Lagrange Points: Mathematical Representation & Important Facts

Lagrange points or Lagrangian points are the points in space at which the force that is exerted between any two given objects becomes equal. The objects at the Lagrangian point experience a neutral kind of force. The two-body systems such as Earth and the sun produce some enhanced regions of attraction as well as repulsion. It plays a very important role in astronomy and can be used by spacecraft to remain in their position without consuming much fuel. There are basically five Lagrange points wherein a body of a small mass can orbit in a pattern with any two larger masses. In this article, we will learn more about the Lagrangian points, their use, and mathematical representations.

Key Takeaways: Lagrangian points, Astronomy, Celestial Bodies, Earth, Sun, Satellite, Gravitation

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What are Lagrangian Points?

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Lagrangian Point or Lagrange Point can be understood as a point that is near two large celestial bodies moving in an orbit in a way that the smaller object is able to maintain its position in relation to the large bodies. Lagrangian points can also be called Lagrange points or Liberation Points, and are denoted by L.

Lagrangian Point

For every known combination of 2 large bodies, there exist up to 5 Lagrangian points from L1 to L5. Leonhard Euler discovered the first three Lagrangian points i.e. L1, L2, and L3. The remaining two points, L4 and L5, were later discovered by Joseph Louis.

Lagrangian Point of a two-body system

Of all the five Lagrangian points, three of them are unstable while two are stable.

• L1, L2, and L3 are the three unstable Lagrange points that lie along the line that connects two of the large masses.
• L4 and L5 Lagrangian points however are stable points that form the apex of 2 equilateral triangles having large masses on their vertices.

In the system of Earth and sun, L1 and L2, the first two Lagrangian points exist at about 1,500,000 km or 900,000 miles from the Earth away and towards the sun. Satellites are located at the Lagrangian of the sun earth system.

Read More: Space Wave Propagation

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Lagrange Points and Mathematical Details

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There are 5 Lagrangian Points which are as follows:

Lagrange Point 1 (L1)

The point existing on the line between two large masses, known as M1 and M2 is the Lagrange Point 1. The gravitational attraction of a mass is partially canceled by the gravitational force of the other one. The mathematical representation of this point is:

\(\frac{M_1}{(R-r)^2}\)= \(\frac{M_2}{r^2}\)+\(\frac{M_1}{R^2}\)-\(\frac{r(M_1+M_2)}{R^3}\)

Here, r refers to the distance of an L1 point from the small object

R refers to the distance that exists between two main objects

M1 and M2 refer to the mass of the large and small object respectively

The L1 point of the Earth-sun system offers is currently home to SOHO (the Solar and Heliospheric Observatory Satellite). The point gives an uninterrupted view of the sun

Lagrange Point 2 (L2)

Lagrange Point 2 is the point that exists beyond the smaller of the two masses and on the line defined by them. The centrifugal effect on a body present at the L2 point is balanced by the gravitational force of both the large masses. The mathematical representation of this point is:

\(\frac{M_1}{(R+r)^2}\)+ \(\frac{M_2}{r^2}\)=\(\frac{M_1}{R^2}\)+\(\frac{r(M_1+M_2)}{R^3}\)

Here, r denotes the distance of the point L2 from the smaller of the two object

R denotes the distance between any two main objects

M1 and M2 denote the mass of the large and small objects respectively.

WMAP spacecraft was located at the L2 point of the Earth-sun system. It is a vital point in astronomy as the spacecraft is close enough to communicate with the Earth in a proper manner, can keep the Sun, moon, and Earth behind it, and with appropriate shielding can provide a clear view for our telescopes to explore the deep space.

Using a satellite at point L2, it was studied and observed that:

• 2.735 K was the average temperature that was recorded in space.
• Space is made up of 73% dark energy, 23% dark matter, and only 4% of the mass.
• Points lying outside the magnetic field of the Earth were also discovered.
• As there is no deflection observed in the position of the satellite at this point because it has no gravitational pull, the data recorded is accurate and reliable.

Lagrange Point 3 (L3)

The point that lies beyond the larger of the two masses and on the line defined by them is known as the Lagrange Point 3. The mathematical representation of this point is:

\(\frac{M_1}{(R-r)^2}\)+ \(\frac{M_2}{(2R-r)^2}\)=(\(\frac{M_2}{M_1+M_2}\)R+R-r)\(\frac{M_1+M_2}{R^3}\)

Here, r denotes the distance between the L3 and the smaller object

R denotes the distance between both the main objects

M1 and M2 denote the mass of the large and small objects respectively.

The point L3 remains hidden behind the sun at all points of time.

Read More: Difference Between Equinox and Solstice

Lagrange Point 4 (L4) and Lagrange Point 5 (L5)

L4 and L5 points exist on the line defined by the centers of both the masses in a way that they lie at the third corner of both the equilateral triangles. Using radial acceleration, the mathematical representation of this point is

a = \(\frac{-GM_1}{r^2}\)sgn(r) + \(\frac{GM_2}{(R-r)^2}\)sgn(R-r) + \(\frac{G((M_1+M_2)r-M_2R}{R^3}\)

Here, a refers to the radial acceleration

r refers to the distance from the larger body 

sgn(x) is the sign function of x

L5 follows while L4 leads the orbit of the Earth. They are home to stable orbits as long as the mass ratio between both the masses is more than 24.96. Objects that orbit at the L4 and L5 points are known as Trojans.

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Applications of Lagrangian Point

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Some applications related to spacecraft and Lagrangian points are:

• For the Earth-sun system, at Lagrangian points objects are never shadowed by either the moon or the Earth.
• For a solar telescope, it provides an uninterrupted view of the space weather and the sun at point L1.
• For the Earth-moon system, lunar orbits and Earth become easily accessible with negligible changes to velocity.

The easiest way to understand the system of Lagrange points is to think of a frame of reference that rotates with the entire system. The forces acting on a body at rest in this frame can be calculated using an effective potential in the same way that wind speeds can be calculated using a weather map. When the contours of the effective potential are close together, the forces are strongest, and when they are far apart, the forces are the weakest.

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Things to Remember

• Lagrangian points are the points in space at which the force that is exerted between any two given objects becomes equal.
• The two-body systems such as Earth and the sun produce some enhanced regions of attraction as well as repulsion.
• For every known combination of 2 large bodies, there exist up to 5 Lagrangian points from L1 to L5.
• The point existing on the line between two large masses, known as M1 and M2 is the Lagrange Point 1. 
• Lagrange Point 2 is the point that exists beyond the smaller of the two masses and on the line defined by them.
• The point that lies beyond the larger of the two masses and on the line defined by them is known as the Lagrange Point 3.
• L4 and L5 points exist on the line defined by the centers of both the masses in a way that they lie at the third corner of both the equilateral triangles.

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