In astronomy, the Lagrangian point plays a vital role. It was essential to learn about the points at which the force exerted between any two objects is equal. The object placed on the Lagrangian point experiences the neutral force.
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What is a Lagrangian Point?
A lagrangian point is defined as the point near two large bodies in orbit such that the smaller object maintains its position relative to the large orbiting bodies. Lagrangian points are also known as L points or Lagrange points, or Libration points.
There are five Lagrangian points from L1 to L5 for every given combination of two large orbital bodies. The first three Lagrangian points L1, L2 and L3 were discovered by Leonhard Euler and L4 and L5 were discovered by Joseph-Louis.
Of the five Lagrange points, three are unstable and two are stable. L1, L2 and L3 are unstable Lagrange points and they lie along the line connecting the two large masses. L4 and L5 are stable Lagrange points and they form the apex of two equilateral triangles that have large masses at their vertices. In the Earth-Sun system, the first (L1) and second (L2) Lagrangian points, occur at 1,500,000 km (900,000 miles) from Earth toward and away from the Sun. The lagrangian of the sun-earth system is home to satellites.
Lagrange Points and Mathematical Details
Lagrange Point 1 (L1)
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 cancelled by the gravitational force of M2. The following is the mathematical representation:
\(\begin{array}{l}\frac{M_{1}}{(R-r)^{2}} = \frac{M_{2}}{r^{2}}+\frac{M_{1}}{R^{2}}-\frac{r(M_{1}+M_{2})}{R^{3}}\end{array} \) |
Where, r is the distance of an L1 point from the smaller object, R is the distance between the two main objects and M1 and M2 are the masses of the large and small objects.
Lagrange Point 2 (L2)
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. The following is the mathematical representation:
\(\begin{array}{l}\frac{M_{1}}{(R+r)^{2}}+\frac{M_{2}}{r^{2}}=\frac{M_{1}}{R^{2}}+\frac{r(M_{1}+M_{2})}{R^{3}}\end{array} \) |
Where, r is the distance of L2 point from the smaller object, R is the distance between the two main objects and M1 and M2 are the masses of the large and small objects.
| Important Facts that were studied using a satellite at point L2 |
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Lagrange Point 3 (L3)
The point that lies on the line defined by the two large masses and beyond the larger of the two. The following is the mathematical representation:
\(\begin{array}{l}\frac{M_{1}}{(R-r)^{2}}+\frac{M_{2}}{(2R-r)^{2}}=(\frac{M_{2}}{M_{1}+M_{2}}R+R-r)\frac{M_{1}+M_{2}}{R^{3}}\end{array} \) |
where r is the distance of the 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 objects.
Lagrange Point 4 (L4) and Lagrange Point 5 (L5)
These points lie on the line defined between the centres of the two masses such that they lie at the third corner of the two equilateral triangles. The following is the mathematical representation using radial acceleration:
\(\begin{array}{l}a=\frac{-GM_{1}}{r^{2}}sgn(r)+\frac{GM_{2}}{(R-r)^{2}}sgn(R-r)+\frac{G((M_{1}+M_{2})r-M_{2}R)}{R^{3}}\end{array} \) |
Where a is the radial acceleration, r is the distance from the large body M1 and sgn (x) is the sign function of x.
Related Physics articles:
| Universal Law Of Gravitation | Motion Of Celestial Bodies In Space |
| Solar System | Earth’s Satellites |
Frequently Asked Questions – FAQs
What are Lagrangian Points or Lagrange points?
It is the point that is near two large bodies in orbit such that the smaller object maintains its position relative to the large orbiting bodies.
How many Lagrangian Points are there?
There are five Lagrangian points.
Who discovered the first three Lagrangian Points?
L1, L2 and L3 points were discovered by Leonhard Euler.
Who discovered L4 and L5?
Name the five Lagrangian points.
L1, L2, L3, L4 and L5 are the five Lagrangian points.
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