Lagrangian Point

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The lagrangian point in the celestial mechanics are the orbital points near two large co-orbiting orbits. The gravitational force in the lagrangian point of two bodies cancel out in such a way that a small object placed in the orbit there is in equilibrium relative to the center of mass of the large bodies. 

Five such points are there which are labeled as L1, L2, L3, L4, L5. These are in all the orbital plans of the two large bodies. L3. L2 and L1 are on the line through centers of the two large bodies, whereas, the L4 and L5 they both act as the third vertex of an equilateral triangle formed with the center of two large bodies. 

The unstable equilibria are L1, L2 and L3 on the other hand the L4 and L5 point are stable which implies that the objects can orbit around them in a coordinate in a rotating system tied to the two large bodies. 

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History of Lagrangian Point

The three lagarian points that are L1, L2, L3 were discovered by Leonhard Euler only a few years before when Joseph-Louis Lagarang discovered the remaining two. 

Lagrangian in 1772 published an essay on three body problems. In its first chapter he considered the three body problem.

In the second chapter from that, he demonstrated two special constant pattern solutions, the equilateral and collinear, for any three masses with circular orbits.  The L1 point lies on the two large masses which are M2 and M1 and between them.

The gravitational attraction of M2 in that point particularly cancels M1s gravitational attraction. 

The L2 point lies on the line through two line masses between the smallest of the two. The two large masses gravitational force balances the centrifugal effect on a body at L2

The point L3 lies on the line defined by two large masses, beyond the larger of the two. Within the earth-sun system, the L3 existes on the opposite side of the sun. 


Neutral Objects at Lagarian Point 

It’s common to find orbiting or objects at L4 and L5 points of natural orbital systems. The other commonly used name of it is trojans.

Asteroids discovered in the 20th century orbiting at sun- Jupiter L4 and L5 were named after the character from Homer's iliad. The asteroid present at the L4 point, which leads Jupiter, is referred to as the greek camp whereas, those which lie at point 5 are referred to as trojan camps. Other examples are:
The Earth and Sun at point L5 and L4. It contains at least one asteroid 2010 TK. 

The Moon-Earth at L5 and L4 points contain interdisciplinary dust  in it, which are called Kordylewski clouds. The hiten spacecraft munich dust counter although detected no increase during it’s passage through the phase in 1992.

It’s presence was confirmed in 2018 by a team of Hungarian physicists and scientists. At these specific points stability is greatly complicated by solar gravitational influence.


Mathematical Details

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The constant pattern solution to the three body problem are the lagrangian points. For example- the two masses given bodies in the orbits around their common barycenter, that are the five positions in the space where the third body which is of comparatively negligible mass could be placed so as to maintain it’s position of two massive bodies. As we can see in the frame of rotation reference that matches the angular velocity of the two co-orbiting bodies. 

The field of gravitational force of the two massive bodies combines providing the centripetal force at the lagrangian point allowing the smaller third body to be relatively stationary with respect to the first two. 

The L1s location is the solution of the following equation, gravitation providing the centripetal force:
M/(R-r)=M2/r+ (M1/M1+M2 * R-r) M1+M2/R3

Where r is the distance between point L1 and smaller object. R is the distance between two objects which are main. And M1 nad M2 are masses of the two objects. 


Some Applications 

Some specific spacecraft applications of lagrangian point are:

The Sun-Earth

Earth and Sun are at the point L1 for the observation of the Sun and Earth system. Here objects are never shadowed by Earth or Moon, when we observe Earth away from the sunlit hemisphere.

The first motion of this type was observed in 1978, from the international Sun Earth explore 3 mission used as an interplanetary early warning storm monitor for solar disturbance. DSCOVR, since june 2015 has orbited the L1 point.

For a space based solar telescope conversely it is useful because it provides an uninterrupted view of the sun and any space weather, which reaches point L1 first on earth. Currently solar telescopes are located around point L1, including the heliospheric observatory and solar and advanced composition explorer. 


Earth-Moon

It comparatively allows easy access to Earth And lunar orbits with minimal charges in velocity and this has an advantage to position a halfway manned space station which minted ro help cargo and personnel to the moon and back. 

FAQ (Frequently Asked Questions)

1. Name the Stable Lagrangian Points.

These are the positions in space where the gravitational force of the two body systems for example the Earth and the Sun produced enhanced regions of attraction and repulsion. The lagrangian point stable- labeled as L5 and L4, from the two equilateral apex triangles that have the large masses at their vertices.

2. In Space How Many Lagrangian Points Are Present?

Lagarian point structure- there are in total 5 lagrangian points around the major bodies such as stars or the planets. Amongst them, three of them lie along the line connecting the two large bodies. In the Sun-Earth system for example the first point L1, lies between the Sun and the Earth at about i million miles from the Earth.

3. What Is the Size of the Lagrangian Point?

Each planet in the solar system is having it’s own lagrangian point. The island of stability gets bigger and farther from the sun and also for more massive planets. The ones which are associated with Earth are roughly 500000 miles around 800000 kilometers wide.

4. What Lagrangian Point Does the Moon Have?

The Moon-Earth lagrange point system- a three object mechanical system the Earth, the Moon, the Sun constitute a three body problem. The lagrangian points L5 and L4 constitute stable equilibrium points. So that the object placed there would be in a stable orbit with respect o]to the planet Earth and Moon.