Lagrangian Point
In astronomy, the Lagrangian point plays a vital role and it was important to learn about the points at which the force exerted between any two objects is equal such that the object placed in Lagrangian point experiences the neutral force.
What Is A Lagrange Point?
Lagrangian point is defined as the point that is near two large bodies in orbit such that the smaller object maintains its position relative to the large orbiting bodies. Lagrangian point is also known as L point or Lagrange points or Libration points.
There are five Lagrangian points from L_{1} to L_{5 }for every given combination of two large orbital bodies. For SunEarth system, there are L_{1} to L_{5 }Lagrangian points such that L_{1}, L_{2} and L_{3} forms a line through the center of two large bodies and L_{4} and L_{5} forms an equilateral triangle with centers of two large bodies. L_{4} and L_{5} are stable making them rotate in a coordinate system that is tied to the two large bodies whereas L_{1}, L_{2, }and L_{3} are unstable. There are planets having trojan satellite near L_{4} and L_{5}.
The first three Lagrangian points L_{1} L_{2}and L_{3} were discovered by Leonhard Euler and L_{4} and L_{5} were discovered by JosephLouis Lagrange.
Lagrange Points And Mathematical Details
L_{1} point: The point that lies between two large masses M_{1} and M_{2} and on the line defined by them. The gravitational attraction of M_{1} is partially canceled by the gravitational force of M_{2}. Following is the mathematical representation:
\(\frac{M_{1}}{(Rr)^{2}} = \frac{M_{2}}{r^{2}}+\frac{M_{1}}{R^{2}}\frac{r(M_{1}+M_{2})}{R^{3}}\) 
Where,
r is the distance of an L_{1} point from the smaller object
R is the distance between the two main objects
M_{1} and M_{2} are the masses of the large and small object
L_{2} point: The point that lies on the line defined by the two large masses and beyond the smaller of the two. The centrifugal effect on a body at L_{2} is balanced by the gravitational force of the two large masses. Following is the mathematical representation:
\(\frac{M_{1}}{(R+r)^{2}}+\frac{M_{2}}{r^{2}}=\frac{M_{1}}{R^{2}}+\frac{r(M_{1}+M_{2})}{R^{3}}\) 
Where,
r is the distance of L_{2} point from the smaller object
R is the distance between the two main objects
M_{1} and M_{2} are the masses of the large and small object
L_{3} point: The point that lies on the line defined by the two large masses and beyond the larger of the two. Following is the mathematical representation:
\(\frac{M_{1}}{(Rr)^{2}}+\frac{M_{2}}{(2Rr)^{2}}=(\frac{M_{2}}{M_{1}+M_{2}}R+Rr)\frac{M_{1}+M_{2}}{R^{3}}\) 
Where,
r is the distance of L_{3} point from the smaller object
R is the distance between the two main objects
M_{1} and M_{2} are the masses of the large and small object
L_{4} and L_{5} points: These points lie on the line defined between the centers of the two masses such that they lie at the third corner of the two equilateral triangles. Following is the mathematical representation using radial acceleration:
\(a=\frac{GM_{1}}{r^{2}}sgn(r)+\frac{GM_{2}}{(Rr)^{2}}sgn(Rr)+\frac{G((M_{1}+M_{2})rM_{2}R)}{R^{3}}\) 
Where,
a is the radial acceleration
r is the distance from the large body M1
sgn (x) is the sign function of x
There are a few facts of these points that were studied using satellite at point L_{2}, such as:

The average temperature of space was recorded to be 2.735K.

The composition of space is 73% of dark energy, 23% of dark matter and 4% of the mass.

Points outside the earth’s magnetic field were discovered.

As there is no gravitational pull at this point, there won’t be any deflection in satellite positioning resulting in accurate data records.