Wednesday, April 24, 2024

Understanding Langrange Points

Written By Keshav Mohta (Grade 11)


We see the Sun every day, rising in the east and setting in the west. It has become a part of our everyday routine. But still, studying it closely is a challenge. If we tried to put a spacecraft closer to the Sun, the Sun’s gravity would suck it in, burning the spacecraft instantly.

Aditya L1 is ISRO’s latest successful mission, designed to study the solar atmosphere. It is one of the few spacecrafts to stay in position near the Sun without getting sucked in. But how does it do so?

To understand that we must go back to 1772, to the days of French mathematician Josephy-Louis Lagrange. He was a naturally brilliant mathematician whose work in number theory, differential equations and mechanics laid the foundations for a lot of the physics and math we study today. One of his notable papers was on the “General Three-Body Problem”, which is about how the forces of gravity play out amongst three objects of different masses.

In the context of Aditya L1, we’ll simplify this to 2 bodies, the Earth and the Sun. The Sun has a much larger mass than the Earth, leading to a much larger gravitational field. It exerts a continuous force perpendicular to the direction of the Earth’s motion. Such a force, called a centripetal force, causes the Earth to move in an almost circular orbit around the Sun.

Image Courtesy – Study.com

The diagram above illustrates this well. ‘Fc’ shows the direction of the force and ‘v’ shows the direction of movement.

However, the diagram above doesn’t take into account the gravity of the Earth. Below is a more representative diagram illustrating how the gravitational fields of the Sun and the Earth behave.

Image Courtesy – NASA

Where the lines become closer together the gravitational field is stronger and where they are further apart the gravitational field is weaker. An object placed in the Earth’s gravitational field (e.g. the Moon) starts orbiting the Earth, and an object in the Sun’s gravitational field starts orbiting the Sun. As you might expect, closer to the two masses, the gravitational field lines are also much closer together. If the Earth was not there, the Sun’s gravitational field would continue to be concentric circles, but because of the Earth, we see that the lines seem to get distorted, bending around the Earth.

Langrange managed to find something very interesting in these interacting gravitational fields. He identified 5 special points (marked above), now called Langrage Points. These can be considered points of “equilibrium”, where the density of field lines is very low. At these points, the gravitational fields of the Sun and the Earth cancel each other out, meaning that an object placed at any of these points will not start orbiting the Sun or the Earth.

For the more geography-inclined students, consider the field lines to be contour lines for terrain. L4 and L5 are like ‘hilltops’. If I place a ball at the apex of a hill, it will stay stationary. But a slight movement to the left or right and the ball will start rolling down. Similarly, L1, L2 and L3 are like ‘valleys’. The ball will remain stationary, enclosed from all sides by mountains, where mountains represent the ends of the gravitational fields. Therefore, the Langrage points can be considered “parking spots” for spacecraft and are extremely useful for us.

Of these 5 points, L4 and L5 are more stable than the others, but also much further away. It is a bit pointless to place anything at L3 because it will always be on the opposite side of the Sun, making it hard to communicate with Earth. L2 is the current location of the James Webb Space Telescope, a perfect place to look out at the rest of the Milky Way.

And L1 gives us a clear observation post for the Sun. With the Earth on one side and the Sun on the other, with minimal fuel, Aditya L1 can observe the Sun and send vital information back to the Earth.

To calculate the position of the L1 point relative to the Earth and the Sun, we can consider the diagram below.

As the Earth rotates around the Sun, the L1 point will also move with the Earth. For Aditya L1 to remain at L1 it will have to ensure it is always in line with the Earth and the Sun, implying that it will have to take the same time to orbit the Sun as the Earth does. By using some physics and algebra, we can calculate the individual orbital periods of Aditya L1 and of the Earth. Upon equating these two time periods we can find the distance between L1 and the Earth and eventually calculate the coordinates of L1.

Aditya L1 joins SOHO (Solar and Heliospheric Observatory), a project by NASA and ESA to be one of the few spacecrafts at L1. Its prime feature is a coronagraph (nothing to do with the Coronavirus!), that can block out the bright glare from the sun, allowing the spacecraft to directly view the surface, also called the corona. This will allow it to send back data about solar activities and their impact on Earth’s climate. Aditya L1 marks an incredible leap in space technology, pushing humanity one more step closer to understanding our surroundings.


Featured Image Courtesy – Hindustan Times



Keshav Mohta
Keshav Mohta
I live in Mumbai, India with my parents. I enjoy coding, robotics, and play the drums. I also write articles and stories. I have written two books which are published and can be found on Amazon and Goodreads. Here is a link to my latest book.

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