Close to the Earth’s orbit there are **five points where a perfect balance is reached** , where the gravitational pull of the Sun and the Earth are perfectly compensated, to provide any object that occupies these points with an **orbit of identical duration to that of the planet,** against of the laws of mechanics. These points are known as **Lagrange points** and were discovered in the 18th century by **Leonhard Euler** . You read that right, Lagrange points were not discovered (not all of them at least) by Lagrange.

What actually happened is that around the year **1750** , Euler found the first three of these five points as solutions to **a very particular case of the three-body problem** . Some 20 years later Lagrange found the remaining two. These points are a clear example of how a simple mathematical curiosity can become a useful and interesting physical reality.

The three-body problem that occupied Euler and Lagrange when they discovered these famous points is intended to describe **the orbits of three** **mutually orbiting** bodies. Although it seems simple, the reality is that even today **we do not know a general solution for this problem** . That is, we do not have an equation that describes the orbit of these three bodies for any possible initial configuration (masses and velocities).

**Only if we establish certain restrictions can we reach a solution** : if we make the three masses identical and give the velocities specific values, for example. Well, the restriction established by Euler and Lagrange for this particular solution was that **two of the objects had to be much more massive than the third of them** . This is not the case if we consider the Sun-Earth-Moon system, but it is if we study a **Sun-Earth system to which we add a small asteroid** or even one of our probes or rockets.

These points will therefore be those places in space where **the gravitational attraction of the two larger bodies** (the Sun and the Earth in the case that concerns us now) **will add or compensate** , so that any object located at one of these points will have **a orbit around the Sun of the same duration as that of the Earth** . Let’s see why this “shouldn’t be” and how exactly it works for each of the five points.

**L**_{1}

This point is located on **the line that directly joins the Sun and the Earth, between the two bodies** . At the first Lagrange point, the gravitational attraction of the star and the planet cancel each other out to reach an **equilibrium** . An object located in the region between the Earth and the Sun **should describe an orbit with a period less than that of the Earth** . As an example we have Mercury and Venus, which take 88 and 225 days respectively to complete an orbit. However, in the line that joins both bodies, **the Earth’s gravity is able to counteract the Sun’s gravity, making the object feel as if it is orbiting a less massive star** , lengthening the period of the orbit for a given distance. The point L _{1} will be the one for which this elongated period coincides exactly with that of the Earth.

**L**_{2}

This point is also **on the line joining the two main bodies, but located beyond the smaller of the two** . That is, located beyond the Earth in the opposite direction to the Sun. In this region, **the orbit of any object should have a period greater than that corresponding to the Earth** . Mars for example takes 687 days to complete one orbit. However, by placing the object on the Earth-Sun line, Earth’s gravity adds to the Sun’s, making the object feel **as if it is orbiting a more massive star** and reducing the period needed to complete one orbit. The point L _{2} will be the one for which **this shortened period coincides exactly with that of the Earth** . This is the spot occupied by the James Webb **Space Telescope** **[link to “This is the first sharp image from the James Webb”]** . This point was chosen because, in addition to being convenient for the telescope to always be close to the Earth, its field of view is free, as it points away from the Sun, the Earth and the Moon, which are always behind the James Webb.

**L**_{3}

The third Lagrange point is also located on the line between the star and the planet, but **located beyond the Sun** , at the point opposite the Earth, but slightly further away than it. An object located on the Earth’s own orbit **should have a period identical to this, if we only consider the gravitational pull of the Sun.** However, by being in this zone, **the Earth itself adds to that** gravitational pull, meaning that the object must be just outside Earth’s orbit to have the Earth’s period. Again at point L _{3} the objects would feel **as if they were orbiting a slightly more massive Sun** , shortening their orbital period from the original value.

**L** _{4}**and L** _{5}

The points L _{4} and L _{5} are defined in the same way, as **the third vertex of the two equilateral triangles whose base is the line that joins the Sun and the Earth** . These points are located in the same orbit as the Earth, but **ahead and behind 60º** with respect to it. These points, unlike the first three, lead to a **stable equilibrium** so that if an object deviates from its position, it returns to where it was. **For points L** _{1}**, L** _{2}**and L** _{3}**this is not the case** and any small separation will increase over time unless corrected, as is the case with the James Webb and other telescopes, which **require corrections to their orbit every few weeks** .

Asteroids have been found sharing an orbit at the L4 and L5 points of various planets. These are called **Trojan asteroids** and although for Earth only two of these bodies are known, which orbit around L4, in the case of **Jupiter** it is known that **there should be more than a million bodies occupying these regions** .

REFERENCIAS:

N. J. Cornish, 1998, The Lagrange Points, NASA, WMAP Education and Outreach

Hui, Man-To et al, 2021, The Second Earth Trojan 2020 XL5, The Astrophysical Journal Letters. 922 (2): L25, doi:10.3847/2041-8213/ac37bf