Lagrangian Point

Lagrangian point refers to a point in space at which a small body, which is experiencing the gravitational influence of two large ones, will approximately stay in a state of rest with relation to them. Furthermore, the French mathematician and astronomer Joseph-Louis Lagrange deduced the existence of such points in 1772.

Introduction to Lagrangian Point

Experts define the Lagrangian point as the point that is in orbit near two large bodies in such a manner that the smaller object maintains its position with relation to the large orbiting bodies. Furthermore, Lagrangian points can also be called as L point or Libration points or Lagrange points.

For every combination of two large orbital bodies, there are five Lagrangian points from L1 to L5. Furthermore, Leonhard Euler made the discovery of the first three Lagrangian points L1, L2 and L3. Also, Joseph-Louis made the discovery of Lagrangian points L4 and L5.

Langrangian Point For Sun-Earth System

As far as the Lagrangian point for the Sun-Earth system is concerned, there are L1 to L5 Lagrangian points. Furthermore, the formation of a line via the centre of two large bodies takes place by the Lagrangian points L1, L2, and L3. Also, the formation of an equilateral triangle with the centre of two large bodies takes place by the Langrangian points L4 and L5.

An important point to remember here is that Langrangian point stability is shown by L4 and L5. Moreover, this makes them rotate in a coordinate system whose connection is to the two large bodies. In contrast, Langrangian points L1, L2, and L3 are unstable.

Formula of Lagrangian Point

L₁ point: The point that lies between M₁ and M₂, which are the two large masses. Furthermore, L₁ lies on the line that is defined by these two large masses.

There is also partial cancellation of the gravitational attraction of M₁ by the gravitational force of M₂. Below is the mathematical representation:

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

Where,

• r is representative of 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 is on the line defined by the two large masses and beyond the smaller mass of the two. The balancing of the centrifugal effect on a body at L₂ takes place by the gravitational force of the two large masses. Furthermore, below is the mathematical representation:

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

Where,

• r is representative of the distance of L₂ point from the smaller object
• R is the distance that exists between the two main objects
• M₁ and M₂ represent the masses of the large and small object

L₃ point: The point that is on the line that is defined by the two large masses and beyond the larger mass among the two. Moreover, below is the mathematical representation:

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

Where,

• r is representative of the distance of L₃ point from the smaller object
• R is the distance that exists between the two main objects
• M₁ and M₂ represent the masses of the large and small object

L₄ and L₅ points: These points lie on the line that experts define as between the centres of the two masses in such a manner that they are at the third corner of the two equilateral triangles. Moreover, below is the mathematical representation by making use of radial acceleration:

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

Where,

• a represents the radial acceleration
• r refers to the distance from the large body M1
• sgn (x) is x’s sign function

FAQs For Langrangian Point

Question 1: How many lagrangian points are there in space according to Physics?

Answer 1: As far as space is concerned, one can take a look at the Sun-Earth System. Here, the Lagrangian points are L1 to L5. Most noteworthy, the Langrangian points L1, L2, and L3 are unstable while the Langrangian points L4 and L5 are stable.

Question 2: Give a mathematical expression for Langrangian points L4 and L5?

Answer 2:  The mathematical expression for Langrangian points L4 and L5 is a = \(\frac{-GM_{1}}{r^{2}}sgn\left ( r \right ) + \frac{GM_{2}}{\left ( R-r \right )^{2}}sgn\left ( R-r \right ) + \frac{G\left ( \left ( M_{1}+M_{2} \right )r-M_{2}R \right )}{R^{3}}\).

Here, a refers to the radial acceleration and r refers to the distance from the large body M1. Also, sgn (x) is x’s sign function.