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 L₁ to L₅ for every given combination of two large orbital bodies. For Sun-Earth system, there are L₁ to L₅ Lagrangian points such that L₁, L₂ and L₃ forms a line through the center of two large bodies and L₄ and L₅ forms an equilateral triangle with centers of two large bodies. L₄ and L₅ are stable making them rotate in a coordinate system that is tied to the two large bodies whereas L₁, L_2, and L₃ are unstable. There are planets having trojan satellite near L₄ and L₅.

The first three Lagrangian points L₁ L₂and L₃ were discovered by Leonhard Euler and L₄ and L₅ were discovered by Joseph-Louis Lagrange.

Lagrange Points And Mathematical Details

L₁ point: The point that lies between two large masses M₁ and M₂ and on the line defined by them. The gravitational attraction of M₁ is partially canceled by the gravitational force of M₂. 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 an L₁ point from the smaller object

R is the distance between the two main objects

M₁ and M₂ are the masses of the large and small object

L₂ 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 L₂ 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 L₂ point from the smaller object

R is the distance between the two main objects

M₁ and M₂ are the masses of the large and small object

L₃ 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: 

\(\frac{M_{1}}{(R-r)^{2}}+\frac{M_{2}}{(2R-r)^{2}}=(\frac{M_{2}}{M_{1}+M_{2}}R+R-r)\frac{M_{1}+M_{2}}{R^{3}}\)

Where,

r is the distance of L₃ point from the smaller object

R is the distance between the two main objects

M₁ and M₂ are the masses of the large and small object

L₄ and L₅ 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: 

\(a=\frac{-GM_{1}}{r^{2}}sgn(r)+\frac{GM_{2}}{(R-r)^{2}}sgn(R-r)+\frac{G((M_{1}+M_{2})r-M_{2}R)}{R^{3}}\)

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 L₂, 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.