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, L2, 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

  1. 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
  2.  

  3. 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
  4.  

  5. 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
  6.  

  7. 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.

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