*-written by Srishti Sharma, reviewed by Ankush Banerjee*

The term ‘Lagrange Point’ has been popularized through various works in science fiction as well as many milestones in space exploration. With the recently launched James Webb Space Telescope by NASA set to enter into orbit at the Sun-Earth L2 point and the highly anticipated Aditya L1 mission by ISRO which will be inserted into orbit around the Sun-Earth L1 point, the buzz around Lagrange Points has reached many ears.

**So, what exactly are these Lagrange Points and what is the physics behind them?**

Mathematically speaking, the Lagrange points represent solutions to a restricted three-body problem wherein two massive bodies in orbit gravitationally influence the behaviour of a third, comparatively lighter body.

From the standpoint of celestial mechanics, Lagrange points are positions in space where an object once sent, tends to stay put. This is because they form an equilibrium point for this object influenced by the gravitational forces of two massive objects in orbit.

To avoid ambiguity, let us consider A and B to be the two massive objects. Furthermore, the mass of A is much greater than the mass of B, resulting in B orbiting around A. Examples of such A-B pairs are the Earth-Moon system, the Sun-Earth system or any other planet revolving around the Sun.

In all these systems, there exist certain points where the combined gravitational force by A and B is equal to the centrifugal force experienced by a much smaller body (for example, a satellite). These very points are known as the Lagrange Points. They are also called the Lagrangian points, L-points, libration points, etc. All of the A-B type systems have 5 such Lagrange points in total, namely – L1, L2, L3, L4 and L5.

Mathematician and astronomer – Joseph-Louis Lagrange introduced his concept of Lagrange points in 1772 in his “Essay on three-body problem” and thus the name ‘Lagrange points’. However, Swiss mathematician and astronomer Leonhard Euler too hypothesized the concept of the first three Lagrange points L1, L2 and L3 a couple of years before Louis.

*Interested in cool scientists who contributed immensely to astronomy? Read our article on: Friedmann’s astounding work: Not just a WW1 pilot*

**Positions of the Lagrange Points: **

Out of the five Lagrange points, the first three – L1, L2 and L3 are collinear. They lie along the imaginary line which joins the centres of mass of A and B. The sequence of their positions is L3 – A – L1 – B – L2.

The other two points L4 and L5 are present along the orbit of B, on either side of it as seen in the figure below. Moreover, L4, L5 and A together form the vertices of an equilateral triangle.

**Lagrange Points and Space Exploration: **

- L1 point: It is positioned between A and B such that it lies closer to object B. The existence of L1 arises as to the gravitational forces on an object at that point due to A and B balancing each other. At L1, the orbital period of an object becomes equal to that of B, i.e., an object placed at L1 will orbit A more or less in sync with B.

This makes the L1 point of the Sun-Earth system especially advantageous for observing the Sun as well as Earth. A satellite observing Earth will always face the sunlit part. Solar observations are also ideal from this point because of ease of communication with Earth and uninterrupted view of the Sun. However, the L1 point is slightly unstable and hence spacecraft at L1 needs to be repositioned around every 23 days. Up until now, the Sun-Earth L1 point has been home to various satellite missions including ACE, SOHO, DSCOVR, Wind. It will also house ISRO’s Aditya-L1 in the near future.

- L2 point: The L2 point lies beyond B. On an object at L2, the gravitational pull due to A and B is balanced by the centrifugal force on the object. Similar to L1, the orbital period of an object at L2 becomes more or less equal to that of B.

The Sun-Earth L2 point is an ideal spot for observatories in space. Placing an observatory at L2 would allow them to be adequately shielded from the sun, allowing only the apt amount of radiation for the spacecraft to harvest solar power. With appropriate shielding mechanisms, the spacecraft can maintain low temperatures for the efficient functioning of all its equipment. However, just like the Sun-Earth L1 point, L2 is also a result of unstable equilibrium and any object stationed at L2 must be capable of course correction manoeuvers. It is home to Gaia – a space observatory launched by ESO in 2013, and soon to accommodate the James Webb Space Telescope launched by NASA on Dec. 25, 2021.

The Earth-Moon L2 point has proved useful to communicate with missions observing the far side of the moon. Queqiao – a relay satellite for CNSA’s lunar far side mission Chang’e 4 is an attestation to that.

- L3 point: Although lying along the orbit of B, the L3 point is positioned such that it can never be visible from B as it will always be eclipsed by A. Apart from being ever-elusive, the L3 point of the Sun-Earth system is also the result of an unstable equilibrium similar to L1 and L2. Moreover, being beyond the sun, communication with any satellite placed at L3 would be difficult due to undesirably heavy interference of solar radiation.

However, this mysterious disposition of L3 point has made it a beloved topic in sci-fi. Imagine the possibilities you have with a hidden, unsuspecting point along the orbit of your home planet! How about a hidden planet that is the complete opposite of Earth? What if it harbours a space station made by highly intelligent aliens to spy on us with technology capable of overcoming the unstable gravitational equilibrium?! Or how shocked would you be if there was a top-secret space military force with the hidden and mysterious L3 point nesting its command centre, keeping away enemy extra-terrestrial forces from reaching Earth?! There is much room for one’s imagination to run wild…

- L4 and L5 points: These Lagrange points harbour stable equilibrium conditions, provided that the ratio of the mass of A to B be at least 25. If this condition is satisfied, the L4 and L5 points become extremely stable zones, unaffected by gravitational perturbations. Any object entering into orbit around L4 or L5 would stay put, without even needing any course correction (as opposed to the case of L1 and L2). Thus, it is unsurprising that dust and asteroids are often found orbiting L4 and L5.

These objects were named ‘Trojans’ in accordance with Greek mythology, as an abundance of them was discovered for the first time at the L4 and L5 points of the Sun-Jupiter system. Now, ‘trojan’ is the generic term used for all such natural objects at the L4 and L5 points of any celestial system. The significance of these two Lagrange points is not just limited to the presence of trojan bodies. The L4 and L5 points Sun-Earth and Earth-Moon systems are considered the most viable sites for large observatories, space stations, or space colonization plans. This factor has also led to the popularity of L4 and L5 points in works of science fiction.

*Enjoyed reading this article? Consider reading another cool article of ours on: 5 scary negatives of Space Travel*

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## 2 Comments

## Palak

Well explained. I read about Lagrange Points for the first time today and I understood it very well.

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