L4 and L5 are sometimes called *triangular Lagrange points* or *Trojan points*. The name *Trojan points* comes from the Trojan asteroids at the Sun–Jupiter L4and L5 points, which themselves are named after characters from Homer's *Iliad* (the legendary siege of Troy). Asteroids at the L4 point, which leads Jupiter, are referred to as the "Greek camp", while those at the L5 point are referred to as the "Trojan camp". These asteroids are (largely) named after characters from the respective sides of the Trojan War.

#### Examples

- The Sun–Earth L4 and L5 points lie 60° ahead of and 60° behind the Earth as it orbits the Sun. The regions around these points contain interplanetary dust and at least one asteroid, 2010 TK7, detected October 2010 by WISE and announced July 2011.[14][15] Watch NASA's animated clip..
- The Earth–Moon L4 and L5 points lie 60° ahead of and 60° behind the Moon as it orbits the Earth. They may contain interplanetary dust in what is called Kordylewski clouds; however, theHiten spacecraft's Munich Dust Counter (MDC) detected no increase in dust during its passes through these points.
- The region around the Sun–Jupiter L4 and L5 points are occupied by the Trojan asteroids.
- The region around the Sun–Neptune L4 and L5 points have trojan objects.[16]
- Saturn's moon Tethys has two much smaller satellites at its L4 and L5 points named Telesto and Calypso, respectively.
- Saturn's moon Dione has smaller moons Helene and Polydeuces at its L4 and L5 points, respectively.
- One version of the giant impact hypothesis suggests that an object named Theia formed at the Sun–Earth L4 or L5 points and crashed into the Earth after its orbit destabilized, forming the Moon.

Credits : WikiPedia | NASA